Formula Library

Har formula animated hai — dekho, samjho, try karo 🎬

⚡ Physics — Quick Formula Cheat Sheet

F = ma v = u+at KE = ½mv² V = IR v = fλ F = Gm₁m₂/r² Q = mcΔT p = mv P = F/A T = 2π√(l/g)

👆 Kisi bhi card pe click karo — full explanation + animation + Try It milega!

🔵 Mechanics

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Newton's Second Law — F = ma

Mechanics · Force, Mass, Acceleration

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F = m × a

Force = Mass × Acceleration

F = Force (Newton)m = Mass (kg)a = Acceleration (m/s²)

🧒 Step 1 — Pehle samjho: Force, Mass, Acceleration kya hote hain?

🔵 Force (F) kya hai?
Force ek push ya pull hai jo kisi cheez ko hilata hai, rokta hai, ya uski direction change karta hai. Jab tum kisi ko dhakka dete ho — woh force hai. Jab magnet pin ko kheenchta hai — woh bhi force hai. Force hamesha kisi cheez ke beech mein hoti hai — akal se nahi aati, koi agent chahiye hoti hai.

Force ka unit hai Newton (N). Ek Newton itni force hai jo 1 kg ki cheez ko 1 m/s² se accelerate kare. Ek apple pe lagri gravity approximately 1 Newton hoti hai! 🍎

🟢 Mass (m) kya hai?
Mass = kisi cheez mein kitna matter (padarth) hai. Zyada mass = zyada "aalsipan" — heavy cheez ko hilana mushkil hota hai, aur ek baar chal pade toh rokna bhi mushkil hota hai. Ise inertia kehte hain. Mass kabhi nahi badlta — chahe tum Earth pe ho, Moon pe ho ya space mein.

Weight aur Mass alag hain! Weight = m × g (gravity ka effect). Moon pe tumhara mass same hoga, par weight 6 guna kam hogi kyunki gravity 6 guna kam hai.

🔴 Acceleration (a) kya hai?
Acceleration = velocity mein change ki rate. Speed badhe → acceleration. Speed ghate → deceleration (jo actually negative acceleration hai). Direction bhi badle, jaise circle mein ghumna — woh bhi acceleration hai kyunki direction change ho rahi hai!

a = Δv / Δt matlab: kitni tezi se velocity badh/ghat rahi hai per second.

🧩 Step 2 — Formula kyun aisa hai? Intuition banao!

Sochte hain practically:

Experiment 1: Ek skateboard pe alag alag force lagate hain, mass same rakhte hain.
  Force 10N → acceleration = 2 m/s²
  Force 20N → acceleration = 4 m/s²
  Force 30N → acceleration = 6 m/s²
Dekho: Force double → acceleration double! Matlab a ∝ F (direct proportion)

Experiment 2: Same force lagate hain, mass alag alag rakhte hain.
  Mass 2 kg, Force 10N → a = 5 m/s²
  Mass 4 kg, Force 10N → a = 2.5 m/s²
  Mass 10 kg, Force 10N → a = 1 m/s²
Dekho: Mass double → acceleration half! Matlab a ∝ 1/m (inverse proportion)

Dono combine karo: a = F/m → yaani F = m × a ✅

📐 Step 3 — Teen roop, ek formula

F = m × a → Force nikalni ho jab mass aur acceleration pata ho
Example: 5 kg object 3 m/s² se accelerate ho raha hai → F = 5×3 = 15 N

a = F ÷ m → Acceleration nikalni ho jab force aur mass pata ho
Example: 20N force, 4 kg object → a = 20÷4 = 5 m/s²

m = F ÷ a → Mass nikalna ho jab force aur acceleration pata ho
Example: 30N force, 6 m/s² acceleration → m = 30÷6 = 5 kg

🌍 Step 4 — Real life mein kahan kahan kaam aata hai

🚗 Car brakes: Braking force = m × a. Bhaari truck ko rokne ke liye car se zyada braking force chahiye — isliye trucks ki brakes itni powerful hoti hain.

🚀 Rocket science: Rocket ka mass fuel jalane se ghatta jaata hai. Same engine thrust (force) se jaise jaise mass ghata, acceleration badhti jaati hai — isliye rocket pehle dheere aur phir bahut tezi se jaata hai!

🏏 Cricket: Bowler ball pe force lagate waqt sirf 0.1 second contact mein hota hai — par uss 0.1 second mein itni force lagti hai ki ball 140 km/h ki speed pakad leti hai.

🛡️ Helmet aur Airbag: Crash mein momentum change hona hi hai. Airbag time badhata hai (Δt) → force ghatti hai (F = Δp/Δt) → injury kam hoti hai. Yahi Newton ka 2nd law practically bachaata hai!

🎓 Step 5 — Advanced: Newton's 2nd Law ka sahi roop

Actually Newton ne F = ma nahi likha! Usne likha: F = dp/dt

Jahan p = momentum = mv, aur dp/dt = momentum ka time ke saath rate of change.

Agar mass constant ho: F = d(mv)/dt = m × dv/dt = m × a → isliye F = ma aata hai.

Par jab mass change ho (jaise rocket — fuel jal raha hai, mass ghata): F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt)

Yahi wajah hai ki rocket equation alag hoti hai — Tsiolkovsky Rocket Equation:
Δv = v_e × ln(m₀/m_f)
Jahan v_e = exhaust velocity, m₀ = initial mass, m_f = final mass.

Vector form: F⃗ = m × a⃗ — force aur acceleration dono vectors hain! Direction matter karta hai. Agar do forces opposite direction mein lage, toh net force ka pata karna padega pehle.

💡 Newton's 1st Law connection: Jab F = 0 → a = 0 → object ya rest mein rahega ya same speed se chalta rahega. Yahi "Inertia ka law" hai — F = ma ka special case jab F zero ho!

⚠️ F = ma sirf inertial reference frames mein valid hai — accelerating frame (jaise ghoomti hui car) mein pseudo-forces add karni padti hain (centrifugal force, etc.)

📌 Numericals ki practice:
1️⃣ 1000 kg car pe 5000N force → a = 5000/1000 = 5 m/s²
2️⃣ Elevator 800 kg, upar 2 m/s² se accelerate ho raha → F = 800×(9.8+2) = 9440 N (gravity + acceleration dono ke liye)
3️⃣ Ball on incline (30°): F along slope = mg sin30° = m×9.8×0.5 → a = 4.9 m/s²

🎬 Mass aur Force badlo — ball kitni tezi jaayegi dekho!

Mass (kg)5

Force (N)15

🧮 Try It

Mass m (kg)

Force F (N)

🚗

Equations of Motion (Kinematics)

Mechanics · v=u+at, s=ut+½at², v²=u²+2as

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v = u + at
s = ut + ½at²
v² = u² + 2as

Teen equations of motion — uniform acceleration ke liye

v = Final velocityu = Initial velocitya = Accelerations = Distancet = Time

🧒 Step 1 — Pehle 5 variables samjho (SUVAT)

Kinematics mein hum 5 cheezein track karte hain — inhe SUVAT kehte hain:

s = displacement (kitni door gayi — meters mein). Distance aur displacement alag hain! Distance = total path. Displacement = seedha A se B (direction ke saath).

u = initial velocity (shuru ki speed). Agar cheez aaraami se khadi thi → u = 0. Agar pehle se chal rahi thi → u = woh speed.

v = final velocity (ant mein speed). Yeh woh speed hai jis waqt tum measure karna chahte ho.

a = acceleration (kitni tezi se speed badh/ghat rahi hai). Agar brakes lagaaye → a negative hai (deceleration). Gravity ki wajah se girna → a = +9.8 m/s² (neeche ki taraf).

t = time (kitne seconds). Hamesha seconds mein rakhna — minutes ko 60 se multiply karo.

🧩 Step 2 — Teeno equations kaise bani?

Equation 1: v = u + at
Bahut simple! Acceleration = speed change per second. t seconds mein kitni speed badhi? → a × t. Usmein initial speed add karo → v = u + at.
Socho: Car 10 m/s se chal rahi thi, 3 m/s² acceleration, 4 second baad: v = 10 + 3×4 = 22 m/s.

Equation 2: s = ut + ½at²
Distance = average speed × time. Average speed = (u + v)/2 = (u + u + at)/2 = u + ½at.
Distance s = (u + ½at) × t = ut + ½at².
Agar sirf gravity se girna (u=0): s = ½gt² → yahi "free fall" ka formula hai!
5 second mein free fall: s = ½ × 9.8 × 25 = 122.5 meter.

Equation 3: v² = u² + 2as
Equation 1 se: t = (v-u)/a. Equation 2 mein daalo:
s = u×(v-u)/a + ½a×(v-u)²/a² → simplify karo → v² = u² + 2as.
Yeh tab use hota hai jab time pata nahi, sirf distance aur speeds pata hain.

📐 Step 3 — Equation choose karne ka smart method

Problem mein 3 quantities diye honge, 4th nikalni hogi.

Method: Jo quantity problem mein mentioned hi nahi — woh equation choose karo jisme woh hai hi nahi!

  • s missing hai → use karo v = u + at
  • v missing hai → use karo s = ut + ½at²
  • t missing hai → use karo v² = u² + 2as
  • a missing hai → use karo s = (u+v)t/2 (4th equation!)

🌍 Step 4 — Real applications

🚂 Train braking distance: Train 72 km/h (= 20 m/s) pe chal rahi hai, brakes lagate hain, a = -0.5 m/s². Rukne mein kitna distance? v=0, u=20, a=-0.5 → 0 = 400 - 2×0.5×s → s = 400 meter! Isliye train ke aage track pe mat baitho.

⬇️ Free fall aur gravity: Koi cheez 10m oopar se giraye — kuch time baad land hogi? u=0, a=9.8, s=10 → v² = 2×9.8×10 = 196 → v = 14 m/s se land karti hai! Aur time: t = v/a = 14/9.8 = 1.43 seconds.

🏎️ Overtaking distance: Car 80 km/h pe chal rahi hai, 120 km/h tak 5 seconds mein pahunchi. Acceleration? u=22.2, v=33.3, t=5 → a = (33.3-22.2)/5 = 2.2 m/s². Overtaking mein kitni door gayi? s = 22.2×5 + ½×2.2×25 = 138.5 m. Isliye highway pe overtaking ke liye clear 300m chahiye.

🎓 Step 5 — Advanced: Graphs se samjho

v-t graph (velocity vs time):
  • Slope = acceleration (a = Δv/Δt)
  • Area under graph = distance (s)
  • Straight line = uniform acceleration

s-t graph (displacement vs time):
  • Slope = velocity (v = Δs/Δt)
  • Flat line = object stationary
  • Parabola = uniform acceleration

Non-uniform acceleration: Jab a = constant nahi, toh calculus use hota hai:
v = ds/dt → s = ∫v dt
a = dv/dt → v = ∫a dt
Yahi Class 11 mein Calculus aur Physics ka connection hai!

💡 Units dhyan se! v m/s mein, u m/s mein, a m/s² mein, s meters mein, t seconds mein. km/h ko m/s mein convert: 18 se divide karo (ya 5/18 multiply).

⚠️ Sign convention: Ek direction positive lo, doosra negative. Usually: upar = +ve, neeche = -ve. Ya right = +ve, left = -ve. Consistent raho poore problem mein!

📌 Numericals:
1️⃣ Ball 20 m/s se oopar throw, u=20, a=-9.8: max height pe v=0 → s = (400)/(2×9.8) = 20.4 m, time to top: t = 20/9.8 = 2.04 s
2️⃣ Car 0 se 60 km/h (16.7 m/s) 10s mein: a = 16.7/10 = 1.67 m/s², distance = ½×1.67×100 = 83.5 m

🎬 Car ko accelerate hote dekho

Acceleration (m/s²)3

Initial velocity u0

🧮 Try It — v = u + at

Initial velocity u (m/s)

Acceleration a (m/s²)

Time t (seconds)

⚡

Kinetic & Potential Energy

Energy · KE = ½mv² · PE = mgh

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KE = ½mv²
PE = mgh

Kinetic = motion energy · Potential = height energy

m = mass (kg)v = velocity (m/s)g = 9.8 m/s²h = height (m)

🧒 Step 1 — Energy kya hoti hai? Bilkul root se.

Energy ek abstract cheez hai — dikhti nahi, par uske effects dikhte hain. Energy ki definition hai: "kaam karne ki capacity".

Koi cheez kaam kar sakti hai agar:
  • Woh chal rahi ho (Kinetic Energy — motion ki energy)
  • Woh kisi height pe ho (Gravitational Potential Energy — height ki energy)
  • Woh kheenchi/dabbi ho (Elastic PE — spring/rubber band)
  • Usmein chemical bonds hon (Chemical energy — khana, petrol, battery)

Hum Physics mein sirf Kinetic aur Gravitational Potential Energy pe focus karte hain iss formula mein.

🧩 Step 2 — KE = ½mv² kyun? Proof karo!

Ek object mass m, rest se (u=0) force F se distance s tak chalaya gaya, final speed v.

Kaam (Work) W = F × s
Newton's 2nd law: F = ma
Kinematics: v² = u² + 2as → v² = 2as (u=0) → s = v²/(2a)

W = F × s = ma × v²/(2a) = ½mv²

Yahi Kinetic Energy hai! Kaam jo laga object ko rest se speed v tak laane mein = ½mv². Ise Work-Energy Theorem kehte hain.

v² wala term kyun? Speed double karo → KE 4 guna barhti hai! Isliye highway pe 100 km/h pe accident 60 km/h se 2.7× zyada dangerous hota hai — speed double nahi, sirf 1.67× badhi par energy 2.8× badh gayi!

🌄 Step 3 — PE = mgh kyun? Intuition aur proof

Kisi cheez ko height h tak uthane mein kitna kaam lagta hai?

Gravity ki force = mg (neeche ki taraf).
Hum upar uthane ke liye mg ke equal force lagate hain (uniform speed se).
Kaam = Force × distance = mg × h = mgh

Yah kaam "stored" ho jaata hai object mein — Potential Energy ke roop mein. Jab girne dete hain, yeh energy wapas Kinetic Energy banta hai.

h kahan se measure karein? Kisi bhi reference point se! Ground se, table se, sea level se — jo convenient ho. Sirf h ka change matter karta hai, actual value nahi.

⚖️ Step 4 — Conservation of Energy — sabse bada physics law

Law of Conservation of Energy: Energy na banti hai na khatam hoti — sirf ek form se doosre form mein convert hoti hai.

Free fall mein (no friction):
  Total E = KE + PE = constant har waqt
  Top pe: KE = 0, PE = mgh → Total = mgh
  Beech mein: KE = ½mv², PE = mgh' → ½mv² + mgh' = mgh
  Neeche: KE = mgh (poori PE convert), PE = 0

Iska matlab: v = √(2gh) → kisi bhi height se girti cheez ki ground pe speed bas height pe depend karti hai, mass pe nahi! Isliye feather aur iron ball vacuum mein ek saath girte hain.

🌍 Step 5 — Real life connections

🎢 Roller Coaster design: Engineers pehle highest point se calculate karte hain kitna PE stored hai. Phir ensure karte hain ki sab points pe enough KE ho taaki car ruka na rahe. Minimum speed at top of loop: v = √(gR) (gravity se centripetal force ke liye).

💧 Hydroelectric dam: Paani ko height pe rok ke PE store karo. Girne do → KE bane → turbine ghoomaye → electricity bane. India mein Tehri dam 260.5m height pe 2400 MW bijli deta hai!

🏹 Bow and Arrow: String kheenchte ho → Elastic PE store hoti hai string mein. Chhodo → PE convert hoti hai arrow ki KE mein → arrow oodta hai.

⚽ Penalty kick physics: Footballer ka leg (mass ~10kg) 10 m/s se swing karta hai. KE = ½×10×100 = 500J. Yeh sab ball (0.43 kg) mein transfer → ball ki speed = √(2×500/0.43) = ~48 m/s = ~172 km/h!

🎓 Step 6 — Advanced concepts

Power aur Energy ka connection: Power = dE/dt = rate of energy change. Zyada power = zyada tezi se energy transfer.

Elastic vs Inelastic collision:
  • Elastic: KE conserved (billiard balls)
  • Perfectly inelastic: maximum KE loss (clay ball collision)
  • Real collisions: always inelastic to some degree

Escape velocity: Kisi planet se escape karne ke liye: ½mv² ≥ mgh (where h→∞, using proper integral: v_esc = √(2GM/R)). Earth ke liye = 11.2 km/s.

Relativistic KE: Bahut fast speeds (near c) pe: KE = (γ-1)mc² jahan γ = 1/√(1-v²/c²). Normal speeds pe ye formula ½mv² mein reduce ho jaata hai.

💡 KE always positive hoti hai (½mv² — v² kabhi negative nahi). PE negative bhi ho sakti hai (agar object reference point se neeche ho).

⚠️ Conservation sirf tab valid jab koi non-conservative force (friction, air resistance) kaam na kare. Real life mein friction hoti hai isliye kuch energy heat mein convert hoti hai.

📌 Numericals:
1️⃣ 2kg ball 10m oopar se girti hai: at 6m height v = √(2×9.8×4) = 8.85 m/s
2️⃣ Car 20 m/s → brakes → 0. m=1200kg. Work done by brakes = KE = ½×1200×400 = 240,000 J = 240 kJ
3️⃣ Spring compressed 0.1m (k=1000 N/m): PE = ½×1000×0.01 = 5 J stored

🎬 Ball girti hai — PE → KE conversion dekho

Mass (kg)2

Height (m)5

🧮 Try It

Mass m (kg)

Velocity v (m/s)

Height h (m)

🟠 Electricity

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Ohm's Law — V = IR

Electricity · Voltage, Current, Resistance

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V = I × R

Voltage = Current × Resistance

V = Voltage (Volts)I = Current (Amperes)R = Resistance (Ohms Ω)

🧒 Step 1 — Bijli kya hoti hai? Bilkul shuru se.

Har cheez atoms se bani hai. Atoms mein electrons hote hain jo nucleus ke around ghoomte hain. Kuch materials mein (jaise copper, silver, aluminium) electrons loosely bound hote hain — woh ek atom se doosre atom pe kood sakte hain.

Jab hum battery lagate hain → battery ek pressure (voltage) create karti hai → yeh loose electrons ek direction mein flow karne lagte hain → yahi electric current hai!

Current (I): Kitne electrons per second ek point se guzar rahe hain. Unit = Ampere (A). 1 Ampere = 6.24 × 10¹⁸ electrons per second! Ek normal bulb mein ~0.5A current hoti hai.

Voltage (V): Electrons ko push karne ki "force" ya "pressure". Unit = Volt (V). Battery ke do terminals ke beech ka potential difference. Ghar mein India mein 220V supply aati hai.

Resistance (R): Material ka "virodh" — electrons ko flow karne se rokta hai. Unit = Ohm (Ω). Metals mein kam resistance (good conductors). Rubber/plastic mein bahut zyada resistance (insulators).

🚰 Step 2 — Paani ki pipe wali analogy — deeply samjho

Socho ek paani ki tank rooftop pe hai aur neeche pipe se paani aa raha hai:

  🔵 Tank ki height (pressure) = Voltage — jitni zyada height, utna zyada pressure, utna zyada paani bahega.
  💧 Paani ka flow rate (litre/second) = Current — kitna paani per second nikal raha hai.
  🚧 Pipe ki choti/badi opening = Resistance — chota hole = zyada resistance = kam paani.

Ab sochte hain:
  • Pressure badhao (zyada V) → paani zyada tezi bahega (zyada I) ✅
  • Hole chota karo (zyada R) → paani dheere bahega (kam I) ✅
  • Yahi Ohm's Law hai: I = V/R

Resistance kyun hota hai? Electrons wire ke atoms se collide karte hain. Har collision mein thodi energy heat ke roop mein nikalti hai. Isliye wire garam hoti hai current flow karne pe!

📐 Step 3 — Teen formulas, ek triangle

V = I × R → Voltage nikalni ho
Example: 2A current, 10Ω resistance → V = 2×10 = 20 Volts

I = V ÷ R → Current nikalni ho
Example: 12V battery, 4Ω bulb → I = 12÷4 = 3 Amperes

R = V ÷ I → Resistance nikalni ho
Example: 9V battery, 0.3A current → R = 9÷0.3 = 30 Ohms

🔗 Step 4 — Series aur Parallel circuits

Series Circuit (ek hi line mein sab):
  • R_total = R₁ + R₂ + R₃ (sab add ho jaate hain)
  • Current same hoti hai sab mein: I₁ = I₂ = I₃
  • Voltage divide hoti hai: V = V₁ + V₂ + V₃
  • Problem: Ek bulb fuse → sab band! (Purani Christmas lights aise thi 😄)

Parallel Circuit (alag alag lines):
  • 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ (total resistance kam hoti hai!)
  • Voltage same hoti hai sab mein: V₁ = V₂ = V₃
  • Current divide hoti hai: I = I₁ + I₂ + I₃
  • Ek bulb fuse → baaki sab chalte hain ✅
  • Ghar ki wiring parallel mein hoti hai — isliye ek switch off karo, doosre chalte hain!

🌍 Step 5 — Real life applications

💡 Bulb ka filament: Tungsten wire ki resistance bahut zyada hoti hai. Current flow karo → filament itna garam hota hai (3000°C!) ki light emit karta hai. LED mein yeh differently kaam karta hai — zyada efficient.

🔋 Phone charging: Charger 5V deta hai. Phone battery 3.7V. Charger ke andar circuit resistance control karta hai current ko. Fast charger = zyada voltage (9V-20V) ya zyada current (3A-5A).

⚡ Fuse kyun lagata hai? Fuse = thin wire with low melting point. Agar circuit mein zyada current aaye (short circuit) → fuse wire garam hoti hai → pighal jaati hai → circuit break → fire nahi lagti!

🔌 Earth wire (grounding): Agar appliance ki casing mein current leak ho (insulation failure) → bina earth wire ke tumhe shock lagega. Earth wire current ko seedha earth (zero potential) mein le jaati hai.

🎓 Step 6 — Advanced: Ohm's Law ki limits aur Kirchhoff's Laws

Ohm's Law sirf "ohmic" conductors ke liye valid hai!
Diode, LED, transistor — inhe "non-ohmic" kehte hain. Inke V-I graph straight line nahi hota.

Kirchhoff's Current Law (KCL): Kisi junction pe anewali currents ka sum = jaanewali currents ka sum.
I₁ + I₂ = I₃ + I₄ (conservation of charge)

Kirchhoff's Voltage Law (KVL): Kisi closed loop mein sab voltages ka sum = 0.
ΣV = 0 (conservation of energy)

Yeh dono laws complex circuits solve karne ke liye use hote hain. Engineering mein almost every circuit problem inhi se solve hoti hai.

Temperature aur Resistance:
R = R₀(1 + αΔT) — zyada temperature → zyada resistance (metals ke liye).
Isliye bulb cold hone pe resistance kam hoti hai → switch on karte waqt zyada current → filament jaldi fuse ho sakta hai!

💡 Resistance ka colour code hota hai resistors pe — Brown Black Red Gold = 1, 0, ×100, ±5% = 1000Ω ± 5% = 1kΩ. Electronics mein yeh zaroori skill hai!

⚠️ Kabhi bhi 220V wire ko seedha haath mat lagao! I = V/R = 220/1000 (body resistance ~1kΩ) = 0.22A — itni current se heart arrest ho sakta hai. 0.1A bhi fatal hai!

📌 Numericals:
1️⃣ 3 resistors (2Ω, 4Ω, 6Ω) series mein: R_total = 12Ω, 12V battery → I = 1A, V across 6Ω = 6V
2️⃣ Same resistors parallel: 1/R = 1/2+1/4+1/6 = 6/12+3/12+2/12 = 11/12 → R = 1.09Ω
3️⃣ Ghar mein 10 bulbs (60W each) 220V pe: I each = 60/220 = 0.27A, Total I = 2.7A

🎬 Voltage badhao → Bulb kitna chamkta hai dekho

Voltage V (Volts)12

Resistance R (Ω)4

🧮 Try It

Find

Value 1

Value 2

🟢 Waves

🌊

Wave Speed — v = fλ

Waves · Speed = Frequency × Wavelength

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v = f × λ

Wave speed = Frequency × Wavelength (lambda)

v = speed (m/s)f = frequency (Hz)λ = wavelength (m)

🧒 Step 1 — Wave kya hoti hai? Ek dum root se.

Wave ek disturbance ka travel hai — matter ka travel nahi! Yeh samajhna bahut zaroori hai.

Stadium mein "Mexican wave" dekha hai? Har banda sirf uthta baithta hai — par wave poore stadium mein travel karti hai. Banda move nahi kiya, energy move ki!

Paani ki lehren bhi aisi hi hain — paani ke molecules sirf oopar neeche hote hain (circle mein actually), par wave aage badhti hai. Ek cork paani pe rakkho — woh aage nahi jaayega, bas oopar neeche hoga!

Waves do tarah ki hoti hain:
🔵 Transverse waves: Particles wave ki direction ke perpendicular hilte hain. Jaise: rope wave, light waves, water surface waves.
🔴 Longitudinal waves: Particles wave ki direction mein hi compress aur stretch hote hain. Jaise: sound waves, spring waves.

🧩 Step 2 — Frequency, Wavelength, Speed — teen pillars

Wavelength (λ — lambda): Ek complete wave ki length. Crest se crest tak, ya trough se trough tak. Unit = meters.

Frequency (f): 1 second mein kitni complete waves pass hoti hain ek point se. Unit = Hertz (Hz). 1 Hz = 1 wave per second.

Time Period (T): Ek complete wave banana ya pass karne mein kitna time lagta hai. T = 1/f. Agar f = 50 Hz → T = 1/50 = 0.02 seconds.

Speed (v): Wave kitni tezi se travel karti hai. Yeh medium pe depend karta hai, frequency pe nahi!

Formula kyun v = fλ hai?
1 second mein f waves pass hoti hain. Har wave ki length = λ.
Toh 1 second mein total distance covered = f × λ = speed!
Itna simple hai — distance = number of waves × length of each wave. ✅

🌍 Step 3 — Sound waves deeply

Sound ek longitudinal pressure wave hai. Koi cheez vibrate karti hai → surrounding air molecules compress aur expand hote hain → yeh disturbance chain reaction ki tarah aage badhta hai → tumhare kaan tak pahunchta hai.

Sound ki speed different mediums mein:
  • Air (0°C): ~331 m/s
  • Air (25°C): ~346 m/s (garam air mein fast — molecules tezi se move karte hain)
  • Water: ~1480 m/s (4× faster than air!)
  • Steel: ~5100 m/s (15× faster!)
  • Vacuum: 0 m/s — sound travel nahi kar sakti (space mein koi nahi sunata!)

Kyun solid mein fastest? Solid mein particles bahut close hote hain → disturbance jaldi transmit hoti hai.

Sonic boom: Jab plane sound ki speed (Mach 1 = ~340 m/s) se fast ho jaata hai → woh apni sound waves se aage nikal jaata hai → ek bada shock wave banta hai → "BOOM"!

💡 Step 4 — Light waves aur Electromagnetic Spectrum

Light ek electromagnetic wave hai — ise medium ki zaroorat nahi! Vacuum mein bhi travel karta hai isliye sun ki roshni Earth tak pahunchti hai.

Speed of light: c = 3 × 10⁸ m/s = 3,00,000 km/s. Itni tez ki 1 second mein Earth ke 7.5 chakkar laga de!

Electromagnetic Spectrum (frequency badhne ke saath):
  📻 Radio waves → λ = km to meters (FM radio, WiFi)
  📡 Microwaves → λ = cm (microwave oven, 5G)
  🌡️ Infrared → λ = mm to μm (TV remote, body heat)
  🌈 Visible light → λ = 400–700 nm (jo dikhta hai)
  ☀️ Ultraviolet → λ = nm (sunburn, sterilization)
  ⚕️ X-rays → λ = pm (medical imaging)
  ☢️ Gamma rays → λ = fm (nuclear reactions, cancer treatment)

Sab same speed (c) se travel karte hain vacuum mein — sirf wavelength aur frequency alag hai!

📻 Step 5 — Doppler Effect — aaj ke zamane ki zaroori cheez

Ambulance aate waqt "WHEEE" high pitch — jaate waqt "WHOOO" low pitch. Kyun?

Ambulance aate waqt: source aur tumhare beech distance ghatta hai → waves "compressed" hoti hain → wavelength choti → frequency zyada → pitch high.
Jaate waqt: distance badhta hai → waves "stretched" → λ badi → f kam → pitch low.

Formula: f_observed = f_source × (v ± v_observer)/(v ∓ v_source)

Doppler Effect ke uses:
  • 🚗 Speed gun: Police ki radar gun Doppler effect use karti hai speeding detect karne ke liye
  • 🌌 Redshift: Galaxies humse door ja rahi hain → unki light redshift (lower frequency) dikhti hai — isi se scientists ne universe expansion discover kiya!
  • ❤️ Echocardiogram: Dil ki blood flow measure karne ke liye ultrasound Doppler use hota hai

🎓 Step 6 — Advanced: Superposition, Interference, Standing Waves

Superposition Principle: Do waves same jagah pe milein → unki amplitudes add/subtract ho jaati hain.

Constructive Interference: Do waves in phase (crest + crest) → zyada badi wave. Noise-cancelling headphones exactly opposite wave banate hain → destructive interference → silence!

Destructive Interference: Waves out of phase (crest + trough) → cancel ho jaate hain → wave ek dum flat!

Standing Waves: Guitar string pe → wave jaati hai, reflect hoti hai → incident aur reflected wave milke standing wave banate hain. Sirf specific frequencies pe resonance hoti hai → musical notes!
f_n = n × v/(2L) jahan n = 1, 2, 3... (harmonics)

💡 Frequency aur wavelength ka inverse relationship: v = fλ mein v constant (same medium) → f badhao → λ ghata! Radio waves ki λ = 100m, visible light ki λ = 500nm — ek hi "family" ki waves!

⚠️ v = fλ mein speed medium change karne pe change hoti hai, frequency nahi! Light jab glass mein jaati hai — speed aur wavelength change hoti hai par frequency same rehti hai.

📌 Numericals:
1️⃣ FM radio 98 MHz: λ = 3×10⁸ ÷ 98×10⁶ = 3.06 m — isliye car ki FM antenna ~30cm hoti hai (λ/10)
2️⃣ Middle C note (262 Hz) in air: λ = 346÷262 = 1.32 m
3️⃣ Ultrasound 5 MHz in tissue (v=1540 m/s): λ = 1540÷5×10⁶ = 0.308 mm — itna chhota to very detailed images!

🎬 Frequency badhao → wavelength choti hoti hai

Frequency (Hz)2

🧮 Try It

Frequency f (Hz)

Wavelength λ (m)

🟠 Gravitation & Pressure

🌍

Gravitation — F = Gm₁m₂/r²

Gravitation · Universal Law of Gravitation

▼

F = Gm₁m₂/r²

Gravitational Force between two masses

G = 6.67×10⁻¹¹ Nm²/kg² m₁,m₂ = masses (kg) r = distance between centres (m) F = Gravitational Force (N)

🧒 Step 1 — Newton ka apple wala moment — poori kahani

1666 mein Newton apne ghar ke baagiche mein baitha tha (plague ki wajah se Cambridge university band thi!). Ek apple gira. Newton ne socha — yeh apple sirf Earth ki taraf kyun gira? Upar kyun nahi gaya? Sideways kyun nahi?

Phir ek aur baat socha: Agar Earth apple ko kheench rahi hai toh kya Earth Moon ko bhi kheenchti hai? Kya Moon bhi "gir" rahi hai Earth ki taraf — bas aage bhi chal rahi hai isliye orbit mein hai?

Yahi gravity ki asli samajh thi! Moon "fall" kar raha hai Earth ki taraf, par apni tangential speed ki wajah se curve karke orbit mein rehta hai. Satellite bhi aisa hi karta hai!

🧩 Step 2 — Formula ka matlab tukda tukda

F = G × m₁ × m₂ / r²

G (Universal Gravitational Constant): G = 6.674 × 10⁻¹¹ N·m²/kg²
Yeh itna chhota kyun hai? Kyunki gravity actually bahut weak force hai! Sirf jab masses bahut badi hoon (planets, stars) tab gravity feel hoti hai. Do log 1m door khade hoon — unke beech gravity = 6.674×10⁻¹¹ × 60 × 60 / 1 = 2.4×10⁻⁷ N — practically zero!

m₁ × m₂ (product of masses): Dono masses multiply hoti hain. Ek mass double karo → force double. Dono double karo → force 4 guna!

r² (distance squared): Yahi "inverse square law" hai. Distance double karo → force ek-chauthaayi (1/4) ho jaati hai. Distance 10× → force 1/100! Isliye Moon aur Sun ka gravity Earth pe feel hota hai par distant stars ka nahi.

⬇️ Step 3 — g = 9.8 m/s² kahan se aaya?

g = G × M_Earth / R_Earth²

Daalo values:
G = 6.674 × 10⁻¹¹
M_Earth = 5.972 × 10²⁴ kg
R_Earth = 6.371 × 10⁶ m

g = (6.674 × 10⁻¹¹ × 5.972 × 10²⁴) / (6.371 × 10⁶)²
g = (3.986 × 10¹⁴) / (4.059 × 10¹³)
g = 9.82 m/s² ≈ 9.8 m/s² ✅

g alag alag jagah par alag kyun hota hai?
  • Mount Everest pe: r slightly zyada → g thoda kam (~9.77)
  • Poles pe: Earth thodi납작 (flattened) → r kam → g thoda zyada (~9.83)
  • Moon pe: M_Moon bahut kam, R_Moon bhi kam → g_moon = 1.62 m/s² (Earth ka 1/6)
  • Mars pe: g = 3.72 m/s²
  • Jupiter pe: g = 24.8 m/s² — wahan tumhara weight 2.5× ho jaayega!

🌍 Step 4 — Orbital Motion aur Satellites

Satellite ko orbit mein kaise rakha jaata hai? Gravity provide karta hai centripetal force!

GMm/r² = mv²/r
→ v = √(GM/r) — orbital speed

ISS ki orbital speed: r = 6.371×10⁶ + 4×10⁵ = 6.77×10⁶ m
v = √(6.674×10⁻¹¹ × 5.97×10²⁴ / 6.77×10⁶) = 7.67 km/s = 27,600 km/h!

ISS 90 minutes mein ek orbit lagaata hai — din mein 16 sunrises dikhte hain astronauts ko!

Geostationary orbit: Agar T = 24 hours → satellite Earth ke saath ghoomta hai → TV dish hamesha ek jagah point karo → yahi ISRO ke communication satellites hain! Height ≈ 35,786 km.

🎓 Step 5 — Advanced: Gravitational Potential Energy aur Escape Velocity

Gravitational PE (near surface): U = mgh (simple)
Gravitational PE (general): U = -GMm/r (negative! infinity pe zero, surface pe most negative)

Escape Velocity: Kitni speed se throw karo ki cheez wapas na aaye?
KE = PE → ½mv² = GMm/r → v = √(2GM/r)
Earth ke liye: v_esc = √(2 × 6.674×10⁻¹¹ × 5.97×10²⁴ / 6.37×10⁶) = 11.2 km/s

Black Hole: Jab v_esc = c (light speed) → kuch bhi escape nahi kar sakta — yahan tak ki light bhi nahi! Yahi black hole hai.
Schwarzschild radius: r_s = 2GM/c² — Sun ke liye = 3km. Matlab sun ko 3km mein compress karo → black hole!

Tidal Forces: Moon ka gravity Earth ke alag alag hisson pe alag hota hai → Earth ke oceans bulge → tides! Yahi wajah hai ki din mein 2 baar high tide aata hai.

💡 Gravity 4 fundamental forces mein sabse weak hai — yet sabse zyada range ki! Electromagnetic force 10³⁶ × stronger hai gravity se, par opposite charges cancel karte hain. Gravity hamesha attractive hai, kabhi repulsive nahi — isliye large scale pe dominant hai.

⚠️ Weightlessness ≠ no gravity! ISS mein astronauts weightless feel karte hain kyunki woh freely fall kar rahe hain (orbit = continuous free fall). Gravity wahan bhi ~88% of Earth's gravity hai!

📌 Numericals:
1️⃣ Two 1000kg objects 10m apart: F = 6.674×10⁻¹¹×10⁶/100 = 6.67×10⁻⁷ N — negligible!
2️⃣ 70kg person on Moon: Weight = 70×1.62 = 113.4 N vs Earth pe 686 N
3️⃣ Satellite 500km above Earth: v = √(GM/(R+h)) = √(3.986×10¹⁴/6.87×10⁶) = 7.61 km/s

🎬 Distance badhao — Force kitni ghatti hai dekho

Distance r (units)4

Mass m₂ (units)3

🧮 Try It

m₁ (kg)

m₂ (kg)

r (m)

💧

Pressure & Archimedes' Principle

Fluids · P = F/A · Buoyancy = ρVg

▼

P = F/A
Fb = ρ × V × g

Pressure = Force per Area · Buoyant Force = weight of displaced fluid

P = Pressure (Pa) F = Force (N) A = Area (m²) ρ = fluid density (kg/m³) V = submerged volume (m³)

🧒 Step 1 — Pressure kya hai? Bilkul shuru se.

Pressure = Force per unit area. Same force, alag area → pressure alag hoti hai.

Heels wali sandal example:
  60 kg ladki → Weight = 60 × 9.8 = 588 N
  Heel ka area ≈ 1 cm² = 0.0001 m²
  Pressure = 588 / 0.0001 = 58,80,000 Pa = 58.8 bar!
  Compare karo elephant ke paon se: 5000 kg, 4 paon, each ~500 cm² → P = 49000/(4×0.05) = 2,45,000 Pa
  High heel > Elephant! Isliye heels pe walk karna soft soil mein mushkil hai.

Pressure unit — Pascal (Pa):
1 Pa = 1 N/m²
1 atm (atmospheric pressure) = 1,01,325 Pa ≈ 10⁵ Pa
Matlab: har 1 cm² pe roughly 1 kg ka weight! Hum yeh pressure feel nahi karte kyunki body ke andar se bhi pressure lagti hai.

💧 Step 2 — Fluid pressure: P = ρgh

Paani mein neeche jaao → pressure badhti hai kyunki upar ka paani tumhare pe press kar raha hai.

P = ρ × g × h
ρ (rho) = liquid ki density (kg/m³)
g = 9.8 m/s²
h = depth (meter)

Paani: ρ = 1000 kg/m³, 10m depth pe P = 1000×9.8×10 = 98,000 Pa ≈ 1 atm extra!
Matlab har 10m neeche jaao → 1 atm extra pressure lagti hai.

Submarines: Military submarine 300m tak jaati hain → 30 atm extra pressure → hull ko crush hone se bachana padta hai! Titanic wreck pe pressure = ~400 atm.

Pascal's Law: Enclosed fluid pe pressure ek jagah lagao → sab jagah equally transmit hoti hai. Yahi hydraulic jack ka principle hai! Chhota force, chhoti piston → large force, badi piston. Car lift, JCB, airplane brakes — sab hydraulics!

🛁 Step 3 — Archimedes Principle — poori kahani

Real story: King Hiero ne sona diya goldsmith ko crown banane. Doubt tha ki goldsmith ne sone mein silver mix kiya. Archimedes ko kaam diya — bina crown todey pata karo!

Nahaate waqt Archimedes ne dekha — jab woh tub mein gaya, paani overflow hua. Uske body ka volume = overflow hua paani ka volume. "EUREKA!" (Greek mein "I have found it!")

Crown paani mein daalo → kitna paani badhta hai = crown ka volume.
Same mass ka pure gold ka volume nikalte hain.
Agar crown ka volume zyada → density kam → silver mix hua hai! Crown nakli nikla, goldsmith ko punish kiya gaya.

Archimedes Principle: Koi bhi object fluid mein daalo → ek upward force lagti hai (Buoyancy/Upthrust) jo displaced fluid ke weight ke equal hoti hai.
F_b = ρ_fluid × V_displaced × g

🚢 Step 4 — Float ya Sink? Density ka game

Object floats jab: F_b ≥ W → ρ_fluid × V_displaced × g ≥ ρ_object × V_object × g

Case 1 — Completely submerged:
V_displaced = V_object
Float karna → ρ_object < ρ_fluid
Sink karna → ρ_object > ρ_fluid

Case 2 — Partially submerged (floating):
Weight = Buoyancy
ρ_object × V_object × g = ρ_fluid × V_submerged × g
V_submerged / V_total = ρ_object / ρ_fluid

Iceberg: ρ_ice = 917, ρ_seawater = 1025
Fraction submerged = 917/1025 = 89.5% — isliye 90% iceberg paani ke neeche hota hai!

Steel ship kyun float karti? Steel ki density = 7800 kg/m³ — paani se 7.8× dense. Par ship hollow hai! Average density = total mass / total volume (including air inside) < 1000 kg/m³ → float!

🎓 Step 5 — Advanced: Atmospheric Pressure aur Barometer

Atmospheric pressure kyun hoti hai? Earth ke upar ki pure air column ka weight → surface pe pressure = 1,01,325 Pa.

Barometer: Mercury tube ko uski taraf se uthao → mercury 76 cm tak oopar rahe (atmospheric pressure support karta hai). Isse 760 mmHg ya 1 atm kehte hain.

Altitude badhne pe pressure ghatti hai:
  • Sea level: 1013 hPa
  • 5500m (Himalaya): ~500 hPa (half!)
  • 10000m (Flight level): ~265 hPa
  Isliye oopar paani 100°C se kam temperature pe boil karta hai (less pressure → lower boiling point). Darjeeling mein chai at 90°C boil hoti hai!

Bernoulli's Principle: Fluid ki speed badhne pe pressure ghatti hai.
P + ½ρv² + ρgh = constant
Airplane wing: Upper surface pe air fast → low pressure. Lower surface pe slow → high pressure. Pressure difference = LIFT!

💡 Dead Sea mein itna namak hai (density ~1240 kg/m³) ki log effortlessly float karte hain bina tairna jane! Ρ_body ≈ 985 kg/m³ < 1240, toh sab float karte hain.

⚠️ Bends disease (scuba diving): Agar diver zyada tezi se surface pe aaye → blood mein dissolved nitrogen bubbles ban jaate hain (pressure suddenly ghatti hai) → bahut painful, deadly bhi ho sakta hai!

📌 Numericals:
1️⃣ 0.5 m³ wood (ρ=600) paani mein: F_b = 1000×0.5×9.8 = 4900N, W = 600×0.5×9.8 = 2940N → Floats, net upward = 1960N
2️⃣ Hydraulic lift: Area ratio 1:100, force in = 50N → force out = 5000N — itne se bhaari car uth jaati hai!
3️⃣ 20m depth mein pressure = 10⁵ + 1000×9.8×20 = 2.96×10⁵ Pa ≈ 3 atm

🎬 Block ko paani mein daalo — buoyancy dekho

Block density (kg/m³)600

Block size3

🧮 Try It — Buoyant Force

Fluid density ρ (kg/m³)

Submerged volume V (m³)

🔴 Magnetism & Electricity

🧲

Magnetic Force — F = qvB & Fleming's Rule

Magnetism · Lorentz Force · Left & Right Hand Rules

▼

F = qvB sinθ

Magnetic force on a moving charge in a magnetic field

F = Force (N) q = charge (C) v = velocity (m/s) B = magnetic field (T) θ = angle between v and B

🧒 Step 1 — Magnetism aur Electricity — ek hi cheez ke do roop!

1820 mein Oersted ne ek experiment kiya — ek current-carrying wire ke paas compass rakha. Compass needle move ki! Matlab current se magnetic field banti hai!

Phir Faraday ne ulta kiya — moving magnet se current banti hai!

Maxwell ne sab prove kiya: Electricity aur Magnetism alag nahi — yeh ek hi force hai — Electromagnetism! Aur light bhi electromagnetic wave hai — yeh sabse bada discovery tha 19th century ka.

Magnetic Force kab lagti hai?
Sirf moving charges pe magnetic force lagti hai — stationary charge pe nahi!
F = q × v × B × sinθ
q = charge, v = speed, B = magnetic field strength, θ = angle between v and B

🧩 Step 2 — Formula F = qvB sinθ deeply samjho

θ = 90° (perpendicular): sin90° = 1 → Maximum force = qvB
Charge field ke perpendicular move kare → max force, circular motion hoga

θ = 0° (parallel): sin0° = 0 → Force = 0!
Charge field ke saath move kare → koi force nahi, straight line mein jaayega

θ = 45°: sin45° = 0.707 → Force = 0.707 × qvB

F = BIL (current-carrying wire):
Current = charges ki flow → har charge pe force lagti hai → total wire pe force lagti hai
F = B × I × L × sinθ
B = field, I = current, L = wire ki length field mein

✋ Step 3 — Fleming's Rules — step by step

Fleming's Left Hand Rule (Motors ke liye — FBI):
Baayen haath uthao. Teeno ungliyaan 90° pe failo:
  ☝️ Forefinger (index) → Field direction (B)
  🖕 Middle finger → Current Idirection
  👍 Thumb → Force/Motion direction

Memory trick: "FBI" — Field, B is middle, I middle, thumb = motion

Fleming's Right Hand Rule (Generators ke liye):
Same position, Daayan haath:
  ☝️ Forefinger → Field (B)
  🖕 Middle finger → Induced Current (I)
  👍 Thumb → Motion of conductor

Easy yaad karne ka tarika:
  Motor = Left hand (L for L-oading electricity → motion)
  Generator = Right hand (R for R-unning → makes electricity)
  Motor = Electricity → Motion | Generator = Motion → Electricity (bilkul ulta!)

⚙️ Step 4 — Motor aur Generator — ghar ki electricity

DC Motor kaise kaam karta hai:
Current-carrying coil magnetic field mein → force lagti hai → coil ghoomti hai → shaft rotate karti hai → fan, mixer, washing machine chalta hai!

AC Generator (Alternator):
Coil ko mechanically ghumaao magnetic field mein → changing flux → induced EMF → AC current!
India mein 50 Hz supply → coil 50 baar per second ghoomti hai power plant mein.

Transformer:
EMF = -N × dΦ/dt (Faraday's Law)
V₁/V₂ = N₁/N₂
Step-up transformer: N₂ > N₁ → voltage badhata hai (power plant → transmission lines → 400,000V!)
Step-down transformer: N₂ < N₁ → voltage ghata hai (substation → ghar → 220V)

🌍 Step 5 — Real Applications

🏥 MRI Machine: Powerful magnetic field (1.5T–7T) body ke hydrogen atoms ko align karta hai. Radio waves se atoms disturb karo → woh signal emit karte hain → computer image banata hai. Sirf soft tissue bhi dikh sakta hai — X-ray se better!

🚄 Maglev Trains: Magnetic levitation — track aur train ke beech repulsion → train oopar uth jaati hai → friction zero → bahut tez (600+ km/h possible). Japan ka SCMaglev 603 km/h world record!

🔊 Speaker: Audio signal → current in coil → magnetic field → coil moves → cone vibrate → sound! Precisely F = BIL.

💳 Magnetic stripe card: Magnetic particles information store karte hain. Swipe karo → particles reader pe force create karte hain → data read hota hai.

🎓 Step 6 — Advanced: Electromagnetic Induction aur Lenz's Law

Faraday's Law: EMF = -dΦ/dt
Φ (phi) = magnetic flux = B × A × cosθ
Flux change karo → EMF induce hoti hai → current flow hoti hai (agar circuit closed ho)

Lenz's Law: Induced current hamesha woh direction mein hoti hai jo flux change ko oppose kare.
(Negative sign in Faraday's law se aata hai)
Example: Magnet coil ke paas laao → induced current aisa magnetic field banati hai jo magnet ko resist kare → coil magnet ko "push back" karti hai

Eddy Currents: Solid conductor mein changing magnetic field → circulating currents → heating effect.
Uses: Induction cooktop, metal detector, electromagnetic braking (MRI mein motion damping)

Self Inductance (L): EMF = -L × dI/dt
Coil apne aap ki current change ko oppose karti hai — inductor voltage "lag" karta hai current se 90°

💡 Earth ka magnetic field compass ko align karta hai — par did you know? Magnetic North aur Geographic North alag hain! Currently ~11° ka difference hai, aur yeh slowly change hota rehta hai. Geomagnetic reversal bhi hoti hai (last 780,000 years ago)!

⚠️ High magnetic fields dangerous hain! MRI room mein steel objects mat laao — 1.5T field pe ek steel chair missile ki tarah khich sakti hai aur fatal injury de sakti hai!

📌 Numericals:
1️⃣ Electron (q=1.6×10⁻¹⁹C) 10⁶ m/s, B=0.5T, θ=90°: F = 1.6×10⁻¹⁹×10⁶×0.5 = 8×10⁻¹⁴ N
2️⃣ Motor coil: I=2A, L=0.1m, B=0.3T: F = 0.3×2×0.1 = 0.06 N per wire
3️⃣ Transformer: V₁=220V, N₁=1000, N₂=50 → V₂ = 220×50/1000 = 11V (step-down)

🎬 Charge particle magnetic field mein kaise move karta hai

Charge q (C)1

Field B (T)2

🧮 Try It

Charge q (C)

Velocity v (m/s)

Field B (T)

Angle θ (°)

⚡

Power — P = W/T = VI = I²R

Work & Energy · Electrical Power

▼

P = W/t = VI = I²R = V²/R

Power = Rate of doing work = Electrical power

P = Power (Watt) W = Work (Joule) t = Time (s) V = Voltage (V) I = Current (A) R = Resistance (Ω)

🧒 Step 1 — Power ka seedha matlab

Power = Kaam ki RATE — kitna kaam kitne time mein hua.

Do logon ko ek hi manzil tak seedhiyaan chadh ke jaana hai. Pehla 10 second mein pahuncha, doosra 1 minute mein. Dono ne same kaam kiya (same weight, same height) — par pehle ne zyada power use ki!

P = W / t
P = Power (Watts)
W = Work done (Joules) = F × d
t = Time (seconds)

1 Watt = 1 Joule per second. Matlab 1W ka source har second 1 Joule energy consume ya produce karta hai.

Ek Watt kitna hota hai?
  • 100W bulb: pehle common tha, ab LED 9W mein same light
  • Human body (rest): ~80W produce karta hai (tabhi room garam hoti hai log hone se!)
  • Cyclist (racing): 400–500W
  • Car engine: 100,000W = 100 kW = ~134 HP
  • Lightning bolt: ~5 billion watts — par sirf 0.0002 seconds!

🧩 Step 2 — Electrical Power ke teen formulas

P = V × I
Voltage × Current. Jab dono pata ho. Ghar ki supply: 220V, 5A → P = 1100W

P = I² × R
Current² × Resistance. Heating effect ke liye. Resistance badhao → zyada heat.

P = V² / R
Voltage² ÷ Resistance. Jab voltage aur resistance pata ho, current nahi.

Yeh teeno kaise ek hain?
P = VI, aur V = IR (Ohm's law) → P = (IR)×I = I²R ✅
P = VI, aur I = V/R → P = V×(V/R) = V²/R ✅

🏠 Step 3 — Bijli ka bill samjho — completely!

1 Unit = 1 kWh = 1000W × 1 hour = 3,600,000 Joules

Ghar ke appliances aur bill:
  💡 LED bulb (9W) × 8h = 0.072 kWh/day = 2.16 unit/month
  🌀 Fan (75W) × 12h = 0.9 kWh/day = 27 unit/month
  ❄️ AC 1.5 ton (1500W) × 8h = 12 kWh/day = 360 unit/month!
  📺 TV 40" (100W) × 6h = 0.6 kWh/day = 18 unit/month
  🫙 Refrigerator (150W) × 24h = 3.6 kWh/day = 108 unit/month

₹8/unit rate pe: AC = ₹2880/month sirf AC se! Isliye AC bill itna aata hai.

5-star rating kyun important hai?
5-star AC (BEE): ~0.75 kW/ton vs 1-star: ~1.5 kW/ton. Same cooling, half bijli → 5-star 3-4 saal mein apna extra cost recover kar leta hai!

🌍 Step 4 — Mechanical Power: P = F × v

P = W/t = (F × d)/t = F × (d/t) = F × v

Iska use: Engine power calculate karne ke liye
Car 60 km/h (16.67 m/s) pe drag force = 500N overcome karne ke liye:
P = 500 × 16.67 = 8335 W ≈ 8.3 kW = 11 HP just to maintain speed!

Efficiency (η):
η = (Useful output power / Total input power) × 100%
  Incandescent bulb: ~5% (95% heat waste!)
  LED: ~50%
  Electric motor: ~90%
  Internal combustion engine: ~25–30% (petrol car)
  Solar panel: ~20%
  Human muscle: ~25%

🎓 Step 5 — Advanced: Power transmission aur three-phase supply

Power transmission loss:
P_loss = I²R (in transmission line resistance R)
Agar P = VI dena hai: zyada V → kam I → I² mein zyada reduction → bahut kam loss!
Isliye power plants 400,000V pe transmit karte hain, ghar pe 220V pe step down karte hain.
Bina iss trick ke electricity produce karne ki 60% energy transmission mein waste ho jaati!

Three-phase supply:
Industrial areas mein 3-phase supply aati hai (3 wires, 120° phase difference).
Single phase: 220V (ghar ke liye)
Three phase: 440V line-to-line (heavy machinery ke liye)
Motors 3-phase pe zyada efficient aur smooth chalte hain.

Power factor (cos φ):
AC mein apparent power = V_rms × I_rms
Real power = V_rms × I_rms × cosφ
Inductive loads (motors, transformers) mein cosφ < 1 → power factor poor → extra current draw → loss.
Capacitors lagake power factor improve karte hain — industries ko penalty lagti hai poor power factor pe!

💡 Horsepower (HP) = 746W. James Watt ne horses se compare kiya apni steam engine ko — ek horse roughly 746W kaam karta hai continuously. Tabse HP unit use hoti hai!

⚠️ High voltage transmission lines ke neeche mat raho! Lines 400kV pe hoti hain — even without touching, extremely high voltage spark over kar sakta hai (air bhi conductor ban jaata hai at sufficient voltage).

📌 Numericals:
1️⃣ 60kg insaan 10m seedhiyaan 20s mein chadhta hai: P = (60×9.8×10)/20 = 294 W
2️⃣ Motor 5kW, 4 ghante: Energy = 5×4 = 20 kWh = 20 units = ₹160 at ₹8/unit
3️⃣ Transmission: 10MW at 220V → I = 10⁷/220 = 45,455A, at 440kV → I = only 22.7A — 2000× less current!

🎬 Voltage aur Current badhao — Power dekho

Voltage V (Volts)12

Current I (Amps)3

🧮 Try It — P = VI

Voltage V (Volts)

Current I (Amps)

🟡 Thermodynamics & Optics

🌡️

Heat Capacity — Q = mcΔT

Thermodynamics · Specific Heat · Calorimetry

▼

Q = mcΔT

Heat absorbed/released = mass × specific heat × temperature change

Q = Heat (Joules) m = mass (kg) c = specific heat capacity (J/kg·K) ΔT = Temp change (K or °C)

🧒 Step 1 — Heat aur Temperature — dono alag hain!

Bahut log confuse karte hain — heat aur temperature same nahi hain!

Temperature: Average kinetic energy of molecules. Jitni tezi se molecules vibrate/move karein, utna zyada temperature. Unit: °C, K, °F

Heat (Q): Total thermal energy jo ek object se doosre mein transfer hoti hai. Yeh depend karta hai mass pe bhi!

Example: Ek thimble bhar boiling water (100°C) aur ek bucket bhar water (60°C).
Thimble ka temperature zyada hai — par bucket mein zyada heat energy hai (zyada mass)!

0 Kelvin (Absolute Zero): -273.15°C. Is temperature pe molecules bilkul bhi move nahi karte — minimum possible energy state. Practically achieve karna impossible (lekin 0.0000001K tak pahuncha gaya hai labs mein!).

🧩 Step 2 — Q = mcΔT formula ka har part

m (mass): Zyada mass → zyada heat chahiye same temperature change ke liye. 10kg paani ko garam karna 1kg se 10× zyada heat lega.

c (Specific Heat Capacity): 1 kg substance ka temperature 1°C badhane ke liye kitni heat chahiye.
  • Water: 4200 J/kg·K (bahut high — isliye paani amazing coolant hai!)
  • Aluminium: 900 J/kg·K
  • Iron: 450 J/kg·K
  • Copper: 385 J/kg·K
  • Lead: 128 J/kg·K (low — jaldi garam, jaldi thanda)
  • Sand: ~840 J/kg·K (desert mein din mein hot, raat mein cold kyunki low c compared to sea)

ΔT (Temperature change): Final temperature minus initial temperature.
ΔT positive → heat absorb ho rahi hai (cheez garam ho rahi hai)
ΔT negative → heat release ho rahi hai (cheez thandi ho rahi hai)

🌊 Step 3 — Paani ki specific heat kyun itni high hai?

Paani ke molecules (H₂O) mein hydrogen bonds hoti hain — yeh special bonds bahut energy absorb karte hain. Zyada energy daalo → molecules tez hote hain → bonds todne mein energy lagti hai → temperature dheere badhti hai.

Consequences:
  🌊 Mumbai vs Jaisalmer: Mumbai ke paas sea hai → high c → temperature moderate (26-32°C). Jaisalmer mein sand hai → low c → din mein 45°C, raat mein 10°C!

  🏎️ Car radiator mein paani: Engine se bahut heat nikalti hai. Paani high c ki wajah se bahut saari heat absorb kar leta hai bina temperature bahut badhaye — best coolant!

  ❄️ Ice ki specific heat: 2100 J/kg·K (paani se half!). Isliye ice jaldi thanda/garam hota hai. Par melting mein Latent heat alag lagti hai.

🔥 Step 4 — Heat transfer ke teeno tarike

1. Conduction (Chalana):
Direct contact se heat transfer. Ek end garam karo → molecules vibrate karein → neighbouring molecules ko disturb karein → aage badhein.
Metals best conductors (free electrons bhi energy carry karte hain).
Wood/plastic poor conductors (isliye handles pe lagate hain).
Formula: Q/t = kA(ΔT)/d, jahan k = thermal conductivity

2. Convection (Bahna):
Fluids (liquid/gas) mein. Garam fluid hafla hota hai → upar uthta hai → thanda fluid neeche aata hai → cycle banta hai (convection current).
Upar ceiling pe zyada garam kyun hota hai → convection!
AC room ke top pe kyun lagaate hain → thandi hawa neeche aayegi → natural convection!
Atmosphere mein weather patterns → convection currents!

3. Radiation (Kirnon se):
Electromagnetic waves se heat transfer — medium ki zaroorat nahi!
Sun se Earth tak heat radiation se aati hai (vacuum mein bhi).
Stefan-Boltzmann law: P = εσAT⁴
T (temperature in K) ka 4th power → temperature thodi bhi badhao → radiation bahut zyada badhti hai!

🎓 Step 5 — Advanced: Thermodynamics ke Laws

Zeroth Law: Agar A, C ke saath thermal equilibrium mein hai, aur B, C ke saath bhi → A aur B bhi equilibrium mein hain. (Thermometer kaam karta hai isi law se!)

First Law: ΔU = Q - W
Internal energy change = Heat added - Work done by system
Energy conservation ka thermodynamic form. Perpetual motion machine impossible!

Second Law: Heat hamesha hot se cold ki taraf flow hoti hai — kabhi ulti nahi. Entropy (disorder) hamesha badhti hai universe mein. AC/refrigerator heat ko opposite direction mein move karte hain — par iske liye external work lagti hai!

Third Law: 0 K pe entropy → constant minimum value. 0K exactly achieve karna impossible.

Carnot Efficiency: Ideal heat engine ki maximum efficiency:
η_max = 1 - T_cold/T_hot
Real engines hamesha isse kam efficient hoti hain.

💡 Why do we blow on hot food? Blowing removes warm air layer near food → replaces with cooler air → convection forced → cools faster. Same principle: PC fans, car radiator fan!

⚠️ Q = mcΔT sirf tab use karo jab koi phase change nahi ho rahi! Jab bhi melting/boiling ho → Q = mL use karo. Temperature wahan stable rehti hai jab tak phase change complete na ho.

📌 Numericals:
1️⃣ 2kg iron (c=450) ko 20°C se 300°C: Q = 2×450×280 = 252,000 J = 252 kJ
2️⃣ Geyser: 15L water (15kg) ko 25°C se 65°C: Q = 15×4200×40 = 2,520,000 J = 2.52 MJ
3️⃣ Mixing: 2kg at 80°C + 3kg at 20°C → Final T = (2×4200×80 + 3×4200×20)/(5×4200) = 44°C

🎬 Mass aur ΔT badhao — Heat kitni lagti hai dekho

Mass m (kg)2

Temp Rise ΔT (°C)50

🧮 Try It

Mass m (kg)

Specific heat c (J/kg·K)

Temp change ΔT (°C)

🔭

Optics — Mirror/Lens Formula & Snell's Law

Light · 1/f = 1/v + 1/u · n₁sinθ₁ = n₂sinθ₂

▼

1/f = 1/v + 1/u
n₁sinθ₁ = n₂sinθ₂

Mirror/Lens formula · Snell's Law of Refraction

f = focal length v = image distance u = object distance n = refractive index θ = angle with normal

🧒 Step 1 — Light kya hai? Aur kaise behave karti hai?

Light ek electromagnetic wave hai jo straight line mein travel karti hai (ray of light).

Teen important phenomena:
  🔵 Reflection: Light kisi surface se tukar ke wapas aati hai. Mirror mein chehra dikhta hai.
  🟢 Refraction: Light ek medium se doosre mein jaaye to bend (mod) khati hai. Straw paani mein tedhi dikti hai.
  🟡 Total Internal Reflection (TIR): Light dense medium se rarer mein critical angle se bade angle pe jaaye → wapas reflect ho jaaye. Optical fibre, diamond ki chamak isi se hai!

Laws of Reflection:
i = r (angle of incidence = angle of reflection)
Normal ke dono taraf incident aur reflected ray same plane mein hoti hain.

🌊 Step 2 — Refraction aur Snell's Law

Kyun bend hoti hai light?
Light ki speed medium pe depend karti hai. Denser medium mein speed slow hoti hai.
Air mein c = 3×10⁸ m/s. Glass mein ≈ 2×10⁸ m/s. Water mein ≈ 2.25×10⁸ m/s.

Refractive Index (n):
n = c / v_medium = Speed in vacuum / Speed in medium
n_air ≈ 1 (speed same as vacuum)
n_water = 1.33
n_glass = 1.5
n_diamond = 2.42 (sabse zyada → isliye diamond itna chamakta hai!)

Snell's Law: n₁ sinθ₁ = n₂ sinθ₂
Dense medium mein jaao → bend toward normal (θ₂ < θ₁)
Rarer medium mein jaao → bend away from normal (θ₂ > θ₁)

Critical Angle aur TIR:
Jab glass → air: kisi angle pe refracted ray 90° ho jaati hai (grazes surface). Yeh critical angle hai.
sin(θ_c) = n₂/n₁ = 1/1.5 = 0.667 → θ_c = 41.8° for glass
Isse bade angle pe: light wapas reflect ho jaati hai completely → TIR!
Optical fibre: light TIR se kisi bhi curve mein travel karti hai → internet cables!

🪞 Step 3 — Mirrors: Concave aur Convex

Mirror Formula: 1/f = 1/v + 1/u
f = focal length (positive for concave, negative for convex)
v = image distance
u = object distance (hamesha negative — object hamesha left side)

Concave Mirror (andar curve):
  • f positive
  • u > 2f: Real, inverted, diminished image
  • u = 2f: Real, inverted, same size (used in: searchlights aise set karte hain)
  • f < u < 2f: Real, inverted, magnified (cinema projector!)
  • u < f: Virtual, erect, magnified (makeup/shaving mirror pe nazdik se dekho!)

Convex Mirror (bahar curve):
  • f negative
  • Hamesha virtual, erect, diminished image
  • Wide field of view → rear-view mirror, shop security mirror!

Magnification: m = -v/u
m negative → inverted image
|m| > 1 → magnified
|m| < 1 → diminished

🔍 Step 4 — Lenses: Convex aur Concave

Lens Formula: 1/f = 1/v - 1/u
(Mirror formula se thoda alag — sign convention change!)

Convex Lens (converging):
  • f positive
  • u > 2f: Real, inverted, diminished (camera!)
  • u = 2f: Real, inverted, same size (photocopier!)
  • f < u < 2f: Real, inverted, magnified (slide projector!)
  • u < f: Virtual, erect, magnified (magnifying glass, reading lens!)

Concave Lens (diverging):
  • f negative
  • Hamesha virtual, erect, diminished (myopia/nearsightedness correction!)

Power of Lens: P = 1/f (in meters), unit = Dioptre (D)
f = +50cm → P = +2D (convex, positive)
f = -25cm → P = -4D (concave, negative)
Aankhon ka number actually dioptre mein hota hai! -3D chashma → f = -33.3cm concave lens.

👁️ Step 5 — Aankhon ki problems aur correction

Normal eye: Image exactly retina pe banti hai. Near point = 25cm, far point = infinity.

Myopia (Nearsightedness / Dheere dikhna):
Eyeball thoda bada ya lens zyada curved → image retina ke aage banti hai → door ki cheezein blur.
Correction: Concave lens (negative power) → image thodi peeche shift karo → retina pe padegi.

Hypermetropia (Farsightedness / Paas nahi dikhta):
Eyeball chota ya lens flat → image retina ke peeche banti hai → paas ki cheezein blur.
Correction: Convex lens (positive power) → image aage shift karo → retina pe padegi.

Presbyopia: Aging mein lens ki flexibility ghatti hai → paas aur door dono blur → bifocal lens chahiye (oopar concave, neeche convex).

Astigmatism: Cornea spherical nahi, cylindrical → alag directions mein alag focus → cylindrical lens se correct hota hai.

🎓 Step 6 — Advanced: Dispersion, Scattering, aur Wave Optics

Dispersion (Rainbow): White light = sab colours ka mix. Prism mein: har colour ki speed thodi alag → alag refraction → spectrum! VIBGYOR (Violet...Red). Violet most deviated, Red least.
Rainbow: Raindrop = prism ka kaam karti hai → sunlight disperse → 42° pe red, 40° pe violet.

Rayleigh Scattering (Sky blue kyun?):
Air molecules chhoti wavelength (blue, violet) zyada scatter karti hain (∝ 1/λ⁴).
Isliye sky blue dikhti hai — scattered blue light sab directions se aata hai.
Sunset red/orange kyun: Sun neeche ho → light zyada atmosphere se guzarti hai → blue scatter ho jaata hai → sirf red/orange bachta hai!

Wave Optics:
Young's Double Slit: Light dono slits se interference → bright (constructive) aur dark (destructive) fringes
Fringe width β = λD/d (λ = wavelength, D = screen distance, d = slit separation)
Diffraction: Light slits/obstacles ke edges pe bend → sirf wave theory explain kar sakti, ray theory nahi → proves light is a wave!

💡 Optical illusions bhi optics se hain! Mirage (desert mein paani dikhna): hot air near ground has lower refractive index → light curves upward → sky ka TIR image neeche dikhta hai jaise paani ho!

⚠️ Mirror aur Lens ki formula same lagti hai par sign convention ALAG hai! Mirror mein: real image ka v negative. Lens mein: real image ka v positive. Dhyan raho — yahi sabse common mistake hai exams mein!

📌 Numericals:
1️⃣ Concave mirror f=15cm, u=-45cm: 1/v = 1/15 + 1/(-45) = 3/45-1/45 = 2/45 → v = -22.5cm (real), m = -(-22.5)/(-45) = -0.5 (inverted, half size)
2️⃣ Convex lens f=10cm, u=-15cm: 1/v = 1/10 - 1/(-15) = 1/10+1/15 = 5/30 → v = +30cm (real, inverted), m = 30/(-15) = -2 (inverted, 2× magnified)
3️⃣ Eye problem: Person cannot see beyond 50cm → f = -50cm, P = -2D (concave lens needed)

🎬 Object move karo — image position aur Snell's refraction dekho

Object distance u (cm)30

Refraction angle θ₁ (°)30

🧮 Try It — Mirror/Lens Formula

Focal length f (cm)

Object distance u (cm, use -ve)

🟣 Momentum & Motion

💥

Momentum & Impulse — p = mv

Mechanics · Conservation of Momentum · J = FΔt

▼

p = m × v
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
J = F × Δt = Δp

Momentum = mass × velocity · Conservation Law · Impulse

p = momentum (kg·m/s)m = mass (kg)v = velocity (m/s)J = impulse (N·s)

🧒 Step 1 — Momentum kya hota hai? Root se samjho.

Socho do situations:

  🚗 Ek cycle 5 m/s se aa rahi hai
  🚛 Ek truck 5 m/s se aa rahi hai

Dono ki speed same hai — par truck ko rokna kitna mushkil hoga? Bahut zyada! Kyunki truck ka mass zyada hai.

Ab:
  🏓 Ek table tennis ball 50 m/s se aa rahi hai
  🎳 Ek bowling ball 5 m/s se aa rahi hai

Table tennis ki speed zyada hai — par bowling ball rokna mushkil hai! Kyunki mass zyada hai.

Momentum = Mass × Velocity = p = mv

Momentum dono factors — mass aur velocity — ko combine karta hai ek single quantity mein jo "motion ka measure" hai.

Unit: kg·m/s
Momentum ek vector hai — direction bhi matter karti hai!

🧩 Step 2 — Newton's 2nd Law aur Momentum ka asli connection

Newton ne actually F = ma nahi likha — usne likha:

F = dp/dt = rate of change of momentum

Jab mass constant ho: F = d(mv)/dt = m(dv/dt) = ma — tab F = ma aata hai.

Par agar mass change ho raha ho (jaise rocket):
F = m(dv/dt) + v(dm/dt) — extra term!

Impulse (J):
J = F × Δt = Δp = m(v − u)

Impulse = momentum mein change = Force × time.

Iska matlab: Same momentum change ke liye,
• Zyada time lagao → kam force chahiye
• Kam time → zyada force lagti hai

Yahi safety engineering ka core principle hai!

⚖️ Step 3 — Conservation of Momentum: sabse powerful law

Law: Agar koi external force system pe nahi lag rahi, total momentum conserved rehta hai.

Proof (Newton's 3rd Law se):
Do objects A aur B collide karte hain.
A pe B ki force = F_AB (kuch direction mein)
B pe A ki force = −F_AB (Newton's 3rd law — equal aur opposite)
Dono forces same time Δt tak lagti hain.
A ka momentum change = F_AB × Δt
B ka momentum change = −F_AB × Δt
Total change = F_AB×Δt − F_AB×Δt = 0
Total momentum same raha! ✅

Mathematical form:
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
(Before collision = After collision)

💥 Step 4 — Collisions ke types

Type 1 — Perfectly Elastic Collision:
Momentum conserved ✅ + Kinetic Energy conserved ✅
Real life mein perfectly elastic nahi hota — par billiard/snooker balls close hain.

Elastic collision formulas (1D):
v₁ = [(m₁−m₂)u₁ + 2m₂u₂] / (m₁+m₂)
v₂ = [(m₂−m₁)u₂ + 2m₁u₁] / (m₁+m₂)

Special case — equal masses (m₁=m₂):
v₁ = u₂ aur v₂ = u₁ → velocities exchange!
Billiard ball perfectly hit karo → pehli ruk jaati hai, doosri usi speed se chali jaati hai!

Type 2 — Perfectly Inelastic Collision:
Momentum conserved ✅ + Maximum KE loss ❌
Dono objects ek saath chipak jaate hain:
m₁u₁ + m₂u₂ = (m₁+m₂)v
v = (m₁u₁ + m₂u₂) / (m₁+m₂)

KE loss = ½m₁u₁² + ½m₂u₂² − ½(m₁+m₂)v²
Yeh energy heat, sound, deformation mein jaati hai.

Type 3 — Partially Inelastic (Real world):
Momentum conserved ✅ + Kuch KE loss ❌
Coefficient of restitution: e = (v₂−v₁)/(u₁−u₂)
e = 1 → perfectly elastic
e = 0 → perfectly inelastic
Real balls: e ≈ 0.7–0.9

🌍 Step 5 — Real life mein momentum everywhere

🚀 Rocket propulsion:
Rocket initially at rest (p = 0). Fuel eject karo backward (momentum = −mv_exhaust).
Rocket forward jaata hai — momentum conservation!
Tsiolkovsky equation: Δv = v_e × ln(m₀/m_f)
Space mein koi wall nahi — sirf mass eject karke move karo. Yahi rocket ka principle hai.

🛡️ Airbags aur Safety:
Car crash: Momentum change hona hi hai (speed → 0).
Δp = FΔt → same Δp ke liye Δt badhao → F ghata!
Airbag 0.05s mein inflate hoti hai → stopping time 0.005s se 0.05s ho jaata hai → force 10× kam → injury bahut kam!
Helmet, crumple zones, seat belts — sab yahi karte hain.

🏏 Cricket — bat aur ball:
Ball 140 km/h aati hai → batsman hit karta hai → ball 120 km/h jaati hai (opposite direction).
Ball ki mass = 0.156 kg. Contact time ≈ 0.001s.
Δp = 0.156 × (38.9 + 33.3) = 11.27 kg·m/s
F = Δp/Δt = 11.27/0.001 = 11,270 N = 1.15 ton force! Itni force bat pe lagti hai ek shot mein!

🔫 Gun recoil:
Initially gun + bullet at rest (p = 0).
Bullet forward → gun backward (recoil) — momentum conserve!
m_bullet × v_bullet = m_gun × v_recoil
0.01 kg × 900 m/s = 3 kg × v → v = 3 m/s recoil speed

🎓 Step 6 — Advanced: 2D Collisions aur Center of Mass

2D Collision:
X aur Y components separately conserve karo:
m₁u₁ cosα = m₁v₁ cosθ + m₂v₂ cosφ (x-direction)
m₁u₁ sinα = m₁v₁ sinθ − m₂v₂ sinφ (y-direction)

Billiards mein 2D collision hoti hai — pro players mentally yeh calculate karte hain!

Center of Mass (COM):
x_com = (m₁x₁ + m₂x₂ + ...) / (m₁ + m₂ + ...)
COM ka velocity = total momentum / total mass = v_com

Important property: Koi external force nahi → COM ki velocity constant!
Explode ho jaaye system → fragments alag alag jaayein, par COM usi straight line pe chalta rahega jisme pehle tha.

Rocket in space: COM hamesha straight line mein — chahe koi bhi direction mein exhaust niklo.

Variable mass system (Rocket equation derivation):
At time t: rocket mass M, velocity v.
dm mass exhaust speed u se eject karo in time dt.
Momentum conservation: Mv = (M−dm)(v+dv) + dm(v−u)
Simplify: M dv = u dm
Integrate: Δv = u ln(M₀/M_f) — Tsiolkovsky equation!

💡 Newton's Cradle (wo dangling balls ka toy) — momentum aur energy conservation ka perfect demo! Ek ball giraao → ek doosri taraf uthti hai. Do giraaon → do uthte hain. Kyunki sirf yahi arrangement dono conservation laws satisfy karta hai simultaneously!

⚠️ Momentum conservation sirf tab valid hai jab net external force = 0. Friction ya gravity jaisi external forces ho toh full momentum conserve nahi hota. Par collision ke bahut thode waqt mein (Δt → 0) external impulse negligible hoti hai → tab bhi approximately conserve hota hai!

📌 Numericals:
1️⃣ 2kg ball 5 m/s, hits 3kg at rest (perfectly inelastic): v = (2×5)/(2+3) = 2 m/s. KE before = 25J, after = ½×5×4 = 10J → 15J lost!
2️⃣ 1kg ball 6 m/s hits 2kg at rest (elastic): v₁ = (1−2)×6/3 = −2 m/s (bounces back!), v₂ = 2×1×6/3 = 4 m/s
3️⃣ Gun: 0.05kg bullet 300 m/s, gun 4kg: v_recoil = 0.05×300/4 = 3.75 m/s

🎬 Collision dekho — momentum conservation live

Mass 1 (kg)5

Mass 2 (kg)3

Initial Speed (m/s)4

🧮 Try It — Elastic Collision

m₁ (kg)

u₁ (m/s)

m₂ (kg)

u₂ (m/s)

🎯

Projectile Motion — Range, Height, Time

Mechanics · R = u²sin2θ/g · H = u²sin²θ/2g

▼

R = u²sin(2θ)/g
H = u²sin²θ / 2g
T = 2u sinθ / g

Range · Maximum Height · Time of Flight

u = initial speed (m/s)θ = launch angleg = 9.8 m/s²

🧒 Step 1 — Projectile kya hota hai? Bilkul root se.

Koi bhi cheez jo launch ki jaaye aur uske baad sirf gravity ke under ho (koi engine, thrust, ya push nahi) — woh projectile hai.

Examples:
  🏏 Cricket ball jo batsman ne hit ki
  ⚽ Football jo kick ki gayi
  💧 Fountain ka paani
  🪖 Cannon ball (tabhi toh "projectile" word cannon se aaya!)
  🏀 Basketball ka free throw

Sabse important idea: Jab cheez hawa mein ho, horizontal aur vertical motion completely independent hain — ek doosre pe effect nahi karte!

Horizontal direction mein: koi force nahi → constant velocity (uniform motion)
Vertical direction mein: gravity lagti hai → uniform acceleration (g = 9.8 m/s² neeche)

🧩 Step 2 — Initial velocity ko tod do: components

Ek ball u speed se θ angle pe launch ki gayi. Yeh ek velocity vector hai — ise do parts mein todo:

Horizontal component: u_x = u cosθ
Yeh throughout flight same rehta hai — horizontal acceleration zero hai (air resistance ignore karo).

Vertical component: u_y = u sinθ
Yeh gravity ki wajah se change hota rehta hai — pehle ghata (jaate waqt), zero hua (top pe), phir badhaa (aate waqt).

At any time t:
  • x-position: x = u cosθ × t
  • y-position: y = u sinθ × t − ½gt²
  • x-velocity: vₓ = u cosθ (constant!)
  • y-velocity: vy = u sinθ − gt (changes every second)

In dono ko combine karo — tumhe kisi bhi waqt ball ki exact position aur speed milti hai!

📐 Step 3 — Teeno key quantities: Range, Height, Time

🕐 Time of Flight (T) — total hawa mein kitna waqt:
Top pe vy = 0 → u sinθ − gt_top = 0 → t_top = u sinθ / g
Symmetrical motion hai → total time = 2 × t_top
T = 2u sinθ / g

🏔️ Maximum Height (H) — kitna oopar jaayegi:
Top pe vy = 0. Kinematics use karo: vy² = uy² − 2gH
0 = (u sinθ)² − 2gH
H = u² sin²θ / (2g)

📏 Horizontal Range (R) — kitni door land karegi:
R = u_x × T = u cosθ × (2u sinθ / g)
R = 2u² sinθ cosθ / g
Using identity: 2 sinθ cosθ = sin(2θ)
R = u² sin(2θ) / g

🎯 Step 4 — 45° pe maximum range kyun? Prove karo!

R = u² sin(2θ) / g

R maximum hoga jab sin(2θ) maximum hoga.
sin ka maximum value = 1, jo tab hota hai jab 2θ = 90°
→ θ = 45°

R_max = u² × 1 / g = u²/g

Complementary angles ka secret:
sin(2θ) = sin(2 × (90°−θ)) = sin(180°−2θ) = sin(2θ)
Matlab 30° aur 60° same range! 20° aur 70° same range!

Proof karo: u = 20 m/s
  30° pe: R = 400 × sin60° / 9.8 = 400 × 0.866 / 9.8 = 35.4 m
  60° pe: R = 400 × sin120° / 9.8 = 400 × 0.866 / 9.8 = 35.4 m ✅ Same!

Par heights alag hain:
  30° pe H = 400 × sin²30° / (2×9.8) = 400 × 0.25 / 19.6 = 5.1 m
  60° pe H = 400 × sin²60° / (2×9.8) = 400 × 0.75 / 19.6 = 15.3 m
60° wali ball zyada oopar jaati hai, par same door girती hai!

🌍 Step 5 — Real life applications

🏏 Cricket — bowling aur batting:
Fast bowler 140 km/h (38.9 m/s) pe ball dalta hai. Pitch 20m lambi hai. Time of flight ≈ 0.51s. Gravity se ball 1.27m neeche aati hai — isliye bowler oopar se dalta hai crease se, aur woh neeche bounce karta hai.

🚀 Artillery aur warfare:
WWI mein German "Paris Gun" 122km door tak shell maar sakti thi (θ ≈ 50°). Itni door ki air resistance aur Earth ki curvature bhi consider karni padti thi! Real long-range projectiles ke liye basic formula kaafi nahi.

⛳ Golf:
Golf ball 70 m/s pe hit hoti hai. Ideal angle = 45°, par golf balls pe dimples hoti hain jo aerodynamic lift create karti hain → effective optimal angle ~35-40°. Physics aur sports engineering milke!

🎆 Fireworks:
Firework rockets projectile motion follow karte hain. Designer calculate karta hai ki exactly kitni height pe aur kitne time baad burst hona chahiye → perfect display!

🌊 Fountain design:
Architect dekhte hain ki paani ka stream kitni height pe jaayega aur kahan gireg — R aur H ke formulas se exactly calculate karte hain nozzle angle aur pressure.

🎓 Step 6 — Advanced: Air resistance, Oblique projectile, aur Trajectory equation

Trajectory (parabola) ka equation:
x se t eliminate karo: t = x / (u cosθ)
y mein daalo:
y = x tanθ − gx² / (2u² cos²θ)
Yeh ek parabola hai (y = ax − bx² form). Har projectile ka path parabola hota hai (no air resistance)!

Projectile from height h:
Agar ball h height se horizontally throw ki jaaye (θ = 0):
  • Time to fall: h = ½gt² → t = √(2h/g)
  • Horizontal range: R = u × t = u√(2h/g)
  • Final speed: v = √(u² + 2gh)

Air resistance ka effect:
Real life mein air resistance (drag) lagti hai jo velocity ke square ke proportional hoti hai.
F_drag = ½ × C_d × ρ_air × A × v²
Is wajah se:
  • Actual range < theoretical range
  • Optimal angle shifts below 45° (~38-42° for most real cases)
  • Trajectory asymmetric ho jaati hai (steeper on way down)
  • Golf ball, cricket ball: Magnus effect bhi aata hai (spin ki wajah se lift/curve)

Projectile on inclined plane:
Agar surface inclined ho θ_i angle pe, launch angle θ se:
Range along incline = 2u² cos θ sin(θ − θ_i) / (g cos² θ_i)
Maximum range on incline = u² / [g(1 + sinθ_i)]

💡 Galileo ne prove kiya tha ki heavy aur light objects same rate se girte hain (air resistance ignore karke). Projectile motion ka yahi core idea hai — horizontal speed gravitational acceleration ko affect nahi karta aur vice versa!

⚠️ Horizontal range formula R = u²sin2θ/g sirf tab valid hai jab launch aur landing point same height pe hoon. Agar cliff se throw karo ya neeche se oopar — alag equations use karni padengi!

📌 Numericals:
1️⃣ Ball 30 m/s, 60° se launch: T = 2×30×sin60°/9.8 = 5.30 s, H = 900×0.75/19.6 = 34.4 m, R = 900×sin120°/9.8 = 79.5 m
2️⃣ Horizontal throw from 80m cliff, u=15 m/s: t = √(2×80/9.8) = 4.04 s, R = 15×4.04 = 60.6 m, final speed = √(225+2×9.8×80) = 40.6 m/s
3️⃣ Same range as 30°: θ = 90°−30° = 60° ✅

🎬 Angle badlao → trajectory change dekho

Launch angle θ°45

Speed u (m/s)20

🧮 Try It

Initial speed u (m/s)

Angle θ (degrees)

🔄

Circular Motion — a = v²/r, F = mv²/r

Mechanics · Centripetal · ω = 2π/T

▼

a = v²/r = ω²r
F = mv²/r
v = ωr · ω = 2π/T

Centripetal acceleration · Force · Angular velocity

a = centripetal acc (m/s²)v = speed (m/s)r = radius (m)ω = angular velocity (rad/s)T = time period (s)

🧒 Step 1 — Circle mein ghoomna = hamesha accelerating!

Pehli baat jo dimaag mein aati hai: "Circle mein speed constant hai toh acceleration kyun hogi?"

Yeh misconception hai! Acceleration sirf speed change nahi — velocity change bhi hai. Velocity ek vector hai — direction bhi hoti hai.

Circle mein ghoomte waqt speed same rehti hai (uniform circular motion mein), par direction hamesha change hoti rehti hai — isliye velocity change ho rahi hai → acceleration hai!

Yeh acceleration hamesha centre ki taraf point karti hai — ise centripetal acceleration kehte hain.

"Centripetal" word kahan se aaya?
Latin se: "centrum" = centre + "petere" = to seek. Matlab "centre-seeking"!

Important: "Centrifugal force" ek pseudo force hai — actually exist nahi karti! Jab tum car mein outer seat pe push feel karte ho, woh sirf tumhare inertia ka result hai (Newton's 1st Law). Real force centripetal hai — tum circle mein kheenche ja rahe ho, bahar push nahi ho rahe.

🧩 Step 2 — Centripetal acceleration: a = v²/r — proof karo!

Ek object radius r pe, speed v se ghoom raha hai. Thoda sa time Δt lao.

Iss time mein object A se B gaya. Dono positions pe velocity vectors draw karo — dono equal magnitude (v) par alag direction.

Velocity change (Δv) = vector subtraction → ek chhota vector jo centre ki taraf point karta hai.

Geometry se: |Δv|/v = arc length / r = v×Δt / r
→ |Δv| = v²Δt/r
→ a = |Δv|/Δt = v²/r (centre ki taraf) ✅

Isliye: F_c = mv²/r (Newton's 2nd law: F = ma)

Angular velocity (ω) se express karo:
v = ωr (linear speed = angular speed × radius)
a = v²/r = (ωr)²/r = ω²r
F_c = mω²r

📐 Step 3 — Angular quantities deeply samjho

Angular displacement (θ): Radians mein angle. 1 radian = wo angle jisme arc length = radius.
360° = 2π radians. 180° = π radians. 90° = π/2 radians.

Angular velocity (ω — omega):
ω = dθ/dt = angle per second (radians/second)
Ek full circle = 2π radians. Agar T = time period:
ω = 2π/T = 2πf
Earth ka ω = 2π/(24×3600) = 7.27 × 10⁻⁵ rad/s

Angular acceleration (α — alpha):
α = dω/dt (angular speed ka change)
Rotational kinematics: Same SUVAT equations with angular quantities!
  ω = ω₀ + αt
  θ = ω₀t + ½αt²
  ω² = ω₀² + 2αθ

Relation between linear aur angular:
  s = rθ (arc length)
  v = rω (linear speed)
  a_tangential = rα (tangential acceleration, direction change ki wajah se nahi — speed change ki wajah se)
  a_centripetal = rω² = v²/r (centre ki taraf, direction change ki wajah se)

🌍 Step 4 — Centripetal force provide karne wali forces

Centripetal force koi alag force nahi hai — jo bhi force centre ki taraf lagti hai woh centripetal force ka kaam karti hai:

🌍 Gravity → Planetary Orbits:
Satellite ke liye: GMm/r² = mv²/r → v = √(GM/r)
Moon ki orbital speed = √(6.67×10⁻¹¹ × 6×10²⁴ / 3.84×10⁸) = 1.02 km/s
ISS ki orbital speed = 7.67 km/s (closer → must go faster to not fall!)

🧵 Tension → Stone on string:
T = mv²/r. String toot jaaye → tension zero → stone tangentially fly karta hai (Newton's 1st Law!)
Isi principle se: Hammer throw (athletics), sling (David vs Goliath!), centrifuge machine

🛣️ Friction → Car turning:
f = mv²/r → μmg = mv²/r → v_max = √(μgr)
Wet road: μ = 0.4, dry road: μ = 0.8. Radius 50m:
Dry: v_max = √(0.8×9.8×50) = 19.8 m/s = 71.3 km/h
Wet: v_max = √(0.4×9.8×50) = 14 m/s = 50.4 km/h — isliye wet road pe slow!

⬆️ Normal Force → Banked roads aur loop-the-loop:
Banked road pe: tanθ = v²/rg (ideal banking angle — friction ki zaroorat nahi!)
F1 tracks, railway curves, highways — sab banking use karte hain

Loop-the-loop: Top pe minimum speed ke liye:
mg = mv²/r (gravity alone provides centripetal) → v_min = √(gr)
Is se kam speed → car/bike track se gir jaayegi!

🎡 Step 5 — Real life examples deeply

🎡 Merry-go-round — weight difference feel karna:
Top pe (loop): N + mg = mv²/r → N = mv²/r − mg
Jab v = √(gr), N = 0 → weightless feel! Speed aur se kam → N negative — possible nahi → gir jaoge!

Bottom pe: N − mg = mv²/r → N = mg + mv²/r → tumhara weight zyada lagta hai!
Isliye roller coaster ke bottom pe "heavy" feel hota hai aur top pe "light".

🌀 Washing Machine spin cycle:
Drum tezi se ghumta hai (ω zyada) → kapdon ko centripetal force ki zaroorat → wet kapdon ka paani drum se bahar nahi ja sakta → holes se bahar → kapde "dry"!
Actually centripetal force inward chahiye — paani drum ke against outward push karta hai holes se bahar.

🚁 Helicopter turn:
Helicopter tilt karta hai → lift force ka horizontal component centripetal force deta hai → turn hota hai.

🧬 Centrifuge (lab mein):
Blood sample tezi se ghumaate hain → RBC (heavy) neeche, plasma (light) upar. 10,000 RPM pe a = ω²r ≈ 10,000 × g! Itni acceleration se particles seconds mein separate ho jaate hain jo gravity se ghante lete.

🎓 Step 6 — Advanced: Moment of Inertia, Torque, aur Angular Momentum

Moment of Inertia (I):
Rotation mein "mass" ka equivalent. I = Σmr² (har particle ka mass × distance from axis squared).
  • Solid sphere: I = 2/5 mr²
  • Hollow sphere: I = 2/3 mr²
  • Solid cylinder: I = 1/2 mr²
  • Ring: I = mr²
Zyada I → rotate karna mushkil (jaise heavy flywheel).

Torque (τ = rF sinθ):
Rotation mein "Force" ka equivalent. τ = I × α (Newton's 2nd law for rotation)
Spanner se nut kholna: zyada r (longer handle) → zyada torque → zyada easily nut khuljata hai.

Angular Momentum (L = Iω):
Rotation mein "linear momentum" ka equivalent. Conservation of Angular Momentum:
L = Iω = constant (agar koi external torque nahi)

Ice skater spin:
Arms bahar → I zyada → ω slow
Arms andar → I kam → ω fast (Iω = constant!)
Iω = I'ω' → ω' = (I/I')ω → zyada speed!

Gyroscope:
Tezi se ghoomta top apni axis maintain karta hai (angular momentum conservation). Isliye ghoomta hua cycle nahi girta (gyroscopic effect)! Aeroplane ke navigation instruments mein bhi yahi use hota hai.

💡 Earth ka rotation dheema ho raha hai bahut slowly (tidal forces se) — 4 billion saal pehle Earth ka din sirf 6 ghante ka tha! Angular momentum Moon ko dheere dheere door dhakela ja raha hai — Moon har saal 3.8 cm door ho raha hai.

⚠️ Centripetal acceleration aur centripetal force ka direction hamesha inward (centre ki taraf) hota hai — kabhi outward nahi! "Centrifugal force" sirf accelerating frame of reference mein feel hoti hai — inertial frame mein exist nahi karti. Exam mein yeh mistake mat karo!

📌 Numericals:
1️⃣ 1500kg car 80m radius curve pe 20 m/s: F_c = 1500×400/80 = 7500 N. Required μ = F_c/mg = 7500/14700 = 0.51 (dry road pe possible, wet pe borderline!)
2️⃣ 500g stone, 1.5m string, 3 revolutions/second: ω = 2π×3 = 18.85 rad/s, v = 1.5×18.85 = 28.3 m/s, T = 0.5×(28.3)²/1.5 = 266.7 N
3️⃣ Satellite 300km above Earth: r = 6.67×10⁶m, v = √(GM/r) = √(3.986×10¹⁴/6.67×10⁶) = 7.73 km/s, T = 2πr/v = 90.5 minutes

🎬 Speed/radius badlao → centripetal force dekho

Speed v (m/s)10

Radius r (m)5

🧮 Try It

Mass m (kg)

Speed v (m/s)

Radius r (m)

🔁

Simple Harmonic Motion — T = 2π√(l/g)

Oscillations · Pendulum · Spring · x = A sinωt

▼

Pendulum: T = 2π√(l/g)
Spring: T = 2π√(m/k)
x = A sin(ωt)

Time period · Displacement from mean position

T = time period (s)l = pendulum length (m)k = spring constant (N/m)A = amplitudeω = 2π/T

🧒 Step 1 — SHM kya hai? Ek dum simple se shuru karo.

Kisi bhi motion ko SHM tab kehte hain jab:
  ✅ Object ek fixed point (equilibrium) ke around to-and-fro move kare
  ✅ Restoring force hamesha equilibrium ki taraf ho
  ✅ Restoring force displacement ke proportional ho: F = −kx

F = −kx ka matlab:
x = displacement (equilibrium se kitni door)
k = spring constant (kitna stiff hai system)
Negative sign = force hamesha displacement ke opposite (wapas kheenchna)

Examples of SHM:
  🪀 Simple pendulum (small angles pe)
  🌀 Spring-mass system
  🎸 Guitar string
  ⚛️ Atoms in crystal lattice
  🌊 LC circuit (electrical SHM!)
  ❤️ Heart ki rhythm (approximately)

Non-examples (NOT SHM):
Pendulum large angle pe → F ≠ −kx exactly (non-linear)
Bouncing ball → restoring force downward only

🧩 Step 2 — SHM ke equations: x, v, a as functions of time

F = ma = −kx
→ ma = −kx
→ a = −(k/m)x
Let ω² = k/m → a = −ω²x

Yeh differential equation hai: d²x/dt² = −ω²x
Solution: x = A sin(ωt + φ)
jahan A = amplitude, ω = angular frequency, φ = initial phase

Velocity:
v = dx/dt = Aω cos(ωt + φ)
v_max = Aω (jab cos = 1, yaani equilibrium pe)

Acceleration:
a = dv/dt = −Aω² sin(ωt + φ) = −ω²x
a_max = Aω² (jab x = A, yaani extreme position pe)

Phase relationships:
x aur a exactly anti-phase hain (180° apart)
v, x se 90° ahead hai (quarter cycle aage)

⏱️ Step 3 — Time Period aur Frequency

Angular frequency: ω = √(k/m)
Time Period: T = 2π/ω = 2π√(m/k)
Frequency: f = 1/T = (1/2π)√(k/m)

Spring-mass system:
T = 2π√(m/k)
Mass badhao → T badhta hai (slow oscillation)
Spring stiff karo (k badhao) → T ghata hai (fast oscillation)
Note: Amplitude se T independent hai! — chahe 1cm kheeencho ya 5cm, same time period!

Simple Pendulum:
Bob pe restoring force = mg sinθ ≈ mgθ (small angle pe sinθ ≈ θ)
Effective k = mg/L
T = 2π√(m/(mg/L)) = 2π√(L/g)

Shocking facts:
  • Mass matter nahi karta! 100g aur 1kg bob same time pe swing karenge agar L same ho
  • Gravity matter karti hai: Moon pe same pendulum → T = 2π√(L/1.62) = 2.46× slow!
  • Galileo ne 1600s mein discover kiya tha yeh (chandelier dekh ke church mein!)

⚡ Step 4 — Energy in SHM

Potential Energy:
U = ½kx² = ½mω²x² = ½mω²A²sin²(ωt+φ)
Maximum at extreme position (x = ±A): U_max = ½kA²
Zero at equilibrium (x = 0)

Kinetic Energy:
K = ½mv² = ½mω²(A²−x²)
Maximum at equilibrium (v = v_max = Aω): K_max = ½mω²A² = ½kA²
Zero at extreme positions

Total Energy:
E = K + U = ½mω²(A²−x²) + ½mω²x² = ½mω²A² = ½kA²
Constant! Energy oscillate karta hai KE aur PE ke beech — total hamesha same!

Energy ∝ A²: Amplitude double karo → energy 4 guna!
Isliye loud sound mein (badi amplitude) zyada energy hoti hai.

🌍 Step 5 — Real life applications

🕰️ Grandfather Clock (Pendulum clock):
T = 2π√(L/g). Agar L = 1m → T = 2.007s ≈ 2s (1 second left, 1 second right).
Clock fast ho rahi hai → L thoda badhao (pendulum neeche karo) → T badhega → slow hogi.
Temperature change → rod expand/contract → L change → clock fast/slow — isliye temperature compensation use hoti hai precise clocks mein!

🎵 Musical instruments:
Guitar string = SHM. Tension T badhao (tune karo) → k_eff badhta hai → f badhti hai → high pitch.
String length L karo (fret press karo) → f badhti hai → higher note.
f = (1/2L)√(T/μ) jahan μ = linear density (mass per length)

🏗️ Building vibration (earthquake engineering):
Har building ki ek natural frequency hoti hai. Agar earthquake ki frequency = building ki natural frequency → resonance → building oscillations bahut badi ho jaati hain → collapse!
Taipei 101 (Taiwan): 660 ton TMD (Tuned Mass Damper) — ek bada pendulum jo building ke opposite direction mein swing karta hai → oscillations cancel! Engineers yeh SHM se calculate karte hain.

⚕️ Medical — MRI:
Hydrogen nuclei ek specific frequency pe SHM jaisi motion karte hain magnetic field mein (Larmor frequency). Radio waves us frequency pe send karo → resonance → nuclei disturb → signal emit → image!

🎓 Step 6 — Advanced: Damping, Resonance, aur Coupled Oscillators

Damped SHM:
Real world mein friction/air resistance energy absorb karti hai → amplitude dheere dheere ghatti hai.
x = Ae^(−γt) cos(ω't + φ)
jahan γ = damping coefficient, ω' = √(ω₀²−γ²) = damped frequency

Types of damping:
  • Underdamped (γ < ω₀): Oscillations gradually kam hoti hain (door hinge).
  • Critically damped (γ = ω₀): Fastest return to equilibrium without oscillating (car suspension!)
  • Overdamped (γ > ω₀): Slow return, no oscillation (heavy door closer)

Forced Oscillations aur Resonance:
External force specific frequency pe lagao → system oscillate karta hai us frequency pe.
Jab driving frequency ≈ natural frequency → RESONANCE → amplitude bahut badi ho jaati hai!

Famous disaster: Tacoma Narrows Bridge (1940) — wind ki frequency bridge ki natural frequency se match ki → oscillations itni badi hoo gayi ki bridge collapse ho gayi! (YouTube pe video hai!)

Positive use: Radio tuning — capacitor-inductor circuit ki natural frequency adjust karo jo broadcast frequency se match kare → resonance → signal receive!

Coupled Oscillators:
Do pendulums ek rod se connect karo → ek ghumaao dono interact karte hain → "normal modes" → beating phenomenon → interesting!

💡 SHM aur circular motion deeply connected hain! Ek circle mein uniform circular motion ka projection (shadow) ek straight line pe = SHM! x = A cosωt exactly circular motion ka x-component hai radius A ke saath. Yahi wajah hai ω (angular frequency) SHM mein use hota hai!

⚠️ Pendulum formula T = 2π√(L/g) sirf small angles (θ < 15°) ke liye valid hai jab sinθ ≈ θ. Bade angles pe actual period longer hoti hai. θ = 90° pe correction ≈ 18% — ghadi 18% slow chalegi! Large angle ke liye elliptic integrals use karne padte hain.

📌 Numericals:
1️⃣ Spring k=400 N/m, m=1kg: ω = √(400/1) = 20 rad/s, T = 2π/20 = 0.314 s, f = 3.18 Hz
2️⃣ Pendulum 0.5m, Earth pe: T = 2π√(0.5/9.8) = 1.42 s. Moon pe: T = 2π√(0.5/1.62) = 3.49 s — 2.46× slow!
3️⃣ A = 5cm, k=200 N/m: Total energy = ½×200×(0.05)² = 0.25 J. At x=3cm: K = 0.25−½×200×(0.03)² = 0.25−0.09 = 0.16 J

🎬 Pendulum swing + SHM graph live dekho

Length l (m)1.0

Amplitude A°20

🧮 Try It

Pendulum length l (m)

Spring constant k (N/m)

Mass m (kg) [for spring]

🧊

Latent Heat — Q = mL

Thermodynamics · Phase Change · No Temperature Change

▼

Q = m × L

Heat = mass × Specific Latent Heat

Q = heat (Joules)m = mass (kg)L = specific latent heat (J/kg)

🧒 Step 1 — Latent Heat: "hidden" energy kya hai?

Ek experiment karo. Ice lao (−10°C), heat karo continuously aur temperature record karo:

  Phase 1: −10°C → 0°C: Temperature badhti hai (Q = mcΔT, ice ka c = 2100)
  Phase 2: 0°C pe RUKO! Temperature nahi badhti — chahe heat dete raho! Yeh "flat region" hai.
  Phase 3: 0°C → 100°C: Temperature phir badhti hai (Q = mcΔT, water ka c = 4200)
  Phase 4: 100°C pe RUKO AGAIN! Temperature stuck! Yeh dusra "flat region" hai.
  Phase 5: 100°C se aage: Steam ki temperature badhne lagti hai

In flat regions pe energy kahan ja rahi hai?
Temperature nahi badh rahi — par heat di ja rahi hai. Yeh energy molecular bonds todne ya banane mein ja rahi hai. Yahi "hidden" (latent) heat hai.

Latent = Latin mein "hidden" — kyunki yeh temperature change mein nahi dikhta!

🧩 Step 2 — Molecular level pe kya hota hai?

Solid (Ice):
Molecules fixed positions pe hain, strong bonds se jude hain. Sirf vibrate karte hain. Bahut ordered structure (crystal lattice).

Melting (Solid → Liquid) — Latent Heat of Fusion (L_f):
Heat do → molecules tezi se vibrate karte hain → ek level pe vibration itni badi ho jaati hai ki bonds toot jaate hain → molecules freely move karne lagte hain → liquid!
Yeh bond-breaking energy = Latent Heat of Fusion.
Water: L_f = 3.36 × 10⁵ J/kg
Matlab: 1 kg ice melt karne ke liye 336,000 Joules chahiye! Itni hi energy 1 kg paani ko 0°C se 80°C tak garam kar sakti thi!

Liquid (Water):
Molecules freely move karti hain, bonds weak hain (hydrogen bonds jo constantly break aur reform hote hain).

Vaporisation (Liquid → Gas) — Latent Heat of Vaporisation (L_v):
Heat do → molecules ki KE aur badhti hai → surface pe molecules itni tez hoti hain ki atmospheric pressure overcome kar ke escape kar jaati hain → steam!
Yeh escape energy = Latent Heat of Vaporisation.
Water: L_v = 22.6 × 10⁵ J/kg
Note: L_v >> L_f kyunki gas mein molecules poori tarah separate hoti hain — bahut zyada energy chahiye!

📐 Step 3 — Formula aur calculation

Q = mL
Q = heat energy (Joules)
m = mass (kg)
L = Latent heat (J/kg) — specific to substance aur type of transition

Important values yaad karo:
  Water: L_f = 3.36×10⁵ J/kg (melting/freezing)
  Water: L_v = 22.6×10⁵ J/kg (boiling/condensing)
  Lead: L_f = 0.25×10⁵ J/kg (isliye lead jaldi melt hota hai)
  Nitrogen (liquid): L_v = 1.98×10⁵ J/kg
  Alcohol: L_v = 8.55×10⁵ J/kg

Complete process calculate karna:
Ice (−10°C) → Steam (120°C) — 1 kg ke liye:
Q₁ = mcΔT = 1×2100×10 = 21,000 J (ice −10 to 0°C)
Q₂ = mL_f = 1×336,000 = 336,000 J (melting at 0°C)
Q₃ = mcΔT = 1×4200×100 = 420,000 J (water 0 to 100°C)
Q₄ = mL_v = 1×2,260,000 = 2,260,000 J (boiling at 100°C)
Q₅ = mcΔT = 1×2010×20 = 40,200 J (steam 100 to 120°C)
Total = 21,000 + 336,000 + 420,000 + 2,260,000 + 40,200 = 3,077,200 J ≈ 3.08 MJ!

🔥 Step 4 — Steam burns more than boiling water — why exactly?

Dono 100°C pe hain. Par skin pe lagte hi:

Boiling water (100°C):
Skin ko heat transfer karta hai → thanda hone lagta hai → Q = mcΔT se cool down → injury

Steam (100°C):
Step 1: Steam skin pe condense hoti hai (gas → liquid)
→ Latent heat release: 2,260,000 J/kg directly skin pe!
Step 2: 100°C paani phir thanda hota hai → aur Q = mcΔT

Steam se total heat transfer = L_v + mcΔT = 2,260,000 + 4200×(100−37) = 2,524,600 J/kg
Water se total heat transfer = mcΔT = 4200×(100−37) = 264,600 J/kg

Steam se 9.5× zyada energy! Isliye steam burns bahut severe hote hain.

🌍 Step 5 — Real life mein Latent Heat everywhere

❄️ Refrigerator aur AC:
Refrigerant (coolant gas) ko compress karo → liquid ban jaata hai → heat release (L_v reverse)
Uss liquid ko expansion valve se expand karo → evaporate hota hai → L_v absorb karta hai surroundings se → surroundings thanda!
Yahi refrigerator ka cycle hai. AC bhi same — andar se heat absorb, bahar release.

🧊 Cold drink mein ice:
Ice melting ke liye L_f absorb karta hai drink se → drink thanda rehta hai bahut der tak!
Ek small ice cube (10g) melting: Q = 0.01 × 336,000 = 3360 J absorb karta hai.
Same 10g pani ko 0°C se 80°C tak garam karne mein: Q = 0.01×4200×80 = 3360 J.
Same energy! Ice ek chhoti si jagah mein bohot zyada cooling power store karta hai!

☁️ Weather aur Monsoon:
Samundar ka paani evaporate hota hai → L_v absorb karta hai → water vapour banke oopar jaata hai.
Oopar thanda hone pe condense hota hai → clouds → rain → L_v release karta hai → atmosphere garam hoti hai!
Monsoon mein itna rain isliye hota hai — oceans mein stored latent heat release hoti hai.

🌡️ Sweating:
Body sweat produce karti hai → sweat evaporate hota hai → L_v skin se absorb → skin thandi hoti hai!
Ek gram sweat evaporate = 2260 J absorbed = significant cooling. 100W heat output pe: 100/2260 = 0.044 g/s sweat chahiye!

🍳 Pressure cooker:
Pressure badhne se boiling point badhti hai (paani 120°C pe boilta hai).
Zyada temperature → faster cooking. L_v bhi thoda zyada hoti hai higher pressure pe.
Mountains pe pressure kam → boiling point 90°C → khana der se pakta hai!

🎓 Step 6 — Advanced: Clausius-Clapeyron, Triple Point, Phase Diagrams

Boiling point pressure pe kyun depend karta hai?
Clausius-Clapeyron equation: dP/dT = L/(TΔV)
Pressure badhao → boiling point badh jaata hai. Isliye pressure cooker mein 120°C pe boiling!

Phase Diagram:
P-T graph pe teen regions: Solid, Liquid, Gas.
Boundaries = phase transitions. Teen boundaries ek point pe milti hain = Triple Point.
Water ka triple point = 273.16K, 611.73 Pa — yahan solid, liquid, gas teeno coexist karte hain!

Critical Point:
Ek temperature pe (water: 374°C) liquid aur gas mein fark khatam ho jaata hai. Iss se oopar → supercritical fluid (na liquid na gas — dono properties!).
Supercritical CO₂ industrial cleaning agent hai (coffee se caffeine nikalna, dry cleaning!).

Sublimation:
Solid seedha gas ban jaaye (liquid phase skip!) → Q = mL_s
Dry ice (solid CO₂): room temperature pe directly gas ban jaata hai → white fog effect!
Naphthalene (moth balls): dheere dheere sublime hote hain.
Freeze drying (food preservation): food freeze karo, phir vacuum mein → ice sublime → dried food without cooking!

💡 Ice cream vendor ke haath kabhi kyun nahi jalta dry ice se? Kyunki dry ice (−78.5°C) sublimate karta hai — CO₂ gas ka layer skin aur ice ke beech act karta hai insulation ki tarah (Leidenfrost effect similar). Par direct prolonged contact se frostbite ho sakta hai!

⚠️ Latent heat sirf phase change pe lagti hai — temperature change pe nahi! Q = mL aur Q = mcΔT kabhi ek saath mat lagao. Graph mein flat = latent heat. Sloped = specific heat. Exam mein graph questions mein yeh sabse common trick hoti hai!

📌 Numericals:
1️⃣ 500g ice (−5°C) → 50°C water: Q = 0.5×2100×5 + 0.5×336000 + 0.5×4200×50 = 5250+168000+105000 = 278,250 J
2️⃣ 200g steam (100°C) condenses aur 50°C tak thanda ho: Q = 0.2×2260000 + 0.2×4200×50 = 452000+42000 = 494,000 J released!
3️⃣ Geyser 10L water 25°C → steam at 100°C: Q = 10×4200×75 + 10×2260000 = 3150000+22600000 = 25.75 MJ — itni energy chahiye!

🎬 Heating curve dekho — plateau = latent heat zone

Heat added (%)30

Mass (kg)1

🧮 Try It

Mass m (kg)

L (J/kg) [Ice=336000, Water=2260000]

🧪 Chemistry — Quick Formula Cheat Sheet

pH = −log[H⁺] n = m/M PV = nRT M = n/V Kc = [P]^p/[R]^r ΔG = ΔH − TΔS Q = It/F (Faraday) N = N₀(½)^(t/t½) P₁V₁/T₁ = P₂V₂/T₂

👆 Kisi bhi card pe click karo — root se advanced tak full explanation + 4 animations + calculator milega!

🟣 Basic Chemistry

🧪

pH Formula — pH = −log[H⁺]

Acids & Bases · 0 to 14 scale

▼

pH = −log₁₀[H⁺]

pH = acidity/basicity ka measure

[H⁺] = Hydrogen ion concentration (mol/L)

📖 pH scale kya hai?

0-7 = Acidic: Strong acid (HCl) ≈ 0, Vinegar ≈ 3, Lemon ≈ 2, Coffee ≈ 5
7 = Neutral: Pure water
7-14 = Basic/Alkaline: Blood ≈ 7.4, Baking soda ≈ 9, Soap ≈ 10, Bleach ≈ 13

Magic of log: Ek unit ka difference = 10x difference in concentration! pH 3 acid is 10x stronger than pH 4.

⚠️ Log scale hai — pH 1 is 1000x more acidic than pH 4!

🎬 pH slider se color change dekho

pH value7

🧮 Try It

[H⁺] concentration (mol/L)

⚗️

Mole Concept — n = m/M

Stoichiometry · Moles, Mass, Molar Mass

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n = m / M

Moles = Mass (g) ÷ Molar Mass (g/mol)

n = molesm = mass in gramsM = molar mass (g/mol)

🧒 Step 1 — Mole kyun chahiye? Counting problem se samjho

Socho tumhe ek glass paani mein kitne molecules hain yeh count karna hai. Problem yeh hai — atoms/molecules bahut bahut chote hote hain. Ek glass paani mein itne molecules hain ki agar tum 1 second mein 1 billion bhi count karo, tab bhi poori zindagi se zyada time lagega sirf count karne mein!

Toh scientists ne ek smart trick nikali — jaise hum "12 cheezein" ko "1 dozen" kehte hain, ya "144 cheezein" ko "1 gross" kehte hain, waise hi chemistry mein bola gaya: "6.022 × 10²³ particles = 1 Mole" — yeh number "Avogadro's Number (Nₐ)" kehlaata hai.

Ab chote-chote atoms ko count karne ke bajaye, hum bas "kitne moles hain" bol sakte hain — bilkul jaise eggs ko "kitne dozen" bolte hain, count karne ke bajaye!

Mole = ek BAHUT BADA counting unit, jo specifically atoms/molecules/ions count karne ke liye banaya gaya.

💡 Andaza lagao kitna bada hai 6.022×10²³: agar tum is number mein rice ke daane count karo, toh poori Earth ko kayi kilometer moti rice ki layer se dhak sakte ho!

📌 1 dozen = 12 cheezein (chote groups ke liye), 1 mole = 6.022×10²³ cheezein (atoms/molecules jaise bahut bahut chote particles ke liye)

📐 Step 2 — n = m/M: Molar Mass periodic table se kaise nikaalo

Molar Mass (M) = ek mole (6.022×10²³ particles) ka total weight, grams mein. Yeh periodic table se directly mil jaata hai — har element ke box mein jo number likha hota hai (atomic mass), wahi uska molar mass hai g/mol mein!

Step by step — kisi bhi molecule ka Molar Mass kaise nikaalo:
1. Molecule ka formula dekho (kaunse atoms, kitni baar)
2. Har element ka atomic mass periodic table se nikaalo
3. Atomic mass × us element ki count = subtotal
4. Sab subtotals add karo = Total Molar Mass

Example 1 — Simple molecule (H₂O):
H = 1, O = 16 → H₂O mein 2 Hydrogen + 1 Oxygen → (2×1) + (1×16) = 2+16 = 18 g/mol

Example 2 — Compound molecule (CaCO₃, calcium carbonate):
Ca=40, C=12, O=16 → (1×40) + (1×12) + (3×16) = 40+12+48 = 100 g/mol

Ab formula use karo: n = m/M
36g paani ke liye: n = 36/18 = 2 moles

💡 Periodic table ko apna "molar mass dictionary" samjho — har element ke box ka neeche wala number (decimal mein) directly us element ka g/mol value hai!

⚠️ Subscript numbers (jaise H₂O mein "2") ko bhool mat jaana multiply karna — yeh batate hain element kitni baar molecule mein hai.

📌 NaCl (table salt) ka molar mass: Na=23, Cl=35.5 → 23+35.5 = 58.5 g/mol. Toh 58.5g salt mein exactly 1 mole hota hai!

🔗 Step 3 — Mole hai connecting bridge: Mass, Particles, aur Gas Volume tak

Sabse important baat: Mole ek "bridge" hai jo 3 alag-alag chizon ko connect karta hai — Mass, Particles count, aur (gases ke liye) Volume. Ek baar moles pata chal jaaye, baaki sab nikal sakte ho!

① Moles ↔ Mass: n = m/M (jo humne abhi seekha)

② Moles ↔ Particles: N = n × Nₐ
(N = total particles, Nₐ = 6.022×10²³)
Example: 2 moles paani mein particles = 2 × 6.022×10²³ = 1.204×10²⁴ molecules

③ Moles ↔ Gas Volume (sirf gases ke liye, STP pe): n = V/22.4
(STP = Standard Temperature & Pressure: 0°C, 1 atm. Ek interesting fact: kisi bhi gas ka 1 mole STP pe HAMESHA 22.4 Litre volume leta hai — chahe woh gas Hydrogen ho ya Carbon Dioxide!)
Example: 44.8L CO₂ gas (STP pe) mein moles = 44.8/22.4 = 2 moles

Poora triangle ek saath: agar mass pata hai → moles nikaalo → phir moles se particles YA gas volume bhi nikal sakte ho, sab connected hai!

💡 Yaad rakhne ka tareeka: "Mole" ek universal translator hai — chahe tumhe grams mein answer chahiye, particles mein, ya gas ka volume — sabse pehle moles nikaalo, phir wahan se kahin bhi convert kar sakte ho!

⚠️ n = V/22.4 formula sirf gases ke liye hai, aur sirf STP conditions pe! Solids/liquids ke liye yeh formula use mat karna — wahan n = m/M hi sahi rahega.

📌 8g O₂ gas (STP pe) ka volume kya hoga? Pehle moles nikaalo: O₂ ka M=32, n=8/32=0.25 moles → Volume = n×22.4 = 0.25×22.4 = 5.6 Litre

🌍 Step 4 — Real life mein Mole Concept kahan use hota hai?

Stoichiometry — chemical equations balance karna:
Jab koi chemical reaction hoti hai (jaise 2H₂ + O₂ → 2H₂O), woh numbers (2, 1, 2) batate hain ki kitne MOLES react ya form ho rahe hain — yeh sabse common use hai mole concept ka!

Real life mein kahan dikhta hai:
• Medicine dosage: Dawai banane mein exact moles calculate karna zaroori hai, taaki sahi quantity ho
• Industrial chemistry: Factory mein kitna raw material chahiye, kitna product banega — moles se calculate hota hai
• Cooking/Baking science: Baking soda (chemical reaction se gas banata hai) — moles se predict kar sakte ho kitna fluffy banega
• Environmental science: Air mein pollutants ki concentration moles/litre mein measure hoti hai

Common mistakes jo students karte hain:
1. Subscript numbers bhool jaana molar mass calculate karte waqt (jaise CaCO₃ mein 3 Oxygen, sirf 1 nahi)
2. n=V/22.4 formula ko solids/liquids pe bhi laga dena (yeh sirf gases ke liye hai!)
3. Avogadro's number ko galat likh dena (6.022×10²³, na ki 6.022×10²² ya 10²⁴)
4. Units confuse karna — grams aur moles ko same maan lena (yeh dono bilkul alag hain!)

💡 Final answer dene se pehle check karo: kya units sahi hain? "Moles" ek pure number hai (unit-less count), jabki "grams" weight hai — yeh kabhi mix nahi karna!

📌 Methane (CH₄) jalne ka reaction: CH₄ + 2O₂ → CO₂ + 2H₂O. Agar 1 mole CH₄ jalti hai, toh exactly 2 moles O₂ chahiye honge — yeh equation ke numbers se directly pata chalta hai!

🎬 Mass badhao → moles count badhte dekho (H₂O, M=18)

Mass of H₂O (g)36

👀 Har "H₂O" dot ek mole ko represent karta hai. Mass slider badhao → dekho dots ki ginti badhti hai, kyunki zyada mass = zyada moles (formula n=m/M se)!

🧮 Try It

Mass (grams)

Molar Mass (g/mol)

💨

Ideal Gas Law — PV = nRT

Gases · Pressure, Volume, Moles, Temperature · Boyle · Charles · Avogadro

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PV = nRT

Pressure × Volume = Moles × Gas Constant × Temperature (Kelvin)

P = Pressure (Pa or atm) V = Volume (m³ or L) n = Moles of gas R = 8.314 J/mol·K T = Temperature in Kelvin (°C + 273)

🧒 Step 1 — Pehle samjho: Gas kya hoti hai? Bilkul basics se!

Sochte hain ek sealed box mein chhoti-chhoti balls hain — woh hamesha tez-tez idhar-udhar uda rahi hain, box ki deewarein se takra rahi hain, ek doosre se takra rahi hain. Yahi ek gas hoti hai — lakho-karodo chote-chote particles (atoms ya molecules) jo bina ruke bhaag rahi hain! 🏃

Inki teen khaas baatein hain:

① Pressure (P) — Deewar pe thappad!
Gas ke particles jab bhi container ki deewar se takrate hain, ek chhota sa push dete hain. Lakho particles ek second mein lakho baar takrate hain → yahi Pressure hai. Jitna tez takrayen (zyada temperature) ya jitne zyada particles hon — utna zyada pressure!

② Volume (V) — Kitni jagah hai?
Gas jitna space gherta hai — woh uska volume hai. Gas ke particles ke beech mein bahut saari khali jagah hoti hai (isliye gas ko compress kar sakte hain, liquid ko nahi). Jab hum gas ko squeeze karte hain, woh particles aur paas aa jaate hain — volume ghata, pressure badha!

③ Temperature (T) — Kitni tezi se bhaagte hain?
Temperature actually gas particles ki speed ka measure hai! Zyada garam = particles zyada tez bhaagte hain = deewarein se zyada thappad = zyada pressure. Bilkul logical!

🌡️ Kelvin kyun use karte hain?
0°C pe bhi gas particles move karte rehte hain — temperature zero matlab "koi movement nahi" nahi hota! Isliye ek alag scale chahiye tha. Kelvin scale = 0K pe particles literally ruk jaate hain (absolute zero)! 0K = −273°C. Toh hamesha T (Kelvin) = °C + 273 use karo formulas mein.

📌 Tire mein hawa: Lakho air molecules tire ki rubber wall se baar-baar takraate hain → yahi 30-35 PSI pressure banata hai! Garmi mein molecules tez bhagte hain → pressure thoda badh jaata hai → isliye summer mein tyre pressure check karte hain! 🚗

🧩 Step 2 — Teeno Gas Laws: Jo milke PV = nRT bante hain

PV = nRT teen alag discoveries ka combination hai. Inhe ek-ek karke samjho:

📜 Law 1 — Boyle's Law (1662): P aur V ka relation (T aur n constant)

P₁V₁ = P₂V₂ (agar T aur n same rahein)
Pressure badhao → Volume ghata. Dono ka product hamesha same!

Real example: Syringe mein hawa bhar lo (Volume = V₁). Nozzle band karo aur push karo — volume ghata (V₂ < V₁) → pressure badha! Scuba divers bhi yahi use karte hain — deep water pe pressure zyada hota hai toh lungs smaller feel hote hain.

📜 Law 2 — Charles's Law (1787): V aur T ka relation (P aur n constant)

V₁/T₁ = V₂/T₂ (agar P aur n same rahein)
Temperature badhao → Volume badha. Dono ka ratio hamesha same!

Real example: Hot air balloon! 🎈 Andar ki hawa garam karo → volume badha → balloon phool ke oopar uth jaata hai. Thanda karo → wapas aaata hai.

📜 Law 3 — Avogadro's Law (1811): V aur n ka relation (P aur T constant)

V₁/n₁ = V₂/n₂ (agar P aur T same rahein)
Zyada gas (moles) → zyada volume. Ratio constant!

Real example: Balloon phuko → zyada molecules (n badha) → balloon bada (V badha)! Isi se aata hai: STP pe kisi bhi gas ka 1 mole = 22.4 Litre — amazing fact!

Combine karo teeno laws → PV = nRT! 🎉
Boyle: PV = constant → P ∝ 1/V
Charles: V ∝ T
Avogadro: V ∝ n
Teeno milao: V ∝ nT/P → V = R × (nT/P) → PV = nRT ✅ Yahi wajah hai formula aisa hai!

🎬 Animation 1: Boyle's Law — Compress karo, Pressure dekho!

Piston slider se gas compress karo — particles zyada cramped ho jaate hain, pressure badhta hai!

Volume (litre) — Piston badhao/ghata10

Particles count15

👀 Volume slider left karo (compress) → box chhotaa hoga → particles cramped → pressure number badhega! Right karo → relax!

📐 Step 3 — PV = nRT kaise use karein? Chaar roop, ek formula

Ek formula — chaar cheezein nikal sakte ho! Jis cheez ka pata nahi, usse ek side le jaao:

P = nRT / V — Pressure nikaalte waqt
Example: 2 mole gas, 300K, 10L → P = (2 × 8.314 × 300) / 0.01 = 4,98,840 Pa ≈ 4.93 atm

V = nRT / P — Volume nikaalte waqt
Example: 1 mole, STP (273K, 101325 Pa) → V = (1 × 8.314 × 273) / 101325 = 0.0224 m³ = 22.4 L ✅

T = PV / nR — Temperature nikaalte waqt
Example: 1 atm (101325 Pa), 22.4L, 1 mole → T = (101325 × 0.0224) / (1 × 8.314) = 273 K = 0°C ✅

n = PV / RT — Moles nikaalte waqt (gas ki quantity)
Example: Cylinder mein 5 atm, 10L, 300K → n = (506625 × 0.01) / (8.314 × 300) = 2.03 moles

Units ka chart — yeh confuse karta hai:
P → Pa (Pascal, SI unit) ya atm (1 atm = 101325 Pa)
V → m³ (SI) ya L (1 L = 0.001 m³ = 10⁻³ m³)
T → hamesha Kelvin! (K = °C + 273)
R → 8.314 J/mol·K (agar P Pa mein aur V m³ mein)
R → 0.0821 L·atm/mol·K (agar P atm mein aur V L mein — yeh easier hota hai!)

⚠️ Sabse common galti: Temperature Celsius mein daal dena! Hamesha + 273 karo. 27°C = 300K. 0°C = 273K. −73°C = 200K.

🎬 Animation 2: Temperature aur Particle Speed — garam = tez bhaagne!

Temperature slider badhao → particles zyada tez bhaagne lagte hain → wall pe zyada force → zyada pressure!

Temperature (K)300

👀 100K pe particles dheere chalk rahi hain (thanda). 600K pe bahut tez bhaag rahi hain (garam)! Speed ∝ √T — temperature 4 guna badhao, speed 2 guna hoti hai.

🌍 Step 4 — Real life mein Ideal Gas Law kahan kaam aata hai?

🎈 Weather Balloons:
Meteorology mein rubber balloons ko helium se bharte hain aur 30km oopar tak chhod dete hain! Jaise-jaise balloon oopar jaata hai, atmospheric pressure bahut ghatta jaata hai. PV = nRT se: P ghata, n aur T approximately same → V badha! Balloon phool ke phool ke finally phut jaata hai jab volume bahut bada ho jaata hai. Phutne se pehle balloon 100× bada ho jaata hai!

🚗 Car Tyre Pressure:
Ghar se 5 km drive ke baad tyre touch karo — geela mahsooos hoga kyunki garam! Friction se tyre garam → gas ka T badha → P badha (V fix hai tyre ka). Isliye tyres inflate karte waqt ek chhota extra dete hain, aur check always thanda tyre mein karo.

❄️ Refrigerator aur AC — Real life PV = nRT:
Refrigerant gas (coolant) ko compressor squeeze karta hai: V ghata → pressure badha → temperature badha (gas garam ho jaati hai). Iss garam gas ko outside se cool karte hain (condense). Phir expansion valve se suddenly expand karo: V badha → pressure ghata → temperature bahut ghata → andar se heat suck kar leti hai! Yahi hai AC ka cycle — pure gas laws!

🏔️ Mountain pe cooking mushkil kyun?
Mountain pe atmospheric pressure kam hota hai (~0.7 atm at 3000m). Charles + Boyle combined: kami pressure pe paani 100°C se pehle boilta hai (~90°C at 3000m). Ek to khana dheere pakta hai, uper se incomplete cooking ka risk!

🫁 Breathing aur Lungs:
Diaphragm neeche aata hai → lung ka volume badha → andar pressure ghata → outside air rush in (Boyle's Law!). Exhale karo: Diaphragm oopar → volume ghata → pressure badha → air bahar. Tum breathing mein Boyle's Law use karte ho — har second!

💉 Scuba Diving (Nitrogen Narcosis):
Samundar ke andar 10m depth pe pressure = 2 atm. Har 10m pe 1 atm badhta hai. PV = nRT mein P zyada → lungs mein gas dissolve zyada hoti hai. Zyada tez surface aao → pressure achanak ghata → dissolved gas bubbles ban jaati hain blood mein → The Bends (decompression sickness) — bahut dangerous! Isliye divers dheere-dheere surface aate hain.

📌 Fun experiment: Balloon ko freeze karo (fridge mein rakkho 10 min). Fridge se nikalo — chhota dikta hai! Bahar warm air mein → phool jaata hai. Yahi Charles's Law live dekh rahe ho! ❄️🎈

🎬 Animation 3: Charles's Law — Garam karo, Balloon bada dekho!

Temperature badhao (constant pressure) → balloon ka volume badhega — exactly V ∝ T!

Temperature (K) — 273K = 0°C300

👀 150K pe balloon chhota, 600K pe do-guna bada. Ratio exactly same hai: V₁/T₁ = V₂/T₂!

🔗 Step 5 — Combined Gas Law: Jab n constant ho

Agar moles change nahi ho rahe (same gas, sealed container) toh:

PV = nRT → nR = constant → PV/T = constant

Toh: P₁V₁/T₁ = P₂V₂/T₂ (Combined Gas Law)

Yeh bahut useful hai exam mein — jab gas ki state change ho (state 1 → state 2):

Example: Ek cylinder mein gas: P₁ = 2 atm, V₁ = 5L, T₁ = 300K. Garam karo T₂ = 400K, Volume expand ho V₂ = 7L. Find P₂?
P₂ = P₁V₁T₂ / (T₁V₂) = (2 × 5 × 400) / (300 × 7) = 4000/2100 = 1.90 atm ✅

Special cases — yaad rakho:
→ Sirf Boyle's chahiye (T constant): P₁V₁ = P₂V₂
→ Sirf Charles's chahiye (P constant): V₁/T₁ = V₂/T₂
→ Sirf Gay-Lussac chahiye (V constant): P₁/T₁ = P₂/T₂
→ Poora ideal gas: PV = nRT (jab n bhi change ho)

💡 Gay-Lussac's Law: Pressure Cooker! Sealed cooker — Volume fix hai. Garam karo → T badha → P badha. Isliye cooker itni tez pakkata hai (zyada pressure = zyada boiling point = zyada temperature = faster cooking)!

🎬 Animation 4: PV/T = nR — Sabko Ek Saath Dekho!

Teeno sliders badhao/ghata ke dekho — PV/T hamesha nR ke barabar rehta hai!

Pressure (atm)1

Volume (L)10

Temperature (K)300

👀 Koi bhi slider badlao — nR value hamesha same rahega! Yahi hai PV = nRT ka magic. Upar chart mein green bar hamesha same height pe hoga.

🎓 Step 6 — Advanced: PV = nRT kahan se aata hai? Kinetic Molecular Theory!

PV = nRT ek experimental law hai — par scientists ne ise theoretically bhi prove kiya! Yeh proof Kinetic Molecular Theory (KMT) se aata hai.

KMT ki 5 manyataen (assumptions):
1. Gas particles ka apna volume zero hota hai (particles bahut chhote hain, inke beech space infinitely bada hai comparatively)
2. Particles ke beech koi attractive ya repulsive force nahi hoti (particles free hain)
3. Particles hamesha straight line mein bhagte hain jab tak koi collision na ho
4. Collisions perfectly elastic hain — kinetic energy conserve hoti hai
5. Average kinetic energy directly proportional hai absolute temperature ko

KMT se P ka derivation (simplified):
Ek particle, mass m, speed v, ek cube box (side L) mein. X-direction mein speed vₓ.
Ek wall se collision pe momentum change = 2mvₓ
Time between collisions = 2L/vₓ
Force by one particle = 2mvₓ / (2L/vₓ) = mvₓ²/L
Pressure = Total force / Area = Nmvₓ²/(L × L²) = Nmvₓ²/V
3D mein: v² = vₓ² + vy² + vz² → average: <vₓ²> = v²/3
→ PV = Nmv²/3 = (2/3)N × (½mv²) = (2/3) × KE_total

KMT se: Average KE = (3/2)kT (k = Boltzmann's constant)
→ PV = N × (2/3) × (3/2)kT = NkT
N = nNₐ (total particles) aur k × Nₐ = R (gas constant!)
→ PV = nRT ✅ Proved from first principles!

RMS Speed of gas particles:
v_rms = √(3RT/M) — jahan M = molar mass (kg/mol)
Air molecules (M ≈ 0.029 kg/mol) at 300K: v_rms = √(3×8.314×300/0.029) ≈ 509 m/s = 1832 km/h!
Gas particles hawa se bhi tez bhaag rahe hain — bas direction random hai isliye net movement slow lagta hai (diffusion).

Maxwell-Boltzmann Distribution:
Saare particles ek hi speed se nahi bhagte — ek speed distribution hoti hai. Curve bell-shape (but skewed) hota hai. Temperature badhao → curve right shift aur flatten hota hai (zyada particles fast, avg speed badhti hai).

⚠️ "Ideal" gas = KMT ki 5 assumptions sahi hain. Real gases high pressure ya low temperature pe deviate karti hain — particles ke beech interaction hota hai! Van der Waals equation real gases ke liye use hoti hai: (P + an²/V²)(V − nb) = nRT.

🔬 Step 7 — Real Gases vs Ideal Gas: Van der Waals Equation

Real gases 2 jagah se ideal se alag hoti hain:

① Particles ka real volume:
Real particles space lete hain — infinitely small nahi hain! Toh effective volume = V − nb (jahan b = particle volume constant per mole).

② Intermolecular attraction:
Real particles ek doosre ko thoda attract karte hain. Wall pe strike karne se pehle thoda slow ho jaate hain → measured pressure thoda kam hoti hai actual se. Toh effective pressure = P + an²/V².

Van der Waals Equation:
(P + an²/V²)(V − nb) = nRT

Jahan a = intermolecular attraction constant, b = volume constant — har gas ke liye different values hain.

Real gas ideal kab behave karta hai?
→ High temperature pe (particles itni tez bhaag rahe hain ki attraction negligible)
→ Low pressure pe (particles itne door hain ki attraction negligible)

Compressibility factor Z:
Z = PV/(nRT) — ideal gas ke liye Z = 1 exactly.
Real gas ke liye: Z > 1 (repulsion dominant, high pressure) ya Z < 1 (attraction dominant, medium pressure).

Critical point — special temperature:
Har gas ke liye ek temperature hoti hai (Critical Temperature, Tc) jiske upar gas ko kitna bhi compress karo, liquid nahi banega. CO₂ ka Tc = 31°C. Isliye CO₂ fire extinguisher room temperature pe compress ho jaata hai, par O₂ cylinder ka gas bilkul liquid nahi banta (Tc = −118°C).

💡 Dalton's Law of Partial Pressures: Mixed gases mein, har gas independently pressure contribute karti hai. P_total = P_A + P_B + P_C ... Yahi weather forecasting mein water vapour pressure (humidity) calculate karne mein use hota hai!

📌 Exam shortcut — Pressure barabar karo units pehle:
1️⃣ 1 atm = 101.325 kPa = 101325 Pa
2️⃣ 0°C = 273K, 25°C = 298K, 100°C = 373K — yeh teen yaad rakho!
3️⃣ STP (0°C, 1 atm): 1 mole any gas = 22.4 L
4️⃣ SATP (25°C, 1 bar): 1 mole any gas = 24.8 L (naya standard)
5️⃣ Molar mass use karo → moles nikalo → phir PV=nRT lagao — sequence mat todna!

🧮 Try It — PV = nRT Full Calculator

Jo variable find karni ho usse empty chhod do, baaki teen fill karo!

Find

n (moles)

T (Kelvin)

V (Litres)

P (atm)

🟤 Solutions & Equilibrium

🫙

Molarity — M = n/V

Solutions · Concentration · mol/L

▼

M = n / V

Molarity = Moles of solute ÷ Volume of solution (in Litres)

M = Molarity (mol/L or M) n = moles of solute V = Volume in Litres

📖 Concentration kya hoti hai?

Molarity batata hai ki ek litre solution mein kitne moles solute dissolved hain.

Examples:
• 1M NaCl = 1 mole NaCl in 1L water = 58.5g/L
• 0.1M HCl = dilute acid (chemistry lab mein common)
• 2M NaOH = concentrated base

Dilution rule: M₁V₁ = M₂V₂ — zyada paani milao toh concentration kam hoti hai lekin moles same rehte hain!

💡 n = M × V (moles nikalo) | V = n/M (volume nikalo)

⚠️ Volume hamesha LITRES mein use karo! mL ko 1000 se divide karo.

📌 4g NaOH (M=40) ko 500mL mein dissolve kiya → n=4/40=0.1 mol, V=0.5L → M = 0.1/0.5 = 0.2 mol/L

🎬 Solute badhao ya volume badhao — Molarity kaise change hoti hai dekho

Moles of solute (n)0.5

Volume (L)1.0

🧮 Try It

Moles of solute (n)

Volume (L)

⚖️

Chemical Equilibrium — Kc Formula

Equilibrium · Concentration ratio · Reversible reactions

▼

aA + bB ⇌ cC + dD
Kc = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ

Equilibrium constant = Products ka product ÷ Reactants ka product (with powers = stoichiometry)

Kc = Equilibrium constant [X] = Molar concentration of X a,b,c,d = stoichiometric coefficients

📖 Equilibrium kab aata hai?

Reversible reaction mein forward aur backward reaction same rate pe chalti hain → Dynamic Equilibrium.

Kc ka matlab:
• Kc > 1 → Products dominant (forward reaction favoured)
• Kc < 1 → Reactants dominant (backward reaction favoured)
• Kc = 1 → Roughly equal amounts

Le Chatelier's Principle: Koi disturbance do → system balance restore karta hai!

💡 Pure solids aur pure liquids ko Kc expression mein NAHI likhte — only aqueous/gaseous species!

⚠️ Temperature change karo → Kc ki value change hoti hai. Pressure/concentration change → Kc same rehta hai!

📌 N₂ + 3H₂ ⇌ 2NH₃ → Kc = [NH₃]² / ([N₂][H₂]³)

🎬 Kc slider se Products vs Reactants ratio dekho

[Products] concentration2.0

[Reactants] concentration1.0

🧮 Try It — N₂ + 3H₂ ⇌ 2NH₃

[NH₃] mol/L

[N₂] mol/L

[H₂] mol/L

⚡ Electrochemistry & Nuclear

🔋

Faraday's Laws of Electrolysis

Electrochemistry · m = ZIt · Z = M/nF

▼

m = Z × I × t
Z = M / (n × F)
F = 96500 C/mol

Mass deposited = Electrochemical equivalent × Current × Time

m = Mass deposited (g) Z = Electrochemical equiv. (g/C) I = Current (Amperes) t = Time (seconds) M = Molar mass (g/mol) n = Valency (electrons transferred) F = Faraday constant = 96500 C/mol

📖 Electrolysis kya hota hai?

Faraday's First Law: Electrode pe deposit hone wala substance directly proportional hai electricity ke → m ∝ Q (charge).

Faraday's Second Law: Same charge se alag substances alag amounts mein deposit hote hain — ratio = equivalent weights ka ratio.

Real use: Gold/silver plating, copper refining, aluminium extraction (Hall-Heroult process), hydrogen production.

Combined formula: m = (M × I × t) / (n × F)

💡 Total charge Q = I × t (Coulombs). 1 Faraday = charge of 1 mole electrons = 96500 C

📌 2A current, 30 min, Cu²⁺ solution (M=64, n=2): m = (64×2×1800)/(2×96500) = 1.193 g Cu deposited

🎬 Current aur time badhao → deposition badhte dekho

Current I (Amperes)2

Time t (minutes)30

🧮 Try It — Copper deposition

Current I (A)

Time t (seconds)

Molar mass M (g/mol)

Valency n

☢️

Radioactive Decay — N = N₀e^(−λt)

Nuclear Chemistry · Half-life · Decay constant

▼

N = N₀ · e^(−λt)
t½ = ln2 / λ = 0.693 / λ

Remaining atoms = Initial atoms × e raised to (−decay constant × time)

N = Atoms remaining at time t N₀ = Initial number of atoms λ = Decay constant (s⁻¹) t = Time elapsed t½ = Half-life (time for N → N/2)

📖 Radioactive decay kya hota hai?

Unstable nuclei spontaneously disintegrate → emit alpha/beta/gamma radiation. Yeh ek first-order process hai — rate depends only on remaining atoms.

Half-life (t½): Jitne time mein original sample aadha reh jaata hai. Har t½ ke baad:
• 1st half-life → 50% remains
• 2nd → 25% remains
• 3rd → 12.5% remains
• n half-lives → (1/2)ⁿ remains

Applications: Carbon dating (C-14, t½=5730 yr), medical isotopes (I-131, t½=8 days), nuclear power (U-235)

💡 After n half-lives: N = N₀ × (1/2)ⁿ — ek aur useful form!

⚠️ λ aur t½ inverse proportional: Bada λ → choti half-life → fast decay

📌 C-14 ka t½ = 5730 yr, λ = 0.693/5730 = 1.21×10⁻⁴ yr⁻¹. 10000 yr baad: N = N₀ × e^(−1.21×10⁻⁴×10000) = N₀ × 0.298 = 29.8% remaining

🎬 Time badhao → exponential decay curve dekho

Decay constant λ0.1

Time elapsed (units)10

🧮 Try It

N₀ (initial atoms)

λ decay constant

Time t

🔴 Geometry

📐

Pythagoras Theorem — a² + b² = c²

Geometry · Right triangle · Hypotenuse

▼

a² + b² = c²

Right triangle mein: do sides ka squares ka sum = hypotenuse ka square

a, b = two shorter sidesc = hypotenuse (longest)

📖 Kyun hota hai aisa?

Geometric proof: a ke square ka area + b ke square ka area = c ke square ka area. Literally squares banao triangle ke teeno sides pe — chhote dono ka area milake bada wala area barabar hota hai!

Sirf right triangle mein kaam karta hai! (90° angle wala)

Famous pairs: 3-4-5, 5-12-13, 8-15-17

📌 GPS mein distance, construction mein right angle check, game graphics — sab jagah use!

🎬 Sides pe squares dekho — areas add hote hain

Side a3

Side b4

🧮 Try It — Find hypotenuse c

Side a

Side b

⭕

Circle — Area = πr², Circumference = 2πr

Geometry · π = 3.14159…

▼

A = πr²
C = 2πr

Area = π × radius² · Circumference = 2 × π × radius

📖 π (pi) kya hai?

π (pi) ≈ 3.14159... ek magical constant hai. Kisi bhi circle ka circumference ÷ diameter = hamesha π!

Area kyun r² hai? Square ka area = side². Circle square ke andar fit hoti hai. Approximately 3.14 chote squares = ek circle (same radius ke). Isliye area = πr².

Pizza, wheel, coin, planet orbit — sab ke liye!

💡 Radius = Diameter/2. Diameter = Radius×2. D = 2r.

🎬 Radius change karo → Area aur Circumference dekho

Radius r5

🧮 Try It

Radius r

📈 Finance Maths

💰

Compound vs Simple Interest

Finance · A=P(1+r)ⁿ vs SI=PRT/100

▼

CI: A = P(1+r/n)ⁿᵗ
SI: A = P + PRT/100

Compound interest → interest pe bhi interest milta hai

P = Principal (jo paisa shuru mein daala) R / r = Rate of interest (% per year) T / t = Time (years) n = saal mein kitni baar compound hota hai A = Amount (final total paisa)

🧒 Step 1 — Interest kya hota hai? Bilkul shuru se samjho

Socho tum apne dost ko ₹100 udhaar dete ho ek saal ke liye. Dost kehta hai "main tumhe ₹110 wapas dunga" — yeh extra ₹10 hi "Interest" hai! Interest matlab paisa "rent" charge karna jab tum apna paisa kisi ko use karne dete ho (bank ko, ya koi tumhe deta hai).

Ab interest calculate karne ke 2 tareeke hain:

1️⃣ Simple Interest (SI): Har saal SAME amount interest milta hai — sirf original paisa (Principal) pe.
2️⃣ Compound Interest (CI): Har saal interest badhta jaata hai — kyunki pichle saal ka interest bhi ab "Principal" mein add ho jaata hai, aur usi pe bhi interest milta hai!

Yehi sabse bada difference hai: SI mein paisa seedhi line (linear) mein badhta hai, CI mein paisa tezi se curve (exponential) mein badhta hai!

💡 Socho ek snowball pahaad se neeche lurhak raha hai — shuru mein chota hai (SI jaisa, slow growth), par jaise jaise lurhakta hai zyada snow chipakta hai aur bada hota jaata hai (CI jaisa, fast growth)!

📌 Bank mein paisa rakhna, EMI/loan lena, fixed deposit (FD) — sab jagah SI ya CI ka concept use hota hai.

➕ Step 2 — Simple Interest (SI) — poora samjho, step by step

SI Formula: SI = (P × R × T) / 100
Total Amount: A = P + SI = P(1 + RT/100)

Kyun "Simple" kehte hain? Kyunki interest hamesha sirf original Principal (P) pe calculate hota hai — chahe kitne saal ho jaayein, interest amount same hi rehta hai har saal.

Step by step example: ₹10,000 ko 10% rate pe 3 saal ke liye invest karo
1. SI = (10000 × 10 × 3) / 100 = ₹3,000 (total interest, 3 saalon mein)
2. Per year interest = 3000/3 = ₹1,000 har saal — same hi rehta hai!
3. Total Amount A = 10000 + 3000 = ₹13,000

Graph pe dekho toh: SI ka growth ek seedhi straight line hai — kyunki har saal equal amount add hota hai.

💡 SI ka shortcut: agar tumhe pata hai 1 saal ka interest, toh sirf usse saalon se multiply kardo — kyunki amount kabhi badalta nahi!

⚠️ SI formula mein "/100" bhoolna mat — Rate already percentage mein hai, isliye divide karna zaroori hai.

📌 ₹5,000 at 8% for 4 years: SI = (5000×8×4)/100 = ₹1,600 | Total Amount = 5000+1600 = ₹6,600

🚀 Step 3 — Compound Interest (CI) — "Snowball Effect" gehraai se

CI Formula: A = P(1 + r/n)ⁿᵗ, jahan CI = A − P

Kyun "Compound" kehte hain? Kyunki har baar jab interest add hota hai, woh nayi Principal ban jaata hai — aur usi badi Principal pe agla interest calculate hota hai. Isi liye paisa "ghoomte hue" (compound) badhta hai!

n ka matlab — kitni baar saal mein compound hota hai:
• n=1 → Yearly (saal mein 1 baar)
• n=2 → Half-yearly (saal mein 2 baar)
• n=4 → Quarterly (saal mein 4 baar)
• n=12 → Monthly (saal mein 12 baar)
Zyada n → zyada baar compound → zyada paisa (chhota sa fayda, par fayda hai!)

Step by step example (yearly, n=1): ₹10,000 at 10% for 3 years
Year 1: 10000 → 10000×1.1 = 11,000
Year 2: 11000 → 11000×1.1 = 12,100 (interest=1100, pehle se zyada!)
Year 3: 12100 → 12100×1.1 = 13,310 (interest=1210, aur bhi zyada!)
Total CI = 13310−10000 = ₹3,310 (SI se ₹310 zyada — chhota lagta hai par lambi muddat mein bada fark padta hai!)

💡 Einstein ne CI ko "8th wonder of the world" bola tha — kyunki yeh time ke saath exponentially badhta hai. Jitna early invest karo, utna zyada fayda!

⚠️ Formula mein "t" ki power "n×t" honi chahiye (total compounding periods), na ki sirf "t" — agar half-yearly hai 3 saal ke liye, toh power = 2×3 = 6 hoga, rate bhi half (r/2) hoga.

📌 ₹10,000 at 10% for 20 years (yearly compounding): CI Amount = 10000×(1.1)²⁰ ≈ ₹67,275 — SI se (₹30,000) almost double!

⚖️ Step 4 — SI vs CI: Side-by-side comparison (Exam shortcut)

Property	Simple Interest	Compound Interest
Interest calculate hota hai pe	Sirf original Principal	Principal + pichla interest
Har saal ka interest	Same rehta hai	Badhta jaata hai
Growth ka shape	Straight line (Linear)	Curve (Exponential)
Kab use hota hai	Short-term loans	FD, Mutual Funds, Savings
Lambi muddat mein behtar	❌	✅ (zyada fayda)

Real life mein kahan use hota hai?
• SI: kuch short-term personal loans, kuch types ke bonds
• CI: Bank Fixed Deposits (FD), Mutual Funds, Recurring Deposits, Credit Card unpaid bills (isliye credit card bill jaldi pay karo, warna CI tezi se badhta hai!)

💡 Quick check trick: agar question mein "every year same interest" ya "principal pe hi interest" likha ho → SI hai. Agar "interest pe bhi interest", "compounded", ya "yearly/half-yearly basis" likha ho → CI hai.

📌 Credit card warning: ₹50,000 unpaid bill at 36% annual CI (3% monthly) → 1 saal mein hi ₹50,000 → almost ₹71,400 ho jaata hai agar kuch pay na karo!

🎬 SI vs CI growth comparison dekho (₹10,000 pe)

Rate (%/year)10

Years10

👀 Neela bar (SI) hamesha seedha badhta hai. Purple bar (CI) curve ki tarah badhta hai — Years slider ko zyada le jaao aur dekho purple bar neele se kitna aage nikal jaata hai!

🧮 Try It — ₹10,000 invest karo

Principal (₹)

Rate (%/year)

Years

🟢 Algebra Essentials

🔣

Quadratic Formula — x = (−b ± √(b²−4ac)) / 2a

Algebra · Roots of ax²+bx+c=0 · Discriminant

▼

x = (−b ± √(b²−4ac)) / 2a
for ax² + bx + c = 0

Quadratic equation ke roots — discriminant decide karta hai kitne real roots hain

a = x² ka coefficient b = x ka coefficient c = constant term D = b²−4ac = discriminant

📖 Discriminant kya bolta hai?

D = b²−4ac — yeh square root ke andar wala part hai.

• D > 0 → 2 distinct real roots (parabola x-axis ko 2 jagah kaategi)
• D = 0 → 1 equal real root (parabola x-axis ko sirf touch karegi)
• D < 0 → No real roots (complex/imaginary — parabola x-axis ko nahi kaategi)

Sum of roots = −b/a Product of roots = c/a

💡 Pehle factoring try karo (faster). Agar nahi bana → quadratic formula use karo.

⚠️ a=0 mat hona chahiye — warna quadratic nahi, linear equation hai!

📌 x²−5x+6=0 → a=1,b=−5,c=6 → D=25−24=1 → x=(5±1)/2 → x=3 or x=2

🎬 a,b,c badhao → parabola aur roots live dekho

b-5

🧮 Try It — Find Roots

a (x² coeff)

b (x coeff)

c (constant)

📉

Logarithm Rules — log(ab), log(a/b), log(aⁿ)

Algebra · log base · Change of base · Natural log

▼

log(ab) = log a + log b
log(a/b) = log a − log b
log(aⁿ) = n·log a
log_b(x) = log(x) / log(b)

Multiplication → Addition · Division → Subtraction · Power → Multiplication

log = base 10 (common log) ln = base e (natural log) log_b(x) = y means bʸ = x

📖 Log kyun use karte hain?

Log multiplication ko addition mein convert karta hai — isliye calculator se pehle log tables se complex calculations hoti theen!

Important values yaad rakho:
• log₁₀(1) = 0 log₁₀(10) = 1 log₁₀(100) = 2
• log₁₀(0.1) = −1 ln(1) = 0 ln(e) = 1
• log_b(b) = 1 (always!) log_b(1) = 0 (always!)

Change of base: log₃(81) = log(81)/log(3) = 1.908/0.477 = 4 (kyunki 3⁴=81)

💡 log(ab) = log a + log b — isliye pH = −log[H⁺] mein products multiply hote hain toh logs add hote hain

⚠️ log(a+b) ≠ log a + log b. Yeh common galti hai!

📌 log(1000) = log(10³) = 3×log(10) = 3×1 = 3 | log(50) = log(100/2) = 2 − log2 = 2 − 0.301 = 1.699

🎬 x slider karo → log curve aur value dekho

x value10

Base b10

🧮 Try It

Value a

Value b

Power n

🎲

Permutation & Combination — nPr, nCr

Counting · Arrangement vs Selection · Factorial

▼

nPr = n! / (n−r)!
nCr = n! / (r! × (n−r)!)
nCr = nPr / r!

Permutation = order matters · Combination = order doesn't matter

n = total items r = items chosen n! = n factorial = n×(n−1)×…×1

📖 Kab kaunsa use karein?

Permutation (order matters): Lock code (1-2-3 ≠ 3-2-1), race positions, seating arrangements.
Combination (order doesn't matter): Choosing team members, lottery numbers, selecting books.

Golden rule: "Select karke arrange bhi karna hai?" → nPr. "Sirf select karna hai?" → nCr.

nCr = nPr / r! — Kyunki same group ke r items r! tareekon se arrange ho sakte hain, lekin combination mein wo sab same maane jaate hain.

💡 nC0 = nCn = 1 nC1 = n nCr = nC(n−r) (symmetry!)

📌 5 logo mein se 3 ka team: nCr = 5!/(3!×2!) = 120/12 = 10 ways | 5 logo ki race top-3 positions: nPr = 5!/(5−3)! = 120/2 = 60 ways

🎬 n aur r slide karo → nPr vs nCr fark dekho

Total items n5

Choose r3

🧮 Try It

n (total)

r (choose)

🔺

Binomial Theorem — (a+b)ⁿ expansion

Algebra · Pascal's Triangle · General term

▼

(a+b)ⁿ = Σ ⁿCr · aⁿ⁻ʳ · bʳ
T(r+1) = ⁿCr · aⁿ⁻ʳ · bʳ

General term (r+1)th = nCr × a^(n-r) × b^r · r starts from 0

n = power/exponent r = term number (0-indexed) ⁿCr = binomial coefficient T(r+1) = (r+1)th term

📖 Pascal's Triangle se shortcut

Pascal's Triangle mein nth row ke numbers = binomial coefficients for (a+b)ⁿ.

Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1

So (a+b)³ = a³ + 3a²b + 3ab² + b³ (coefficients: 1 3 3 1)

Middle term trick: n even → middle term = T(n/2+1). n odd → 2 middle terms = T((n+1)/2) aur T((n+3)/2)

💡 nth term mein b ki power = r = n−1 se dhundho. e.g. 4th term of (a+b)⁶ → r=3 → ⁶C₃·a³·b³ = 20a³b³

📌 (1+x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴ | (a−b)³ = a³ − 3a²b + 3ab² − b³ (odd powers of b are negative)

🎬 Pascal's Triangle — n badhao → rows aur expansion dekho

Power n4

🧮 Try It — Find rth Term

Power n

Term number r+1

📊 Matrices & Coordinate Geometry

🔢

Matrices — Basic Operations & Determinant

Algebra · Addition · Multiplication · det · Inverse

▼

[A+B]ᵢⱼ = Aᵢⱼ + Bᵢⱼ
[AB]ᵢⱼ = Σ Aᵢₖ × Bₖⱼ
det[[a b][c d]] = ad − bc
A⁻¹ = (1/det A) × adj A

Matrix ops: element-wise addition · row×col multiplication · 2×2 determinant

m×n = matrix with m rows, n cols det(A) = ad−bc for 2×2 A⁻¹ = inverse (exists only if det≠0)

📖 Matrix operations ke rules

Addition: Sirf same size matrices add ho sakti hain — element by element.
Multiplication: A(m×n) × B(n×p) = C(m×p). A ki cols = B ki rows hona chahiye!

2×2 Matrix:
[a b] [e f] [ae+bg af+bh]
[c d] × [g h] = [ce+dg cf+dh]

Determinant: det = ad−bc. Tells if matrix is invertible. det=0 → singular (no inverse).
Inverse: A⁻¹ = (1/det)×[d −b; −c a] for 2×2.

💡 AB ≠ BA generally! Matrix multiplication is NOT commutative.

⚠️ A×B karne ke liye A ki column count = B ki row count hona MUST hai.

📌 det[[3,1],[2,4]] = 3×4 − 1×2 = 12 − 2 = 10 → Inverse exists!

🎬 2×2 matrices enter karo → multiplication aur determinant live dekho

A: a2

A: b1

B: e3

B: f0

🧮 Try It — 2×2 Matrix Multiply & Determinant

A = [a b; c d]

B = [e f; g h]

📍

Coordinate Geometry — Distance, Section & Slope

Geometry · Distance formula · Midpoint · Slope · Section formula

▼

d = √[(x₂−x₁)² + (y₂−y₁)²]
M = ((x₁+x₂)/2, (y₁+y₂)/2)
m = (y₂−y₁)/(x₂−x₁)
P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Distance · Midpoint · Slope · Section formula (internal division)

d = distance between 2 points M = midpoint m = slope/gradient m:n = ratio for section formula

📖 Har formula ka use

Distance formula: Pythagoras theorem ka coordinate version! Horizontal + vertical distances se diagonal distance.

Slope (m): = rise/run = (y change)/(x change). Slope tell karta hai line kitni steep hai:
• m=0 → horizontal line
• m undefined → vertical line
• m positive → line upar jaati hai
• m negative → line neeche jaati hai

Section formula: Point P, line segment AB ko m:n ratio mein divide karta hai. m:n=1:1 → midpoint formula!

Line equation: y−y₁ = m(x−x₁) (point-slope form)

💡 Parallel lines ka slope equal. Perpendicular lines: m₁ × m₂ = −1

📌 A(1,2) B(4,6): d=√[(4−1)²+(6−2)²]=√[9+16]=√25=5 | Slope=(6−2)/(4−1)=4/3

🎬 Points drag karo → distance, slope, midpoint live dekho

x₁1

y₁2

x₂5

y₂6

🧮 Try It

x₁

y₁

x₂

y₂

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📈 Limits

lim

Limits — lim(x→a) f(x)

Calculus · L'Hôpital's Rule · sinx/x → 1

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lim(x→a) f(x) = L
lim(x→0) sinx/x = 1
lim(x→∞) (1+1/x)ˣ = e

Function ki value jab x kisi point ke paas jaata hai

x→a = x, a ke paas jaata hai (equal nahi) L = limit ki value e = 2.71828... (Euler's number)

🧒 Step 1 — Limit kya hai? Bilkul shuru se, ek kahani se samjho

Socho tum apne ghar ki taraf walk kar rahe ho. Tum gate se 10 meter door ho, phir 5 meter, phir 2 meter, phir 1 meter, phir 0.5 meter... Tum gate ke kareeb aur kareeb aa rahe ho — chahe abhi tak gate ko touch na kiya ho.

Yehi Limit hai! Maths mein hum poochte hain: "Jab x kisi value 'a' ke bahut bahut paas jaata hai (chahe exactly 'a' na bane), tab function f(x) kis value ke paas jaata hai?"

Isko likhte hain: lim(x→a) f(x) = L
Padhte hain: "x tends to a, f(x) ki limit L hai"

Sabse bada confusion clear karo: Limit yeh nahi batata ki x=a pe kya hota hai. Yeh batata hai ki x, a ke bahut bahut kareeb jaane par f(x) kahan "settle" hota hai. Kabhi kabhi yeh same hota hai jo x=a pe hai, par kabhi kabhi bilkul alag hota hai!

💡 Yaad rakho: Limit = "approach karna" (kareeb jaana), "reach karna" (exactly wahan pohochna) nahi! Jaise gate ke kareeb aana, gate ko touch karna zaroori nahi.

📌 Roz ki life mein: Ek glass mein paani daal rahe ho, paani level 100% ke "kareeb" aata jaata hai par kabhi exactly nahi chhalakta — yeh bhi ek limit jaisi feeling hai!

🔍 Step 2 — 0/0 ka mystery aur Left-Right limit

Real example lete hain: f(x) = sinx/x. Agar x=0 seedha daal do → sin(0)/0 = 0/0 → yeh undefined hai (calculator bhi error de dega)!

Par jab x sirf 0 ke bahut kareeb jaata hai (jaise x=0.1, x=0.01, x=0.001...) — har baar answer 1 ke zyada se zyada kareeb aata jaata hai. Isi liye lim(x→0) sinx/x = 1 — chahe x=0 pe khud function defined nahi hai!

Left limit aur Right limit — do taraf se aana:
• x→a⁻ (left limit) = x, 'a' ke baayi taraf se aata hai (a se chhota values, jaise a=2 ke liye 1.9, 1.99, 1.999...)
• x→a⁺ (right limit) = x, 'a' ke daai taraf se aata hai (a se bada values, jaise 2.1, 2.01, 2.001...)

Limit exist karne ka rule:
Agar Left limit = Right limit → tabhi poori limit exist karti hai!
Agar Left limit ≠ Right limit → limit exist nahi karti (function us point pe "jump" karta hai)

💡 Trick yaad rakhne ka: socho tum ek pahaad pe baayi taraf se chadh rahe ho aur tumhara dost daayi taraf se. Agar dono same peak height pe milte ho → limit exist karti hai. Agar alag jagah ruk jaate ho → limit exist nahi karti.

📌 f(x) = |x|/x ke liye x→0: Left side (x negative) pe answer = −1, Right side (x positive) pe answer = +1. Dono alag hain → limit exist nahi karti yahan!

📐 Step 3 — Famous standard limits — kaise aate hain, gehraai se

1️⃣ lim(x→0) sinx/x = 1
Jab x bahut chota ho (radians mein), curve sinx aur seedhi line x ek dusre ke bahut kareeb ho jaate hain. Isi liye unka ratio 1 ke paas jaata hai. Yeh formula sirf radians mein kaam karta hai, degrees mein nahi — kyunki radian measurement khud arc-length se directly judi hai.

2️⃣ lim(x→∞) (1 + 1/x)ˣ = e
Yeh formula compound interest se nikla hai! Agar tum apna paisa "infinite baar" chote-chote intervals mein compound karo (continuous compounding), toh growth factor e = 2.71828... (Euler's number) ke paas pohoch jaata hai. Banking, population growth, radioactive decay — sab jagah 'e' dikhta hai!

3️⃣ Polynomial limits (seedha substitute karo):
Agar function continuous hai (koi gap/break nahi), toh seedha x=a daal do! lim(x→2) (x²+3x) = 4+6 = 10

4️⃣ Indeterminate forms (0/0, ∞/∞):
Jab seedha substitute karne se 0/0 ya ∞/∞ aaye, factorize karo aur cancel karo — jaise neeche example mein!

⚠️ lim(x→0) sinx/x = 1 sirf radians mein sahi hai! Calculator degree mode mein ho toh galat answer aayega — ALWAYS radian mode check karo.

📌 lim(x→2) (x²−4)/(x−2): seedha x=2 daalo toh 0/0 aata hai → factorize karo: (x−2)(x+2)/(x−2) → cancel (x−2) → lim(x→2) (x+2) = 4

🚀 Step 4 — L'Hôpital's Rule — jab factorize na ho paaye

Kabhi kabhi 0/0 ya ∞/∞ form aata hai par factorize karna mushkil ya impossible hota hai (jaise trig ya exponential functions ke saath). Tab L'Hôpital's Rule ek shortcut hai!

Rule: Agar lim f(x)/g(x), 0/0 ya ∞/∞ form de raha hai, toh:
lim f(x)/g(x) = lim f'(x)/g'(x) (upar aur neeche, dono ko alag-alag differentiate karo!)

Step by step kaise use karo:
1. Check karo — kya 0/0 ya ∞/∞ form hai? (Agar nahi, toh L'Hôpital use mat karo!)
2. Numerator ko differentiate karo (f' nikaalo)
3. Denominator ko differentiate karo (g' nikaalo)
4. Naya limit lo: f'(x)/g'(x) ka
5. Agar phir bhi 0/0 ya ∞/∞ aaye, dobara L'Hôpital lagao (jab tak clear answer na mile)

Worked example: lim(x→0) sinx/x → 0/0 form → differentiate: f'(x)=cosx, g'(x)=1 → lim(x→0) cosx/1 = cos(0) = 1 ✓ (same answer jo humne pehle nikala!)

💡 L'Hôpital ek "emergency tool" hai — pehle factorize/simplify try karo, woh aksar fast hota hai. L'Hôpital tab use karo jab baaki tareeke fail ho jaayein.

⚠️ L'Hôpital sirf 0/0 ya ∞/∞ form pe lagao! Kisi normal limit (jo seedhe substitute se nikal jaaye) pe galti se differentiate mat kar dena.

📌 lim(x→∞) (3x²+1)/(x²+5) → ∞/∞ form → differentiate top-bottom: lim 6x/2x = lim 3 = 3 (top aur bottom ki highest power coefficients ka ratio)

🎬 x ko 0 ke paas le jaao — dekho sinx/x kaise 1 ke "kareeb" aata hai

x value (slider ko left le jaao → 0 ke kareeb)1.0

👀 Dekho: jaise jaise slider ko left taraf khischo (x chota hota jaata hai), peela dot upar ki taraf "limit=1" wali hari dashed line ke bilkul kareeb chala jaata hai — par x=0 pe curve khud break ho jaati hai (kyunki 0/0 undefined hai)!

🧮 Try It — L'Hôpital: lim(x→0) sinx/x

x (radians, close to 0)

📉 Differentiation

d/dx

Differentiation — Rules & Standard Derivatives

Calculus · dy/dx · Power Rule · Chain Rule · Product Rule

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d/dx(xⁿ) = nxⁿ⁻¹
d/dx(uv) = u'v + uv'
d/dx(u/v) = (u'v − uv') / v²
dy/dx = dy/du × du/dx

Power · Product · Quotient · Chain Rule

d/dx = x ke respect mein differentiate f'(x) = derivative of f(x) slope = derivative = tangent ki steepness

🧒 Step 1 — Derivative kya hai? Speedometer se samjho

Socho tum car chala rahe ho. Car ka speedometer batata hai ki abhi is exact second tum kitni fast chal rahe ho — yeh hai "instantaneous speed".

Ab socho graph banaya jaaye: x-axis pe time, y-axis pe distance tumne tay ki. Yeh curve banayega. Curve ki steepness (slope) har point pe — wahi tumhari speed hai us moment pe!

Derivative ka matlab: kisi function ki curve, kis exact point pe kitni tezi se badh ya ghat rahi hai — yani uska instant slope.

Likhte hain: f'(x) ya dy/dx (padhte hain: "dee-y bai dee-x" — "y ka chota change, x ke chote change ke respect mein")

• Slope positive → function badh raha hai (car aage badh rahi hai)
• Slope negative → function ghat raha hai (car peeche/brake lagi)
• Slope zero → us point pe function flat hai (car ruki hui, ya curve ka top/bottom point)

💡 Speedometer ki tarah hi, derivative har point pe "abhi kya ho raha hai" batata hai — poori journey ka average nahi, sirf woh ek exact moment.

📌 Agar distance function s(t) = t² hai (t seconds mein, s meters mein), toh speed = s'(t) = 2t. t=3 sec pe speed = 2×3 = 6 m/s

📐 Step 2 — Power, Product aur Quotient Rule — har ek alag se samjho

① Power Rule (sabse aasan, sabse zyada use hota hai):
d/dx(xⁿ) = n·xⁿ⁻¹ — power ko aage le aao, phir power ko 1 se ghata do.
Example: d/dx(x⁵) = 5x⁴

② Product Rule (jab 2 functions multiply ho rahe hain):
d/dx(u·v) = u'v + uv' — "pehle ko differentiate karo × doosra jaisa hai, PLUS pehla jaisa hai × doosre ko differentiate karo"
Example: d/dx(x²·sinx) → u=x², v=sinx → u'=2x, v'=cosx → answer = 2x·sinx + x²·cosx

③ Quotient Rule (jab ek function doosre se divide ho raha hai):
d/dx(u/v) = (u'v − uv') / v² — yaad rakhne ka tareeka: "low-dee-high minus high-dee-low, over low-squared" (neeche wala × upar ka derivative, minus upar wala × neeche ka derivative, sab divide by neeche ka square)
Example: d/dx(sinx/x) → u=sinx, v=x → u'=cosx, v'=1 → answer = (cosx·x − sinx·1)/x²

💡 Power Rule trick yaad rakho: "power ko age le aao, ek ghataao." d/dx(x³)=3x², d/dx(x⁷)=7x⁶ — pattern same rehta hai!

⚠️ Product Rule mein bhool se sirf u'v' mat likh dena (yeh galat hai!) — humesha u'v + uv' poora formula use karo.

📌 f(x) = x³ + 2x² − 5x + 7 → har term ko alag-alag power rule se differentiate karo → f'(x) = 3x² + 4x − 5 → f'(2) = 12+8−5 = 15 (yeh hai slope at x=2)

🔗 Step 3 — Chain Rule — "function ke andar function" ka raaz

Chain Rule tab use hota hai jab ek function doosre function ke andar "chhupa" hota hai — jaise sin(x²): yahan x² pehle "andar" hai, sin "bahar" hai.

Formula: dy/dx = dy/du × du/dx (agar y, u ka function hai, aur u, x ka function hai)

Aasan trick — "Outside-Inside" method:
1. Bahar wale function ko differentiate karo (andar wale ko abhi waise hi rakho)
2. Phir uska multiply karo andar wale ke derivative se

Worked example: d/dx(sin(x²))
• Bahar: sin(◻) → differentiate hoke cos(◻) banta hai → cos(x²)
• Andar: x² → differentiate hoke 2x banta hai
• Multiply karo: cos(x²) × 2x

Doosra example: d/dx((3x+1)⁵)
• Bahar: (◻)⁵ → 5(◻)⁴
• Andar: 3x+1 → derivative = 3
• Answer: 5(3x+1)⁴ × 3 = 15(3x+1)⁴

💡 Test karo: kya function "layered" hai (jaise kisi cheez ke "andar" doosri cheez)? Agar haan, toh chain rule zaroor lagega.

⚠️ Chain rule bhoolna sabse common mistake hai exams mein! Composite function dikhe (jaise sin(2x), (x+1)³, e^(x²)) → always chain rule use karo.

📌 d/dx(e^(3x)) → bahar e^(◻) → e^(◻) hi rehta hai → andar 3x → derivative 3 → answer = 3e^(3x)

📋 Step 4 — Standard derivatives yaad karne wali table + real life mein use

f(x)	f'(x)	Yaad rakhne ka tip
sin x	cos x	sin → cos (positive)
cos x	−sin x	cos → −sin (negative!)
tan x	sec² x	tan → sec squared
eˣ	eˣ	eˣ kabhi nahi badalta — magic!
ln x	1/x	log → 1 upon x
constant (jaise 7)	0	constant kabhi badalta nahi → slope=0

Real life mein derivatives kahan dikhte hain?
• Speed/Acceleration: distance ka derivative = speed, speed ka derivative = acceleration
• Business: Profit function ka derivative = marginal profit (1 extra item bechne se kitna profit badhega)
• Physics: Curve pe maximum/minimum point dhundna (jaise rocket ki maximum height) — wahan derivative = 0 hota hai!

💡 Maximum ya minimum point dhundne ka trick: f'(x) = 0 set karo aur x solve karo — us point pe curve "flat" ho jaati hai (top ya bottom).

📌 Ball ko upar throw kiya: height h(t) = 20t − 5t². Maximum height kab? h'(t)=20−10t=0 → t=2 sec → h(2)=40−20=20 meters

🎬 Curve pe point move karo — tangent line (slope) live dekho

x position1.0

Functionx³

👀 Peeli dashed line wo tangent hai — curve ko us point pe sirf "chhoo" rahi hai. Uska tilt (slope number) hi f'(x) hai. Curve change karo (3 alag functions try karo) aur dekho slope kaise badalta hai!

🧮 Try It — Power Rule: d/dx(xⁿ) at x

Power n

at x =

∫ Integration

∫

Integration — ∫f(x)dx · Area under curve

Calculus · Indefinite · Definite · Integration by Parts

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∫xⁿ dx = xⁿ⁺¹/(n+1) + C
∫ₐᵇ f(x)dx = F(b) − F(a)
∫u dv = uv − ∫v du

Indefinite · Definite integral · Integration by Parts

C = constant of integration F(x) = antiderivative of f(x) ∫ₐᵇ = a se b tak area

🧒 Step 1 — Integration kya hai? Do simple tareekon se samjho

Tareeka 1 — "Ulta karna" (Reverse Engineering):
Socho tumhe pata hai ki kisi cheez ko differentiate karne se 2x milta hai. Ab sawaal ulta poocha jaaye: "Woh ORIGINAL cheez kya thi jiska derivative 2x hai?" Answer: x² (kyunki d/dx(x²)=2x). Yehi Integration hai — differentiation ka bilkul ulta process! Isi liye integration ko "antiderivative" bhi kehte hain.

Tareeka 2 — Squares count karke Area nikalna:
Socho ek curve ke neeche, x-axis tak, ek "ghumawdaar" (curvy) shape hai jiska area nikalna hai. Seedhi shapes (rectangle, triangle) ka area aasaan hai, par curvy shape ka?
Trick: us area ko bahut saari patli-patli vertical strips (rectangles) mein todo. Har strip ka area = height × bahut chota width. Sabko add karo → approx area milega! Ab agar strips ko infinitely thin bana do — exact area mil jaata hai. Yehi Integration hai — "infinite chote rectangles ka sum"!

Isi liye integration sign ∫ ek lamba "S" jaisa dikhta hai — S for "Sum"!

💡 Yaad rakho: Differentiation = slope nikalna (curve kitni steep hai). Integration = area nikalna (curve ke neeche kitna space hai). Dono ek dusre ke "ulte" operations hain.

📌 Real life: ek car ki speed-time graph ka area = total distance jo car ne tay ki! (Speed × Time = Distance, area integration se yehi karta hai)

📐 Step 2 — Indefinite Integration aur "+C" ka raaz

Power Rule for Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C — (power mein 1 jodo, phir nayi power se divide karo)
Example: ∫x² dx = x³/3 + C

"+C" kyun zaroori hai? Socho d/dx(x²)=2x, par d/dx(x²+5)=2x bhi hai, aur d/dx(x²−100)=2x bhi hai! Kyunki constant ka derivative hamesha 0 hota hai. Toh jab hum "ulta" karte hain (integrate), hume nahi pata ki original function mein konsa constant tha — isliye hum generic "C" likh dete hain jo koi bhi number ho sakta hai.

Standard integrals yaad rakho:
• ∫sinx dx = −cosx + C (negative dhyaan rakho!)
• ∫cosx dx = sinx + C
• ∫eˣ dx = eˣ + C (eˣ kabhi nahi badalta — integration mein bhi magic!)
• ∫(1/x) dx = ln|x| + C
• ∫sec²x dx = tanx + C

💡 Trick: integration ki table, differentiation ki table ko "ulta" karke hi banti hai! Jo d/dx(sinx)=cosx tha, woh ban jaata hai ∫cosx dx = sinx + C.

⚠️ Indefinite integral mein "+C" HAMESHA likhna! Exam mein bhool gaye toh half marks kaat lete hain — yeh choti si galti bahut common hai.

📌 ∫(3x² + 2x − 1)dx → har term alag se integrate karo → 3·(x³/3) + 2·(x²/2) − x → = x³ + x² − x + C

📏 Step 3 — Definite Integral — exact area nikalna, step by step

Jab integration ki upar aur neeche ki limit (a aur b) di gayi ho, toh woh ek exact number (area) deta hai — ab koi "+C" nahi chahiye, kyunki C dono jagah cancel ho jaata hai!

Formula: ∫ₐᵇ f(x)dx = F(b) − F(a), jahan F(x), f(x) ka antiderivative hai.

Step by step kaise solve karo:
1. Pehle indefinite integral nikaalo: F(x) (bina +C ke, kyunki yeh khud cancel ho jaayega)
2. F(x) mein upper limit b daalo → F(b) nikaalo
3. F(x) mein lower limit a daalo → F(a) nikaalo
4. Subtract karo: F(b) − F(a) → yeh hai tumhara final area

Worked example: ∫₁³ x² dx
• F(x) = x³/3
• F(3) = 27/3 = 9
• F(1) = 1/3
• Area = 9 − 1/3 = 26/3 ≈ 8.67 square units

💡 Definite integral ek "number" deta hai (area), indefinite integral ek "function" deta hai (formula with +C). Dono alag cheez hain — confuse mat hona!

⚠️ Agar curve x-axis ke neeche jaaye (negative region), toh area "negative" aa sakta hai mathematically — par actual physical area hamesha positive hota hai. Exam mein dhyaan rakhna kaunsa poocha gaya hai.

📌 ∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 2.667 square units (yeh hai x² curve ke neeche ka area, x=0 se x=2 tak)

🧩 Step 4 — Integration by Parts — jab 2 functions multiply ho rahe ho

Kabhi kabhi integrate karna hai ek function jo 2 alag functions ka product hai (jaise x·eˣ, ya x·sinx). Aise mein normal rules kaam nahi karte — tab use hota hai Integration by Parts!

Formula: ∫u dv = uv − ∫v du

Sabse bada confusion — "u" kya choose karoon? Iske liye LIATE rule use karo (priority order, jo pehle aaye usse "u" banao):
Log functions (jaise lnx)
Inverse trig (jaise sin⁻¹x)
Algebraic (jaise x, x², x³)
Trig (jaise sinx, cosx)
Exponential (jaise eˣ)

Step by step:
1. LIATE se decide karo kaun u banega, kaun dv
2. u ka derivative nikaalo (du), dv ka integral nikaalo (v)
3. Formula mein daalo: uv − ∫v du
4. Bacha hua integral solve karo

Worked example: ∫x·eˣ dx → LIATE se: x=Algebraic (u), eˣ=Exponential (dv) → u=x, dv=eˣdx → du=dx, v=eˣ → Formula: x·eˣ − ∫eˣdx = xeˣ − eˣ + C

💡 LIATE yaad karne ka tareeka: "Log Is Always The Easiest" — jo word pehle aata hai (uska priority zyada), usi ko u banao.

⚠️ Galat u choose karne se integral aur complicated ho jaata hai (kabhi infinite loop bhi ban jaata hai!). LIATE order follow karna bahut zaroori hai.

📌 ∫x·sinx dx → u=x, dv=sinx dx → du=dx, v=−cosx → x(−cosx) − ∫(−cosx)dx = −xcosx + sinx + C

🎬 Limits badlo — curve ke neeche ka area fill hote dekho

Lower limit a-2.0

Upper limit b2.0

👀 Wo hara shaded region — wahi answer hai ∫ₐᵇ x²dx ka! Lower limit (a) aur upper limit (b) ko slide karo, aur dekho region kaise bada/chota hota hai — sath mein number bhi badalta hai upar.

🧮 Try It — Definite Integral ∫ₐᵇ xⁿ dx

Power n

Lower limit a

Upper limit b

📏 Geometry — Quick Formula Cheat Sheet

A=πr² C=2πr A=½bh A=l×b a²+b²=c² V=πr²h V=⅓πr²h V=(4/3)πr³ Sector=(θ/360)πr² Tan²=d²−r²

👆 Kisi bhi card pe click karo — full root-to-advanced explanation + animation + Try It milega!

🔺 Triangles

🔺

Triangle — Area = ½ × base × height

Geometry · All triangles · Scalene, Isosceles, Equilateral

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A = ½ × b × h

Area = half of base times height

b = base (koi bhi side)h = perpendicular height

📖 ½ kyun?

Koi bhi triangle ek rectangle ka exactly aadha hota hai! Rectangle banao same base aur height se — usse diagonal se kaato — triangle milega. Rectangle ka area = b×h, toh triangle = ½×b×h.

Perimeter = a + b + c (teeno sides ka sum)

💡 Height hamesha base pe perpendicular (90°) honi chahiye — triangle ke andar ya bahar dono ho sakti hai.

🎬 Triangle ka rectangle connection dekho

Base b6

Height h4

🧮 Try It

Base (b)

Height (h)

🔺

Equilateral Triangle — A = (√3/4)a²

Geometry · Teeno sides equal · Teeno angles 60°

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A = (√3/4) × a²
P = 3a

Teeno sides equal (a) · Height = (√3/2)×a

📖 Special triangle kyun?

Equilateral triangle mein teeno angles hamesha 60° hote hain. Height formula se: h = (√3/2)×a. Phir area = ½×a×h = ½×a×(√3/2)×a = (√3/4)×a².

Yeh nature mein bahut milta hai — snowflakes, honeycomb, crystal structures.

📌 Side = 6cm → Area = (√3/4)×36 = 9√3 ≈ 15.59 cm²

🎬 Side badlao → Area aur Height dekho

Side a6

🧮 Try It

Side a

⬜ Quadrilaterals

⬜

Rectangle — A = l × b, P = 2(l+b)

Geometry · 4 right angles · Opposite sides equal

▼

A = l × b
P = 2(l + b)
d = √(l² + b²)

Area · Perimeter · Diagonal

📖 Diagonal kyun Pythagoras se aata hai?

Rectangle ka diagonal ek right triangle banata hai length, breadth aur diagonal ke saath. Isliye d = √(l²+b²) — seedha Pythagoras!

Square ek special rectangle hai jahan l = b.

💡 Room ka carpet area, field ka area, page ka area — sab rectangle formula!

🎬 Length/Breadth badlao → diagonal dekho

Length l8

Breadth b5

🧮 Try It

Length (l)

Breadth (b)

🟪

Square — A = a², P = 4a, d = a√2

Geometry · Teeno sides equal · Teeno angles 90°

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A = a²
P = 4a
d = a√2

Area · Perimeter · Diagonal

📖 a√2 kyun?

Square ka diagonal right triangle banata hai jahan dono legs = a. Pythagoras se: d = √(a²+a²) = √(2a²) = a√2.

√2 ≈ 1.414 — isliye diagonal hamesha side se 41.4% badi hoti hai!

📌 Chess board, Rubik's cube face, tiles — sab square!

🎬 Side badlao → area squares visually dikho

Side a5

🧮 Try It

Side a

◼

Parallelogram — A = base × height

Geometry · Opposite sides parallel · Rhombus bhi iska type

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A = b × h
P = 2(a + b)

Area = base × perpendicular height

📖 Rectangle se connection

Parallelogram ko dekho — uska ek side kaatke doosre taraf lagao → rectangle banta hai! Same base, same height → same area.

Rhombus: Special parallelogram jahan sab sides equal. A = (d1 × d2)/2 (diagonals se bhi calculate kar sakte ho)

💡 Height slanted side nahi hai — perpendicular distance between parallel sides!

🎬 Parallelogram → Rectangle transformation dekho

Base b8

Height h4

Slant2

🧮 Try It

Base (b)

Height (h)

⭕ Circles

⭕

Circle — Area, Circumference, Arc, Sector, Segment, Tangent

Geometry · Beginners se Advanced · π = 3.14159 · Chord · Arc · Sector · Theorem

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A = πr² C = 2πr = πd d = 2r

Arc = (θ/360) × 2πr Sector A = (θ/360) × πr²

Segment A = Sector A − Triangle A Tangent² = d² − r²

Area · Circumference · Diameter · Arc length · Sector area · Segment area · Tangent length

r = Radius d = Diameter π ≈ 3.14159 θ = Angle (degrees) C = Circumference A = Area

🧒 Step 1 — Circle kya hota hai? Dum se basic se shuru!

Ek pencil lo, ek point pe rakkho — woh hai Centre (केंद्र). Ab pencil ko ek fixed distance pe rakhte hue ghoomao — jo line bane woh hai Circle! 🔵

Circle ki sabse khaas baat: Centre se har jagah ki doori same hoti hai — isi fixed doori ko Radius (r) kehte hain.

Real life mein circle kahan hai?
🍕 Pizza — round hai (aur slices = sectors!)
🕐 Clock — round face, hands radius jitne
🔵 Coin — perfectly circular
🌙 Full Moon — circle dikhti hai sky mein
⚽ Ball cross-section — circle
🍩 Donut hole — circle
🎡 Ferris wheel — ek bada circle

Centre (O): Circle ka middle point — jahaan se saari radii shuru hoti hain.
Radius (r): Centre se kisi bhi edge point tak ki doori. Circle mein infinite radii hain, sab equal!
Diameter (d): Circle ke ek end se doosre end tak, centre se guzarte hue. d = 2r (hamesha!)

💡 Ek bachche ke liye trick: Compass se circle draw karte time jo doori set karte ho — woh radius hai! Compass needle = centre, pencil tip = edge.

🎬 Animation 1 — Circle ke Andar Kya Kya Hota Hai?

Neeche buttons click karo — ek ek part highlight hoga!

👆 Koi bhi button click karo!

🥧 Step 2 — π (Pi) kya hai? Yeh magic number kahan se aaya?

Ek bahut hi interesting cheez: Duniya ki kisi bhi circle ka Circumference ÷ Diameter = hamesha same number aata hai!

Chahe pizza bada ho ya chhota, chahe wheel ho ya coin — C/d hamesha 3.14159265... aata hai. Isi number ko hum π (Pi) kehte hain!

Khud try karo: Koi bhi round cheez lo (glass, tin). Dhaga circle ke bahar lapeto → length = Circumference. Phir diameter napo. C ÷ d karo → ~3.14 aayega! 🤩

π ek "irrational number" hai — iska decimal kabhi khatam nahi hota, kabhi repeat nahi karta:
π = 3.14159265358979323846...

Common approximations:
π ≈ 3.14 (school ke liye kaafi!)
π ≈ 22/7 (fraction roop — thoda better approximation)
π ≈ 3.14159 (zyada precise)

📌 Real example: Agar ek bicycle wheel ka diameter = 70 cm hai, toh ek chakkar mein kitni doori? C = π × d = 3.14 × 70 = 219.8 cm ≈ 2.2 meter! Matlab 100 chakkar = ~220 meter!

💡 Mnemonic for π: "May I have a large container of coffee" — count letters: 3.1415926!

🔵 Step 3 — Circumference = 2πr — Yahaan se kaise aaya?

Humne abhi jaana: C ÷ d = π
Toh: C = π × d
Aur d = 2r, toh: C = π × 2r = 2πr

Dono roop yaad rakho:

C = 2πr
Jab radius pata ho
Example: r=7 → C = 2×3.14×7 = 43.96

C = πd
Jab diameter pata ho
Example: d=14 → C = 3.14×14 = 43.96

r = C ÷ (2π)
Jab circumference pata ho
Example: C=31.4 → r = 31.4÷6.28 = 5

⚠️ Common mistake: C = 2πr mein r sirf ek baar hai! C = 2 × π × r. Kai log 2πr² likh dete hain — woh GALAT hai (woh area ka formula hai!).

🟣 Step 4 — Area = πr² — Kyun r² hai? Proof samjho!

Method 1 — Orange Slices se samjho (10 saal ke bachche ke liye):
Ek orange lo, usके slices karo — bahut saare patte jaisi strips milti hain (jaise pizza slices). Ab un slices ko alternate karte hue rakkho — ek upar ek neeche. Ek almost-rectangle banta hai!

Rectangle ki width = r (radius) hogi
Rectangle ki length = πr (circumference ka aadha = C/2 = 2πr/2)
Area of rectangle = r × πr = πr² ✅

Method 2 — Concentric rings se samjho:
Circle ko bahut saari patli-patli rings (annuli) mein todo. Har ring ko seedha karo — ek thin rectangle milega jiska width = thickness (dr) aur length = 2πr (circumference). Sab rectangles ka area add karo → ∫₀ʳ 2πr dr = πr².

Formula variations:
Area = πr² (jab r pata ho)
Area = π(d/2)² = πd²/4 (jab diameter pata ho)
Area = C²/(4π) (jab circumference pata ho)

📌 Pizza example: 8 inch pizza ka area = π × 4² = 50.27 sq inch. 16 inch pizza ka area = π × 8² = 201.06 sq inch. Notice: Diameter double → Area FOUR guna! (2² = 4 times bigger 🤯)

🎬 Animation 2 — Area Proof: Circle → Rectangle kaise banta hai!

Slices badhao → dekho kaise circle ek rectangle mein badal jaata hai!

Slices (n) 8

Radius r 5

🌈 Step 5 — Chord, Arc, Sector, Segment — Sab alag alag kya hain?

🟡 Chord: Circle ke andar ki koi bhi straight line jo do points ko jorti hai. Diameter sabse badi chord hai (centre se guzarti hai). Chhoti chord doosri jagah hoti hai.
Formula: Chord length = 2r × sin(θ/2) — jahaan θ wo angle hai jo chord centre pe banaati hai.

🟠 Arc: Circle ki boundary (circumference) ka koi bhi part. Do points ke beech ka curved portion.
Arc length = (θ/360) × 2πr — yaani puri circumference ka θ/360 waan hissa.
Agar θ = 360° → full circle → arc = 2πr ✓
Agar θ = 180° → semicircle → arc = πr ✓
Agar θ = 90° → quarter circle → arc = πr/2 ✓

🟣 Sector: Circle ka pizza-slice jaisa hissa. Do radii aur ek arc se bana hota hai.
Sector Area = (θ/360) × πr²
Agar θ = 360° → full circle → Sector A = πr² ✓

🌙 Segment: Chord aur arc ke beech ka region (bina radii ke). Do types hote hain:
Major Segment = bada hissa (zyada area)
Minor Segment = chhota hissa (kam area)
Segment Area = Sector Area − Triangle Area
= (θ/360) × πr² − (1/2) × r² × sin(θ)

💡 Memory trick: Sector = "S" like Slice (pizza slice) 🍕 | Segment = "S" like Small cut (chhota tukda) 🌙 | Arc = curve (circumference ka hissa) 🌈

🎬 Animation 3 — Sector aur Arc: Angle badlao, Pizza slice dekho!

Slider ghoomao → sector, arc length, aur segment sab change hoga!

Radius r 6

Angle θ° 90

📐 Step 6 — Tangent kya hoti hai? Circle ke Theorems!

Tangent: Woh line jo circle ko sirf ek point pe touch karti hai (andar nahi jaati). Tangent hamesha us point pe radius ke perpendicular (90°) hoti hai!

Tangent length formula: Ek point P se circle ke centre O ki doori d ho, aur radius r ho, toh:
Tangent length = √(d² − r²)
(Pythagoras se: tangent² + r² = d²)

📜 Important Circle Theorems:

1. Angle at Centre = 2 × Angle at Circumference:
Ek arc jo centre pe angle θ banaye, wahi arc circumference pe θ/2 angle banata hai.

2. Angles in same segment = equal:
Same chord se same side ke saare angles equal hote hain.

3. Angle in semicircle = 90°:
Diameter pe jo angle circumference pe bane — woh hamesha 90° hota hai! (Thales' Theorem)

4. Opposite angles of cyclic quadrilateral = 180°:
Ek cyclic quadrilateral (jiske sab corners circle pe ho) ke opposite angles ka sum = 180°.

5. Tangents from external point are equal:
Ek bahari point se do tangents draw karo → dono ki length hamesha equal hogi!

6. Perpendicular from centre bisects chord:
Centre se chord pe perpendicular line chord ko exactly aadha karta hai.

📌 Thales Theorem example: Ek diameter ki endpoints A aur B hain. Circumference pe koi bhi point P lo — angle APB hamesha 90° hoga! Try it on paper! 🤯

🎬 Animation 4 — Tangent aur Circle Theorems dekho!

Mode buttons se alag theorems dekho. Point move karo!

Point angle 60°

🌍 Step 7 — Real Life Mein Circle Formulas Kahan Kaam Aate Hain?

🍕 Pizza Delivery: 10 inch vs 12 inch pizza — kitna zyada milega? A₁ = π×5² = 78.5 sq in, A₂ = π×6² = 113 sq in. 12 inch pizza 44% zyada bada hai! (Sirf 20% diameter bada, par 44% area bada — isliye bade size pe zyada value milti hai!)

🛞 Tyre aur Odometer: Car ke tyre ka circumference = ek chakkar mein kitna aage jaayega. Odometer isi se distance count karta hai: total distance = C × number of rotations.

🌍 Earth ki Size: Eratosthenes ne 200 BC mein sirf ek stick aur circle geometry se Earth ka circumference nikaala! Usne 2 cities mein shadows ke angle measure kiye → arc angle nikala → full circle proportion se circumference! Amazing! 🤯

⚙️ Gears aur Machinery: Do gears ke teeth ka ratio = unke radii ka ratio. Sector area se gear tooth calculations hoti hain.

🏟️ Stadium Track: 400m athletic track mein 2 semicircles hote hain. Inner track ka radius chhota, outer track ka bada — isliye outer lane runners thoda aage se shuru karte hain (stagger = outer C − inner C ka difference).

🌡️ Thermometer Gauge: Circular gauge mein sector angle se temperature ya pressure reading display hoti hai — ek full sector sweep = full range.

🔭 Planets aur Orbits: Planetary orbits approximately circular hoti hain. Orbital circumference = 2πr se planet ki speed nikaali jaati hai (v = C/T, jahaan T = orbital period).

🎓 Step 8 — Advanced: Radian, Equation of Circle, 3D Connection

🔄 Radian kya hai? (Class 11+ ke liye)
Degree ek manmana unit hai (kisi ne 360 decide kiya). Radian natural unit hai:
1 radian = woh angle jahan arc length = radius
Full circle = 2π radians = 360°
π radians = 180°
π/2 radians = 90°

Radians mein formulas zyada simple ho jaate hain:
Arc = r × θ (sirf!)
Sector Area = (1/2) × r² × θ (sirf!)

📍 Equation of Circle:
Centre (h, k) aur radius r wali circle ki equation:
(x − h)² + (y − k)² = r²

Agar centre origin (0,0) pe ho:
x² + y² = r²

General form: x² + y² + 2gx + 2fy + c = 0
Centre = (−g, −f), Radius = √(g² + f² − c)

🌐 3D Connection:
Sphere ka SA = 4πr² = exactly 4 circles of radius r!
Cylinder ka CSA = 2πrh (ek circle ko h height tak roll karo)
Cone slant height l se: CSA = πrl (ek circle ka "unrolled" sector)

💡 Radian conversion: Degrees → Radians: multiply by π/180. Radians → Degrees: multiply by 180/π. Example: 90° = 90 × π/180 = π/2 radians.

⚠️ Advanced gotcha: Circle equation mein coefficient of x² aur y² = same (dono 1) hona chahiye. Agar alag hain, toh ellipse banti hai, circle nahi!

🎬 Animation 5 — Segment vs Sector: Fark dekhein live!

Angle badlao — Sector aur Segment dono alag alag colors mein dikhenge.

Radius r 6

Angle θ° 90

🧮 Try It — Saare Formulas ek saath calculate karo!

Radius r

Sector angle θ°

External point dist (for tangent)

📦 3D Shapes

📦

Cuboid — V = l×b×h, SA = 2(lb+bh+hl)

3D Geometry · Box shape · 6 rectangular faces

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V = l × b × h
SA = 2(lb + bh + hl)
d = √(l²+b²+h²)

Volume · Surface Area · Space diagonal

📖 Surface Area kyun 2(lb+bh+hl)?

Cuboid ke 6 faces hote hain — 3 pairs of opposite identical faces:
• Front/Back = l×h (×2)
• Left/Right = b×h (×2)
• Top/Bottom = l×b (×2)
Total = 2(lb + bh + hl)

Room paint karna ho — walls aur ceiling ka area = SA (minus floor)

📌 Room 5×4×3m: Volume = 60m³, SA = 2(20+12+15) = 94m²

🎬 Dimensions badlao → Volume aur SA dekho

Length l5

Breadth b4

Height h3

🧮 Try It

Length

Breadth

Height

🥫

Cylinder — V = πr²h, CSA = 2πrh

3D Geometry · Circular base · Can shape

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V = πr²h
CSA = 2πrh
TSA = 2πr(r+h)

Volume · Curved Surface Area · Total Surface Area

📖 CSA aur TSA mein fark

CSA (Curved SA): Sirf curved part — jaise can ka label. Rectangle ko roll karo → cylinder milta hai! Width = 2πr, Height = h → CSA = 2πrh

TSA (Total SA): CSA + 2 circular bases = 2πrh + 2πr² = 2πr(r+h)

Pipe, water tank, glass, candle — sab cylinders!

💡 Label sirf CSA, paint/cover karna ho toh TSA.

🎬 Radius aur Height badlao → Volume dekho

Radius r3

Height h6

🧮 Try It

Radius (r)

Height (h)

🍦

Cone — V = ⅓πr²h, CSA = πrl

3D Geometry · Circular base · Pointed top · Ice cream cone!

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V = ⅓πr²h
CSA = πrl
TSA = πr(l + r)
l = √(r² + h²)

Volume · Curved SA · Total SA · Slant height

r = base radius h = vertical height l = slant height

📖 ⅓ kyun aata hai Volume mein?

Cone ka volume exactly ⅓ of cylinder hota hai same r aur h ke saath. Ek cylinder mein teen cones fit hote hain!

Slant height l: Base rim se tip tak ka straight distance. Pythagoras se: l = √(r²+h²). Yeh CSA ke liye important hai — like how much paper to wrap an ice cream cone.

Real life: Ice cream cone, birthday cap, funnel, rocket nose.

💡 CSA mein 'l' (slant) use hoti hai, Volume mein 'h' (vertical) — confuse mat karna!

📌 r=3cm, h=4cm → l=5cm, V=⅓×π×9×4=37.7cm³, CSA=π×3×5=47.1cm²

🎬 Radius/Height badlao → Cone visualize karo

Radius r4

Height h6

🧮 Try It

Radius r

Height h

🌐

Sphere — V = (4/3)πr³, SA = 4πr²

3D Geometry · All points equidistant · Ball shape

▼

V = (4/3)πr³
SA = 4πr²

Volume · Surface Area

r = radius (centre se surface)

📖 SA = 4πr² kyun magical hai

Sphere ka Surface Area = 4 circles of same radius! 4 × πr²

Volume formula mein r³ — isliye volume bahut tezi se badhta hai radius ke saath. Radius double karo → volume 8 guna hoti hai!

Hemisphere: V = (2/3)πr³, CSA = 2πr², TSA = 3πr²

Real life: Cricket ball, earth, balloon, soap bubble, basketball.

💡 Sphere ka no flat face, no edge — perfect symmetry. Minimum surface area for given volume!

⚠️ Hemisphere mein TSA = CSA + base = 2πr² + πr² = 3πr² (base circle mat bhoolna!)

📌 r=7cm → SA=4π×49=615.75cm², V=(4/3)π×343=1436.76cm³

🎬 Radius badlao → Sphere aur Hemisphere dekho

Radius r5

🧮 Try It

Radius r

⬛ Special Quadrilaterals

🔷

Trapezium — A = ½(a+b)h

Geometry · One pair of parallel sides · Area formula

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A = ½(a + b) × h
P = a + b + c + d

Area = half × sum of parallel sides × height

a, b = parallel sides h = perpendicular height

📖 Formula ka logic — average of parallel sides!

Trapezium = ek side chhoti, ek side badi — dono parallel. Formula ka logic: agar dono sides equal hoti → rectangle banta. Toh formula average of both sides × height = ½(a+b)×h

Real life: Handbag shape, cross-section of riverbank, architecture, embankment.

Isosceles Trapezium: Non-parallel sides (legs) equal hote hain — symmetric hota hai.

💡 ½(a+b) = average of parallel sides. Rectangle mein a=b → ½(a+a)h = ah ✓

📌 Parallel sides = 8cm aur 5cm, height = 4cm → A = ½×(8+5)×4 = ½×13×4 = 26 cm²

🎬 Parallel sides aur height badlao → Trapezium dekho

Side a (top)5

Side b (bottom)9

Height h5

🧮 Try It

Side a (parallel)

Side b (parallel)

Height h

♦️

Rhombus — A = (d1 × d2)/2

Geometry · All sides equal · Diagonals bisect at 90°

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A = (d₁ × d₂) / 2
A = b × h
P = 4a

Area via diagonals · Area via base×height · Perimeter

d₁, d₂ = diagonals a = side (all equal) h = perpendicular height

📖 Rhombus = tilted square, aur ek khaas property

Rhombus ek special parallelogram hai jiske sab sides equal hote hain. Iske diagonals:
• Bisect each other at right angles (90°)
• Equal nahi hote (square mein hote hain)

Area formula: Diagonals se 4 right triangles bante hain → A = 4 × ½×(d₁/2)×(d₂/2) = (d₁×d₂)/2

Side from diagonals: a = √((d₁/2)² + (d₂/2)²) — Pythagoras!

💡 Square bhi ek rhombus hai jahan d₁ = d₂. Area = d²/2 (square)

📌 d₁=10cm, d₂=8cm → A = (10×8)/2 = 40 cm² | side = √(25+16) = √41 ≈ 6.4cm

🎬 Diagonals badlao → Rhombus aur 4 triangles dekho

Diagonal d₁10

Diagonal d₂8

🧮 Try It

Diagonal d₁

Diagonal d₂

📐 Basic Trigonometry

sin

sin, cos, tan — Unit Circle

Trigonometry · SOH CAH TOA · Values at all angles

▼

sin θ = Opp/Hyp
cos θ = Adj/Hyp
tan θ = Opp/Adj

SOH — CAH — TOA (yaad karo!)

θ (theta) = angle (jo humein chahiye) Opp = Opposite side (angle ke saamne wali side) Adj = Adjacent side (angle ke paas wali side) Hyp = Hypotenuse (sabse badi side, right angle ke saamne)

🧒 Step 1 — sin, cos, tan kya hai? Right triangle se bilkul shuru se samjho

Socho ek right-angled triangle hai (jisme ek angle exactly 90° hai). Trigonometry ek simple sawaal poochti hai: "Agar mujhe ek angle pata hai, toh kya main sides ka ratio nikal sakta hoon?" Aur jawab hai — haan!

Pehle teeno sides ko naam do (kisi bhi non-90° angle θ ko dekh kar):
• Hypotenuse (Hyp) = sabse badi side, jo right angle (90°) ke exactly saamne hoti hai — yeh hamesha fix rehta hai, kisi bhi angle se identify karne ki zaroorat nahi
• Opposite (Opp) = woh side jo humare angle θ ke "saamne" hai (jaise koi cheez seedhi aage dikh rahi ho)
• Adjacent (Adj) = woh side jo humare angle θ ke "paas mein" hai (touch kar rahi hai, par hypotenuse nahi)

Trick — sides badal jaate hain jab angle badalta hai! Agar tum doosre angle se dekho, toh "Opposite" aur "Adjacent" switch ho jaate hain — sirf Hypotenuse hamesha same rehta hai.

Ab teeno ratios define karte hain — yehi SOH-CAH-TOA hai:
Sin = Opp/Hyp | Cos = Adj/Hyp | Tan = Opp/Adj

💡 SOH-CAH-TOA yaad karne ka mazedaar tareeka: "Some Old Horses Chew Apples Happily Throughout Old Age" — har word ka pehla letter match karta hai!

📌 Ek triangle mein θ=30°, Opposite=5, Hypotenuse=10 hai. Toh sin(30°) = 5/10 = 0.5 (yeh exactly match karta hai standard value se!)

🔵 Step 2 — Unit Circle: sin/cos ko 90° se aage extend karna

Problem: right-triangle wala SOH-CAH-TOA sirf 0° se 90° tak kaam karta hai (kyunki triangle ke angles 90° se zyada nahi ho sakte). Par exam mein 120°, 180°, 270° jaise angles bhi poochte hain — unke liye kya karein?

Solution: Unit Circle — ek circle jiska radius exactly 1 hai, center origin (0,0) pe. Iske paas ek rotating point hai jo angle θ ke according circle pe ghoomta hai (anticlockwise, x-axis se shuru karke).

Naya, powerful definition:
• x-coordinate of that point = cos θ
• y-coordinate of that point = sin θ
• tan θ = y/x = sin θ/cos θ

Yeh definition kisi bhi angle ke liye kaam karta hai — chahe 45° ho ya 200° ho ya −60° ho — kyunki point hamesha circle pe kahin na kahin hoga, hypotenuse ki tarah triangle pe restricted nahi hai!

Kyun "radius=1" use karte hain? Kyunki Hyp=1 hone se formula simplify ho jaata hai: sin θ = Opp/Hyp = Opp/1 = seedha y-coordinate. Isi liye Unit Circle naam pada — "unit" matlab radius=1.

💡 Socho ek clock ki second-hand circle pe ghoom rahi hai. Uski height (kitni upar/neeche hai) = sin θ, aur uski sideways position (kitni left/right hai) = cos θ — bilkul wahi jo animation mein dikh raha hai!

⚠️ 90° se zyada angles pe Opposite/Adjacent/Hypotenuse naming confuse ho jaati hai — isliye Unit Circle ka x/y-coordinate approach hi sahi tareeka hai bade angles ke liye.

📌 θ=90° pe point seedha upar (0,1) pe hota hai → cos(90°)=0, sin(90°)=1. θ=180° pe point seedhe left (−1,0) pe hota hai → cos(180°)=−1, sin(180°)=0.

📐 Step 3 — Standard angle values yaad karne wali table + Quadrant sign rule

Yeh table EXAM mein sabse zyada use hoti hai — bilkul gehraai se yaad karo:

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞ (undefined)

Yaad karne ka trick — "0,1,2,3,4" pattern:
sin ke liye: √(0)/2, √(1)/2, √(2)/2, √(3)/2, √(4)/2 → simplify karo → 0, 1/2, 1/√2, √3/2, 1 (yehi 0°,30°,45°,60°,90° ke sin values hain!) cos ke liye seedha ulta order hai!

ASTC Rule — kis quadrant mein kya positive hai:
• Quadrant 1 (0°-90°): All positive (sin, cos, tan sab +ve)
• Quadrant 2 (90°-180°): Sin positive (sirf sin +ve, cos & tan −ve)
• Quadrant 3 (180°-270°): Tan positive (sirf tan +ve, sin & cos −ve)
• Quadrant 4 (270°-360°): Cos positive (sirf cos +ve, sin & tan −ve)

💡 ASTC yaad karne ka tareeka: "Add Sugar To Coffee" — A,S,T,C — har quadrant (anticlockwise, Q1 se shuru) ka apna "positive" letter hai!

⚠️ tan(90°) undefined hai (infinity), kyunki cos(90°)=0 hai aur tan=sin/cos — zero se divide nahi kar sakte. Calculator bhi error dega!

📌 sin(150°) ka sign kya hoga? 150° Quadrant 2 mein hai (90°-180°) → ASTC se Quadrant 2 mein sirf Sin positive hai → sin(150°) = sin(180°−150°) = sin(30°) = +1/2

🌍 Step 4 — Real life mein Trigonometry kahan kahan use hoti hai?

Roz ki life mein trig ka use:
• Building/Tower height: Zameen se angle measure karke, building ki height calculate karna (bina upar chadhe!)
• Navigation (GPS/Ships): Distance aur direction calculate karne mein trig hi base hai
• Satellite positioning: Satellites ki exact location track karna trig se hota hai
• Music & Sound waves: Sound waves ko sin/cos curves se represent karte hain (Physics mein SHM bhi yehi use karta hai)
• Construction & Architecture: Roof ka angle, bridge ka design — sab trigonometry use karta hai
• Video games & Animation: Characters ko rotate/move karna unit circle math se hi hota hai!

Common mistakes jo students karte hain:
1. Calculator ko "Degree mode" ke bajaye "Radian mode" mein chhod dena (ya ulta) — answer bilkul galat aa jaata hai
2. Opposite aur Adjacent confuse kar dena — hamesha pehle angle θ identify karo, phir sides naam do
3. ASTC rule bhool jaana aur sirf positive answer likh dena (jabki kuch angles negative values dete hain)
4. tan(90°) ko "0" ya "1" likh dena — yeh actually undefined hai!

💡 Exam shuru karte hi sabse pehle calculator check karo — "DEG" ya "RAD" mode mein hai? Yeh ek second ka kaam bahut saari galtiyan bacha sakta hai!

📌 Ek 20m lambi seedhi, deewar se 60° angle banati hai zameen se. Deewar ki height kya hogi? height = seedhi × sin(60°) = 20 × (√3/2) ≈ 17.32 meters

🎬 Angle rotate karo → sin/cos/tan change dekho

Angle θ (degrees)30

👀 Hari line = sin θ (height), Peeli line = cos θ (sideways), Purple line = hypotenuse (radius=1). Slider ko 90°, 180°, 270° pe le jaao aur dekho point kis quadrant mein hai! ("sin²θ+cos²θ=1" ko deeply verify karne ke liye "Pythagorean Identities" card dekho.)

🧮 Try It

Angle θ (degrees)

🏗️

Height & Distance Problems

Applied Trig · h = d × tan(θ)

▼

h = d × tan(θ)
d = h / tan(θ)

📖 Real life applications

Angle of elevation (θ): Jis angle se tum kisi building/tower ko oopar dekhte ho.

Formula logic: tan θ = height/distance → height = distance × tan θ

Uses: Building height, mountain elevation, flagpole height, tree height — bina paas gaye measure kar sakte ho!

📌 30m door se building ka angle 60° → h = 30×tan(60°) = 30×1.732 = 51.96m

🎬 Distance ya angle change karo → height calculate dekho

Distance d (m)30

Angle θ (°)45

🧮 Try It

Distance (m)

Angle θ (°)

🔗 Trig Identities

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Pythagorean Identities

Trigonometry · sin²θ + cos²θ = 1 · Always true

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sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Teen fundamental identities — proofs mein hamesha use honge

sec θ = 1/cos θ cosec θ = 1/sin θ cot θ = 1/tan θ

📖 Teen identities, ek source

Teeno identities unit circle se aati hain. Point P(cos θ, sin θ) circle pe hota hai → x²+y² = 1 → cos²θ + sin²θ = 1.

Identity 2: Pehli ko cos²θ se divide karo → 1 + tan²θ = sec²θ
Identity 3: Pehli ko sin²θ se divide karo → cot²θ + 1 = cosec²θ

Use: sin²θ ko (1−cos²θ) se replace karo proofs mein — bahut kaam aata hai!

💡 Yaad karo: "1 + tan² = sec²" aur "1 + cot² = cosec²" — dono mein "1 + something = big reciprocal squared"

📌 sin θ = 3/5 → cos²θ = 1−9/25 = 16/25 → cos θ = 4/5. Then tan θ = 3/4, sec θ = 5/4

🎬 θ rotate karo → sin²θ + cos²θ = 1 verify dekho

Angle θ (°)30

🧮 Try It — Verify Identities

Angle θ (degrees)

➕

Compound Angle Formulas

sin(A±B), cos(A±B), tan(A±B) · Most used in exams

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sin(A+B) = sinA cosB + cosA sinB
sin(A−B) = sinA cosB − cosA sinB
cos(A+B) = cosA cosB − sinA sinB
cos(A−B) = cosA cosB + sinA sinB
tan(A+B) = (tanA+tanB)/(1−tanA tanB)

Two angles ke sum/difference ke trig values

📖 Memory trick aur Double Angle

sin(A+B): "sin cos + cos sin" — cross product pattern
cos(A+B): "cos cos − sin sin" — same sign but minus!

Double Angle (A=B):
• sin 2A = 2 sinA cosA
• cos 2A = cos²A − sin²A = 1−2sin²A = 2cos²A−1
• tan 2A = 2tanA/(1−tan²A)

Use: sin 75° = sin(45°+30°) = sin45 cos30 + cos45 sin30 = (√6+√2)/4

💡 sin mein (+B) → signs same as outer. cos mein (+B) → sign FLIPS (minus). Easy trap in exams!

⚠️ cos(A+B) ≠ cosA + cosB. Yeh bahut common mistake hai!

📌 cos(A−B) − cos(A+B) = 2 sinA sinB (prove karo — subtract the two cos formulas!)

🎬 A aur B badlao → sin(A+B) verify dekho

Angle A (°)45

Angle B (°)30

🧮 Try It

Angle A (°)

Angle B (°)

sin⁻¹

Inverse Trigonometry — sin⁻¹, cos⁻¹, tan⁻¹

arcsin · arccos · arctan · Domain & Range · Principal values

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sin⁻¹(x) → [−π/2, π/2]
cos⁻¹(x) → [0, π]
tan⁻¹(x) → (−π/2, π/2)
sin⁻¹(x) + cos⁻¹(x) = π/2

Inverse functions — angle dhundho jab ratio pata ho

Domain sin⁻¹: [−1, 1] Domain cos⁻¹: [−1, 1] Domain tan⁻¹: (−∞, ∞)

📖 Inverse kya hota hai?

Sin θ = x → θ = sin⁻¹(x). Matlab: "Konse angle ka sin x hai?"

Principal Value Branch: Multiple angles hote hain jinka sin same hota hai (e.g., 30° aur 150° dono ka sin = 0.5). Isliye ek fixed range define karte hain:
• sin⁻¹: [−90°, 90°] — sirf pehla ya chautha quadrant
• cos⁻¹: [0°, 180°] — pehla ya doosra quadrant
• tan⁻¹: (−90°, 90°) — open interval

Key identities:
• sin⁻¹(x) + cos⁻¹(x) = π/2
• tan⁻¹(x) + tan⁻¹(1/x) = π/2 (x>0)
• 2tan⁻¹(x) = sin⁻¹(2x/(1+x²))

💡 sin⁻¹(sin θ) = θ SIRF tab jab θ ∈ [−π/2, π/2]. Warna principal value lena padega!

⚠️ sin⁻¹(x) ≠ 1/sin(x) — yeh reciprocal nahi hai! 1/sin(x) = cosec(x)

📌 sin⁻¹(√3/2) = 60° = π/3 | cos⁻¹(0) = 90° | tan⁻¹(1) = 45°

🎬 x badlao → sin⁻¹, cos⁻¹, tan⁻¹ values dekho

x value0.50

🧮 Try It

x (for sin⁻¹/cos⁻¹, use −1 to 1)

🔢 Number System

HCF

HCF & LCM

Number Theory · HCF×LCM = a×b

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HCF × LCM = a × b
LCM = (a×b) / HCF

HCF = sabse bada common factor · LCM = sabse chhota common multiple

HCF = Highest Common Factor LCM = Lowest Common Multiple a, b = two numbers

🍕 Step 1 — HCF kya hota hai? Pizza se samjho!

Socho tumhare paas 12 chocolate aur 18 biscuits hain. Tumhe inhe equal groups mein baantna hai — bina kuch bacha. Kitne se baat sakte ho?

12 ko divide karte hain: 1, 2, 3, 4, 6, 12
18 ko divide karte hain: 1, 2, 3, 6, 9, 18

Common factors: 1, 2, 3, 6 ← yahi sabse bada common factor hai!

✅ HCF(12, 18) = 6 → matlab 6 groups mein baant sakte ho (har group mein 2 chocolate + 3 biscuit)

HCF = "Sharing ki problem" ka answer! Jab bhi "equal groups mein baantna" aaye → HCF use karo.

🏃 Step 2 — LCM kya hota hai? Race se samjho!

Ram har 4 din mein gym jaata hai. Shyam har 6 din mein jaata hai. Dono aaj saath gaye — kitne din baad phir saath jaayenge?

Ram ke gym days: 4, 8, 12, 16, 20, 24...
Shyam ke gym days: 6, 12, 18, 24...

Sabse pehle common number = 12

✅ LCM(4, 6) = 12 → 12 din baad dono saath milenge!

LCM = "Same time pe milna" ka answer! Jab bhi "pehli baar saath honge" aaye → LCM use karo.

⚡ Step 3 — Teen methods, teen speed

Method 1 — Listing (Bachon ke liye)
Factors ya multiples list karo aur common dhundho.
HCF(8,12): Factors of 8 = {1,2,4,8}, Factors of 12 = {1,2,3,4,6,12} → HCF = 4

Method 2 — Prime Factorisation (Exam ka favourite)
Dono numbers ko prime factors mein todo.
HCF = common primes ka product (lowest power se)
LCM = sabhi primes ka product (highest power se)
12 = 2²×3 · 18 = 2×3² → HCF = 2¹×3¹ = 6 · LCM = 2²×3² = 36

Method 3 — Euclidean Algorithm (Sabse FAST 🚀)
Bada ÷ Chhota = quotient + remainder. Remainder ko divisor banao. Repeat!
HCF(48, 18): 48 = 2×18 + 12 → 18 = 1×12 + 6 → 12 = 2×6 + 0 → HCF = 6 ✅

🔮 Step 4 — Magic formula: HCF × LCM = a × b

Yeh ek powerful shortcut hai! Agar HCF pata ho toh LCM seedha nikal sakte ho bina listing ke:

HCF(12, 18) = 6
LCM = (12 × 18) / 6 = 216 / 6 = 36

Proof kyun kaam karta hai?
a = HCF × p, b = HCF × q (jahan p aur q coprime hain)
a × b = HCF² × p × q
LCM = HCF × p × q (kyunki p,q coprime hain)
→ HCF × LCM = HCF × (HCF × p × q) = HCF² × p × q = a × b ✅

⚠️ Yeh formula sirf 2 numbers ke liye kaam karta hai! Teen ya zyada numbers ke liye alag approach chahiye.

🌍 Step 5 — Real life mein kab use karte hain?

🎵 HCF use cases:
• Fraction simplify karna: 24/36 → HCF(24,36)=12 → 2/3
• Tiles ki size: 12m × 18m room mein square tiles without cutting → side = HCF(12,18) = 6m
• Equal gifts: N items ko maximum groups mein baantna

⏰ LCM use cases:
• Dono buses kab ek saath aayengi (timing problems)
• Fraction add karna: 1/4 + 1/6 → LCM(4,6) = 12 → 3/12 + 2/12 = 5/12
• Kab dono sath kaam khatam karenge
• Traffic lights ka cycle alignment

🎓 Step 6 — Advanced: Teen numbers ka HCF/LCM

Teen numbers a, b, c ke liye:

HCF(a,b,c) = HCF(HCF(a,b), c)
LCM(a,b,c) = LCM(LCM(a,b), c)

Example: HCF(12, 18, 24)
→ HCF(12,18) = 6
→ HCF(6, 24) = 6 ✅

LCM(4, 6, 10)
→ LCM(4,6) = 12
→ LCM(12,10) = 60 ✅

Co-prime numbers: Jab HCF(a,b) = 1 hota hai — jaise 8 aur 9. Inke beech koi common factor nahi. Aise numbers ka LCM = a × b seedha!

💡 Exam trick: "Koi bada number X, dono numbers ko completely divide kare" → HCF dhundho. "Koi chhota number Y, dono numbers se completely divide ho jaaye" → LCM dhundho!

⚠️ HCF × LCM = a × b formula sirf 2 numbers ke liye hai. Teen numbers ke liye galat answer aayega — tabhi step-by-step approach use karo!

📌 Classic problems:
1️⃣ Tiles: 96m × 72m room → HCF(96,72) = 24m side tiles → 12 tiles
2️⃣ Buses: A har 15 min, B har 20 min → LCM(15,20) = 60 min baad saath aayengi
3️⃣ HCF = 12, LCM = 144, ek number = 36 → doosra = (12×144)/36 = 48

🎬 Venn diagram mein common factors — drag karo dekho!

Number A12

Number B18

🧮 Try It — HCF & LCM Calculator

Number A

Number B

🚀 Speed, Work & Time

🚆

Speed, Distance & Time

Arithmetic · D = S×T · Avg Speed = 2ab/(a+b)

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D = S × T
S = D ÷ T
T = D ÷ S
Avg Speed = 2ab / (a+b)

Distance = Speed × Time · Harmonic mean for same-distance trips

D = Distance (km or m) S = Speed (km/h or m/s) T = Time (hours or seconds) a, b = speeds of two trips

🚗 Step 1 — D = S × T bilkul root se samjho

Socho tum cycle pe ho aur 10 km/h ki speed se chal rahe ho.

1 ghante mein kitna chaloge? → 10 km
2 ghante mein? → 20 km
3 ghante mein? → 30 km

Pattern dikha? Distance = Speed × Time — bada hi seedha!

Ab ek magic triangle yaad karo:

[D]
─────────
[S] × [T]

Jis cheez ko dhundna ho usse haath se dhako — baaki ka formula milega!
• D dhundna → S × T
• S dhundna → D ÷ T
• T dhundna → D ÷ S

🔄 Step 2 — Units convert karna sikho

Exams mein speed km/h mein hoti hai par time seconds mein — mix mat karo!

km/h → m/s: × (5/18) [ya ÷ 3.6]
m/s → km/h: × (18/5) [ya × 3.6]

🧠 Trick yaad karo: "18 se divide karo" ya "18 se multiply karo"

72 km/h = 72 × 5/18 = 20 m/s
15 m/s = 15 × 18/5 = 54 km/h

Relative speed:
• Same direction mein (ek hi taraf): Speeds subtract karo
• Opposite direction mein (aamne saamne): Speeds add karo
🚂 Train A: 60 km/h, Train B: 40 km/h same direction → relative = 20 km/h
🚂 Opposite direction → relative = 100 km/h

🤯 Step 3 — Average Speed ka sabse bada trap!

Sawal: Train Delhi se Agra 60 km/h se jaati hai, wapas 40 km/h se aati hai. Average speed kitni?

90% log galat jawab dete hain: (60+40)/2 = 50 km/h ❌

Sahi kyun nahi? Kyunki tum zyada time 40 km/h pe spend karte ho (slower = more time)!

Maano distance = 120 km (dono taraf same)
Jaane mein time = 120/60 = 2 hours
Aane mein time = 120/40 = 3 hours
Total distance = 240 km, Total time = 5 hours
Actual avg speed = 240/5 = 48 km/h

Formula se: Avg = 2ab/(a+b) = 2×60×40/(60+40) = 4800/100 = 48 km/h ✅

Yahi "harmonic mean" hai — jab same distance do speeds se travel karo!

🚂 Step 4 — Train problems ka special twist

Trains ka length important hai — pole cross karna vs bridge cross karna alag hai!

Train pole cross karne mein:
Distance = Train ki length
Time = Train length / Speed

Train bridge/tunnel cross karne mein:
Distance = Train length + Bridge length
Time = (Train + Bridge length) / Speed

Do trains ek dusre ko cross karna (opposite):
Distance = Train1 length + Train2 length
Time = (L1 + L2) / (S1 + S2)

🧮 Example: 100m train, 60 km/h (=50/3 m/s). 400m bridge cross karne ka time:
= (100+400) / (50/3) = 500 × 3/50 = 30 seconds

🌍 Step 5 — Real life aur Advanced concepts

🚤 Boats & Streams:
Still water speed = u, Stream speed = v
Downstream (saath dhara): Speed = u + v
Upstream (dhara ke virudh): Speed = u − v
u = (Downstream + Upstream) / 2
v = (Downstream − Upstream) / 2

🏃 Circular track:
Same direction: Miling time = Track / Relative speed
Opposite direction: Miling time = Track / Sum of speeds

⏱️ Meeting problems:
A aur B saath chalte hain opposite direction mein: Milenge kab?
Time = Total distance / (A's speed + B's speed)

🎓 Important shortcuts:
• Speed ratio = Time ratio ka inverse (same distance pe)
• 3 speeds same time ke liye: Avg = (a+b+c)/3 (Arithmetic mean — valid!)
• Sirf same distance ke liye harmonic mean use hota hai!

💡 SDT Triangle trick: D oopar → dhako D → S×T. Dhako S → D/T. Dhako T → D/S. Haath se cover karo jise dhundna hai!

⚠️ Average speed = 2ab/(a+b) SIRF tab jab same distance dono trips mein. Agar same time ho toh simple average (a+b)/2 sahi hoga!

📌 Practice problems:
1️⃣ Car 3 hours mein 180 km → Speed = 180/3 = 60 km/h
2️⃣ 400m train, 200m bridge, 72 km/h → time = (400+200)/20 = 30 sec
3️⃣ Boat: 16 km/h downstream, 8 km/h upstream → Still = (16+8)/2 = 12 km/h, Stream = (16−8)/2 = 4 km/h
4️⃣ Avg speed: 30 km/h jaana, 20 km/h aana → 2×30×20/50 = 24 km/h

🎬 Car move karti hai — Speed change karo, distance aur time live dekho

Speed (km/h)60

🧮 Try It — Average Speed Calculator

Speed going a (km/h)

Speed returning b (km/h)

👷

Work & Time — 1/A + 1/B = 1/T

Arithmetic · Combined work · Pipes & Cisterns

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1/A + 1/B = 1/T
T = AB / (A+B)
n workers → T/n time

A aur B milke kaam karein → Combined rate = sum of individual rates

A = A akela kitne din mein khatam kare B = B akela kitne din mein khatam kare T = Dono milke kitne din mein

🏗️ Step 1 — "Work Rate" concept: ek dum seedha socho

Socho pizza delivery wale se:
Raju akele 10 deliveries 1 ghante mein karta hai → Per minute = 10/60 deliveries
Vijay akele 15 deliveries 1 ghante mein karta hai → Per minute = 15/60 deliveries

Dono saath → Per minute = 10/60 + 15/60 = 25/60 deliveries

Math language mein:
A ek poori job 10 din mein karta hai → 1 din mein karta hai = 1/10 kaam
B ek poori job 15 din mein karta hai → 1 din mein karta hai = 1/15 kaam

Dono saath 1 din mein karte hain = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6

Matlab: Dono milke 6 din mein poora kaam khatam karenge!

Formula: T = AB/(A+B) = 10×15/(10+15) = 150/25 = 6 ✅

🔑 Step 2 — Shortcut: LCM method (exam mein fastest!)

Total kaam ko ek fixed number maano — A aur B ka LCM!

A = 10 days, B = 15 days → LCM(10,15) = 30 → Maano total kaam = 30 units

A 1 din mein karta hai = 30/10 = 3 units/day
B 1 din mein karta hai = 30/15 = 2 units/day
Dono saath = 3 + 2 = 5 units/day

Time = Total / Combined rate = 30/5 = 6 days ✅

Kyun LCM method better hai? Fractions avoid hote hain → calculation tez aur galti kam!

💧 Step 3 — Pipes & Cisterns (same logic, ulta application)

Tank bhar na aur work khatam karna — same formula, alag context!

Inlet pipe (filling): Tank bhar ta hai → Rate positive (+)
Pipe A 6 hrs mein bhare → rate = +1/6 per hour

Outlet pipe (emptying): Tank khaali karta hai → Rate negative (−)
Pipe B 9 hrs mein khaali kare → rate = −1/9 per hour

Net rate = 1/6 − 1/9 = 3/18 − 2/18 = 1/18 per hour
Tank bharne mein = 18 hours

Classic trap: Both pipes open ho aur tank already half bhara ho → Remaining = half tank, rate same → time = 18/2 = 9 hours!

🔀 Step 4 — Tricky variations jo exam mein aate hain

Case 1 — A kuch dino ke baad chhod de:
A aur B saath shuru karte hain, A x din baad chhod deta hai, B poora karta hai.
→ A ne kiya = x × (1/A), B ne kiya = baaki. B ko kitne din lagenge = baaki ÷ (1/B)

Case 2 — Alternate din kaam:
Day 1: A karta hai, Day 2: B karta hai, Day 3: A... repeat.
2 din ka kaam = 1/A + 1/B. Kitne pairs chahiye? Total/pair rate = pairs. Remaining days check karo.

Case 3 — Man-Days concept:
5 log, 8 din mein kaam → Total man-days = 40
8 log honge toh = 40/8 = 5 din
Workers × Time = Constant (agar kaam same ho)

Case 4 — Efficiency concept:
"A, B se double efficient hai" → A ka 1 din = B ke 2 din ka kaam
Agar B 30 din mein kare → A = 15 din mein. Saath = 10 din (by formula).

🌍 Step 5 — Real life aur Advanced

Construction: 20 workers, 30 din ka project. 10 din baad 5 workers resign kare → Kitna time aur lagega?
Kaam hua = 20×10 = 200 man-days
Baaki kaam = 20×30 − 200 = 400 man-days
Baaki workers = 15
Time baaki = 400/15 = 26.67 days ≈ 27 days

🎓 Advanced: Rate of work + wages:
Agar payment kaam ke ratio mein ho:
A aur B ka ratio of work = (1/A) : (1/B) = B : A (reverse!)
To wages bhi B:A ratio mein hongi — ULTA lagta hai par sahi hai!

Example: A 20 din mein, B 30 din mein → Work ratio = 1/20 : 1/30 = 3:2
Total ₹5000 mein A ko = 3/5 × 5000 = ₹3000, B ko = ₹2000

💡 LCM Method sabse safe: Always total work = LCM(A,B) maano. Fractions avoid honge, calculation fast hogi, exam mein galti nahi hogi!

⚠️ "A, B se 2× efficient" → A ka time = B/2 (aadha). Lekin "A, B se 2× zyada time leta" → A slow hai, B = A/2 time letaa hai. Efficient aur time ULTE proportional hain!

📌 Practice problems:
1️⃣ A=12 days, B=18 days → LCM=36, A=3u/d, B=2u/d, Together=5u/d → Time = 36/5 = 7.2 days
2️⃣ Inlet: 4hrs, Outlet: 6hrs both open → Net = 1/4−1/6 = 1/12 → 12 hrs to fill
3️⃣ 12 men, 15 days project. After 5 days, 4 men leave → Work done=60, Remaining=120, Workers=8 → 15 more days

🎬 A aur B milke kaam karte hain — progress bar live dekho!

A completes in (days)10

B completes in (days)15

🧮 Try It — Work & Time Calculator

A completes in (days)

B completes in (days)

💹 Percentage & Commerce

Percentage — % Change Formula

Arithmetic · Increase/Decrease · Base Value

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% = (Part / Whole) × 100
% Change = (Change / Original) × 100
New = Original × (1 ± %/100)

Percentage · Percentage Change · New value after change

Part = jo mila ya change hua Whole / Original = base (100% hamesha yahi hota hai) %/100 = decimal form

🍕 Step 1 — "Per Cent" ka matlab: 100 mein se kitna?

"Per cent" = "per hundred" (Latin: per centum). Matlab har 100 mein se kitna?

Pizza ke 100 tukde karo. Agar tumne 30 kha liye → tumne 30% khaya!

Class mein 40 bacche hain, 10 absent hain:
Absent % = (10/40) × 100 = 25%

Key insight: Percentage ek comparison hai jo sab cheez ko 100 ke scale pe laata hai — isliye alag alag cheezein compare ho sakti hain!

90/100 marks vs 45/50 marks — kaun better? %: 90% vs 90% — Equal! Bina % ke compare nahi hota.

🔢 Step 2 — Teen types ke sawal (aur dono taraf se solve karna)

Type 1 — "% OF" nikalna (Part dhundho)
30% of ₹500 = (30/100) × 500 = ₹150
Shortcut: 10% = 500÷10=50. 30% = 3×50 = 150 ✓

Type 2 — "KITNA %" hai (% dhundho)
45 out of 180 kitna %? = (45/180) × 100 = 25%
Shortcut: 45/180 = 1/4 = 25% ✓

Type 3 — "ORIGINAL" nikalna (reverse %)
20% increase ke baad value = ₹720 → Original kya tha?
Original × 1.2 = 720 → Original = 720/1.2 = ₹600
ya: Original = 720 × 100/120 = ₹600

⚡ Step 3 — % Change: increase aur decrease

Formula: % Change = (New − Old) / Old × 100

Positive answer → Increase 📈
Negative answer → Decrease 📉

New value shortcut:
• 20% increase → × 1.20 (100% + 20% = 120% = 1.2)
• 15% decrease → × 0.85 (100% − 15% = 85% = 0.85)
• 8.5% increase → × 1.085

Consecutive changes:
10% increase phir 10% decrease:
Original × 1.1 × 0.9 = Original × 0.99 → net 1% loss!

Formula: Net% = a + b + ab/100
= 10 + (−10) + (10×(−10))/100 = 0 − 1 = −1% ✅

🧠 Step 4 — Smart shortcuts jo exam mein time bachate hain

Swap trick: x% of y = y% of x
18% of 50 = 50% of 18 = 9 (50% = half → 18/2 = 9) 🚀
4% of 75 = 75% of 4 = 3 (75% = 3/4 → 4×3/4 = 3) 🚀

Fraction ↔ % table (must memorise!):
1/2=50% · 1/3=33.33% · 1/4=25% · 1/5=20%
1/6=16.67% · 1/8=12.5% · 1/10=10% · 3/4=75%

Successive % change — multiplier method:
20% up → 10% up → 15% down:
= 1.2 × 1.1 × 0.85 = 1.122 → 12.2% net increase

Population / depreciation problems:
Population after n years @ r% growth = P × (1 + r/100)ⁿ
Machine value after n years @ r% depreciation = P × (1 − r/100)ⁿ

📊 Step 5 — Real life mein % kahan kahan aata hai

• Exam marks: 450/600 = 75% (Good!)
• GST: ₹1000 item pe 18% GST → Total = ₹1180
• Bank interest: 8% per year pe ₹5000 invest → ₹400 interest/year
• Discount: MRP ₹2000 pe 25% off → Pay ₹1500
• Election: 60% voter turnout → kul votes mein se 60% ne vote kiya
• Nutrition: "Daily value 15%" label → ek serving mein daily zaroorat ka 15%

🎓 Advanced — % of % (percentage point vs percentage change):
Agar interest rate 5% se 6% ho jaaye:
Change in % points = 1 percentage point
But % change in rate = (1/5)×100 = 20% increase — ye alag hai!
Ye confusion politics/economics mein bahut hoti hai.

💡 Swap trick: x% of y = y% of x. Isliye 18% of 50 = 50% of 18 = 9. Ye exam mein bahut time bachata hai!

⚠️ % change hamesha Original (Old) value se calculate hoti hai — new value se nahi! Yahi sabse common galti hai.

📌 Classic problems:
1️⃣ ₹500 → ₹650: % increase = (150/500)×100 = 30%
2️⃣ 20% up phir 20% down: Net = 1.2×0.8 = 0.96 → 4% loss
3️⃣ After 25% increase, value = ₹750 → Original = 750/1.25 = ₹600
4️⃣ 3% of 500 = 500% of 3 = 5×3 = 15 (swap trick!)

🎬 Original bar aur % change live dekho — increase green, decrease red

Original value500

% Change30

🧮 Try It — % Change Calculator

Original value

New value

💰

Profit & Loss — CP, SP, % Profit

Arithmetic · % Profit = (P/CP)×100 · Discount = MP−SP

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Profit = SP − CP | Loss = CP − SP
% Profit = (Profit / CP) × 100
SP = CP × (100 + P%) / 100
CP = SP × 100 / (100 + P%)

Selling Price · Cost Price · Marked Price · Profit/Loss Percent

CP = Cost Price (khareedne ki kimat) SP = Selling Price (bechne ki kimat) MP = Marked Price (tag pe likhi kimat) P%/L% = hamesha CP ke basis pe!

🛒 Step 1 — CP, SP, MP ka chakkar: dukaan se samjho

Socho ek dukandaar hai — Ramesh Bhaiya:

1️⃣ Wo ek shirt factory se lata hai ₹400 mein → yahi CP (Cost Price) hai
2️⃣ Apni dukaan mein tag lagata hai ₹700 ka → yahi MP (Marked Price) hai
3️⃣ Tum bargain karte ho, wo 20% discount deta hai → SP = 700 × 0.8 = ₹560

Ab calculate karo:
Profit = SP − CP = 560 − 400 = ₹160
% Profit = (160/400) × 100 = 40% on CP

⚠️ Bahut important: % Profit/Loss hamesha CP se nikalo — SP se nahi!

🔄 Step 2 — SP se CP nikalna aur CP se SP banana (reverse formulas)

CP se SP banana:
SP = CP × (100 + P%) / 100     [profit mein]
SP = CP × (100 − L%) / 100     [loss mein]

SP se CP nikalna:
CP = SP × 100 / (100 + P%)     [profit case]
CP = SP × 100 / (100 − L%)     [loss case]

Example: 25% profit pe SP = ₹750. CP = ?
CP = 750 × 100/125 = 750 × 4/5 = ₹600

Multiplier method (fastest):
25% profit → CP × 1.25 = SP
Reverse: SP / 1.25 = CP
750 / 1.25 = 600 ✓

🏷️ Step 3 — Discount: MP se SP kaise milta hai

Discount hamesha MP (Marked Price) pe hota hai!

SP = MP × (100 − d%) / 100
ya SP = MP × (1 − d/100)

Example: MP = ₹1000, 30% discount:
SP = 1000 × 0.70 = ₹700

Successive discounts (bahut important trap!):
20% then 10% ≠ 30% discount!
Net multiplier = 0.80 × 0.90 = 0.72 → 28% effective discount

Formula: Effective discount = 1 − (1−d₁/100)(1−d₂/100) × 100
= 1 − 0.8×0.9 = 1 − 0.72 = 0.28 = 28%

Remember: Har baar discount kam ho jaata hai kyunki base already reduce ho chuki hai!

🎭 Step 4 — Tricky exam questions

Classic trick — Same SP, ek profit ek loss:
Ek item x% profit pe bechi, doosra x% loss pe bechi — same SP pe.
→ Net hamesha LOSS = x²/100 % hoga!
Example: 20% profit + 20% loss same SP pe:
Net loss = 20²/100 = 4% loss

Discount + Profit (dono saath):
MP pe 25% discount dene ke baad bhi 20% profit:
SP = CP × 1.20 aur SP = MP × 0.75
→ MP/CP = 1.20/0.75 = 1.6 → MP = 60% above CP

Dishonest shopkeeper:
Actual weight 800g, claim karta hai 1000g (false weight):
Effective profit = (1000−800)/800 × 100 = 25% profit (bina kuch badle!)

CP nikalo when:
"Ek agar 10% profit pe beche aur doosra 10% loss pe, difference ₹20 hai" → 2 equations banao!

🌍 Step 5 — Real life P&L aur advanced

Manufacturing cost structure:
CP = Raw material + Labour + Overhead
Then MP = CP + desired profit + estimated discount buffer

Break-even point:
Jab SP = CP → 0% profit, 0% loss. Is se kam SP pe becho → loss zone!

Trade discount vs Cash discount:
Trade discount: Bulk buying pe MP se direct cut
Cash discount: Jaldi payment karne pe extra cut

Stock market analogy:
Share ₹100 pe kharida (CP), ₹140 pe becha (SP):
% return = (40/100)×100 = 40% profit
Lekin agar 2 saal hold kiya → Annual return = (√1.4 − 1)×100 ≈ 18.3% per year (CAGR)

💡 Same SP trick: Jab bhi same SP pe ek profit ek loss ho — hamesha NET LOSS hoga = x²/100%. X = 10% pe: 100/100 = 1% loss. X = 20% pe: 400/100 = 4% loss. Yaad raho!

⚠️ Discount MP pe hota hai, profit/loss CP pe hota hai. Ye dono alag base hain — kabhi mix mat karo!

📌 Practice problems:
1️⃣ CP=₹800, SP=₹1000 → Profit=₹200 → %P = 200/800×100 = 25%
2️⃣ 20% + 10% successive discount → Net = 1−(0.8×0.9) = 28% off
3️⃣ 25% profit, SP=₹750 → CP = 750×100/125 = ₹600
4️⃣ Same SP, 15% profit + 15% loss → Net = 15²/100 = 2.25% loss

🎬 CP aur SP set karo — profit/loss aur % live dekho

Cost Price (₹)800

Selling Price (₹)1000

🧮 Try It — Full P&L Calculator

Cost Price (₹)

Selling Price (₹)

Discount % (on MP)

⚖️

Ratio & Proportion — a:b :: c:d

Arithmetic · Direct/Inverse Variation · Cross Multiply

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a : b = c : d → a × d = b × c
Direct: y/x = k (constant)
Inverse: x × y = k (constant)
Compound: (a:b) × (c:d) = ac : bd

Proportion · Direct variation · Inverse variation · Cross multiplication

a:b = ratio (comparison of two quantities) a:b :: c:d = proportion (two equal ratios) k = constant of variation a, d = extremes · b, c = means

🍊 Step 1 — Ratio kya hai? Juice se samjho!

Tumne juice banaya — 3 glass orange + 4 glass water.

Ratio of juice to water = 3:4
Matlab: har 3 glass juice ke liye 4 glass paani.

Agar 6 glass juice ho toh? → Paani = 8 glass (double karo dono)
Agar 9 glass juice ho toh? → Paani = 12 glass (triple karo dono)

Ratio same rehta hai — sirf scale badlata hai!

3:4 = 6:8 = 9:12 = 15:20 → sab equivalent ratios!
Simplest form: Dono ko HCF se divide karo → 3:4 (HCF=1, already simplest)

Key points:
• Ratio mein unit same hona chahiye (3 kg : 4 kg, not 3 kg : 4 m)
• a:b ≠ b:a (order matters!)
• Ratio ek pure number hai — no units

✂️ Step 2 — Proportion: do ratios ko equal set karna

a : b :: c : d padho as "a is to b as c is to d"

Yahan:
Extremes = a aur d (baahri wale)
Means = b aur c (andar wale)

Golden rule: Product of extremes = Product of means
a × d = b × c → Cross multiplication!

Example: 3:4 :: x:20 → 3×20 = 4×x → x = 60/4 = 15

Types of proportion:
• Continued proportion: a:b :: b:c → b² = ac (b is "mean proportional")
• Fourth proportional: a:b :: c:x → x = bc/a
• Third proportional: a:b :: b:x → x = b²/a

↕️ Step 3 — Direct vs Inverse variation: kab kya?

🔼 Direct Variation (ek badhega, doosra bhi badhega):
y ∝ x → y = kx → y/x = constant

Examples:
• Zyada kaam → zyada time (same speed pe)
• Zyada items → zyada cost (same rate pe)
• Petrol zyada → zyada distance

Solve karo: 5 items = ₹200 → 8 items = ?
200/5 = x/8 → x = 8×40 = ₹320

🔽 Inverse Variation (ek badhega, doosra ghatega):
y ∝ 1/x → xy = constant

Examples:
• Zyada workers → kam time (same kaam)
• Zyada speed → kam time (same distance)
• Zyada pipes → jaldi tank bhare

Solve karo: 5 workers, 6 din → 10 workers = ?
5×6 = 10×d → d = 30/10 = 3 din

🔗 Step 4 — Compound ratio aur teen numbers mein baantna

Compound Ratio:
(a:b) compounded with (c:d) = ac : bd
Example: (3:4) × (5:6) = 15:24 = 5:8

Duplicate ratio: a²:b² (ratio ka square)
Sub-duplicate ratio: √a:√b (ratio ka square root)
Triplicate ratio: a³:b³

Amount baantna ratio mein:
₹1200 ko 3:4:5 mein baanto:
Total parts = 3+4+5 = 12
Each part = 1200/12 = ₹100
→ ₹300 : ₹400 : ₹500

Tricky: Ratio change karna:
A:B = 3:4. A mein 6 add karo → A:B = 5:4. B = ?
A = 3k, B = 4k. New A = 3k+6 = 5k' → solve → k=3, B=12

🌍 Step 5 — Real life aur advanced applications

Map scale: 1:50000 → 1 cm on map = 50000 cm = 500 m real distance

Recipe scaling: 4 log ke liye 200g flour, 6 log ke liye = 200×(6/4) = 300g (direct)

Partnership: A:B = 3:5 invest karte hain → profit bhi 3:5 mein baatenge

Ages problem:
Abhi A:B = 3:5, 8 saal baad ratio = 5:7
3k+8 / 5k+8 = 5/7 → 21k+56 = 25k+40 → 4k=16 → k=4
Abhi A=12, B=20 years

Mixture problems (alligation):
₹40/kg aur ₹60/kg mix karo taaki ₹48/kg mile:
Cheaper : Costlier = (60−48) : (48−40) = 12:8 = 3:2

K (constant) find karo:
Agar y ∝ x² aur y=12 jab x=2 → 12 = k×4 → k=3 → y=3x²
Jab x=5 → y = 3×25 = 75

💡 Division trick: Total ko ratio mein baantna? Sum of parts nikalo, ek part ki value = Total/Sum. Phir multiply karo. ₹1200 in 3:4:5 → sum=12, one part=100 → ₹300, ₹400, ₹500.

⚠️ Direct variation mein y/x = constant. Inverse mein xy = constant. Pehle decide karo kaunsa hai — phir calculate karo. Galat type choose kiya toh answer poora ulta aayega!

📌 Practice problems:
1️⃣ 3:4 :: 9:x → x = 4×9/3 = 12 (cross multiply)
2️⃣ 8 machines, 6 hours → 12 machines = ? → 8×6/12 = 4 hours (inverse)
3️⃣ ₹2100 in 2:3:2 → sum=7, part=300 → ₹600 : ₹900 : ₹600
4️⃣ A:B = 2:3, B:C = 4:5 → A:B:C = 8:12:15 (B ko common banao → LCM of 3,4=12)

🎬 Ratio bars aur proportion balance — sliders se explore karo

Part A3

Part B4

Total (₹)700

🧮 Try It — Proportion Solver (find x)

d = ? (a:b :: c:d)

🏷️

Discount — SP = MP × (100−d)/100

Arithmetic · Marked Price · Successive Discounts · Profit+Discount

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SP = MP × (100−d%) / 100
Discount Amount = MP − SP
Effective Discount (d₁,d₂) = d₁ + d₂ − (d₁×d₂)/100

SP = Selling Price | MP = Marked Price | d = Discount %

MP = Tag pe likha price SP = Actual payment d = Discount % CP = Cost Price (dukandaar ka)

🧒 Step 1 — Sab se pehle: Discount hota kya hai? (Ekdum simple)

Socho tumhare dost ke paas ek toy hai jisko wo ₹100 mein bechna chahta hai. Lekin tum kehte ho "yaar thoda sasta kar", toh wo kehta hai "theek hai, ₹20 chhoot de raha hoon — ₹80 mein le lo".

Bas yahi hai Discount! 🎉

✅ Marked Price (MP) = ₹100 (tag pe likha hota hai, ya "original price")
✅ Discount = ₹20 (jo chhoot mili, 20%)
✅ Selling Price (SP) = ₹80 (tumne actually kitna diya)

Simple formula: SP = MP − Discount Amount
Ya fir percent se seedha: SP = MP × (100 − 20) / 100 = MP × 80/100

📌 Real life: Flipkart pe shirt ka MP = ₹500, "20% OFF" likha hai → SP = 500 × 80/100 = ₹400. Tumne ₹100 bachaye!

📐 Step 2 — Formula samjho, proof ke saath

Discount % hamesha MP pe hota hai — yeh rule yaad rakhna hai.

Discount Amount = MP × d/100
SP = MP − Discount Amount
SP = MP − MP × d/100
SP = MP × (1 − d/100)
SP = MP × (100 − d) / 100 ← yahi final formula hai

Reverse formula: Agar SP aur discount% pata hai, MP nikalo:
MP = SP × 100 / (100 − d)

Discount % nikalna: Discount% = (MP − SP) / MP × 100

💡 Shortcut trick: "d% chhoot" matlab tumhara multiplier = (100−d)/100. 15% off → multiply by 0.85. 25% off → multiply by 0.75. Easy!

📌 Jeans MP=₹800, 35% off → SP = 800 × 65/100 = 800 × 0.65 = ₹520

⚠️ Step 3 — Successive Discounts ka TRAP (Ye EXAM mein bahut aata hai!)

Question: Ek item pe pehle 20% discount, phir aur 10% discount mila. Total discount kitna?

❌ GALAT jawab: 20 + 10 = 30% (yeh WRONG hai!)
✅ SAHI jawab: 28% — aur neeche dekho kyon!

Kyon? Dusra discount pehle wale ke baad ke price pe lagta hai, original MP pe nahi!

MP = ₹1000
After 20% off → SP₁ = 1000 × 80/100 = ₹800
After 10% off on ₹800 → SP₂ = 800 × 90/100 = ₹720
Total saved = 1000 − 720 = ₹280 = 28% off (NOT 30%!)

Direct Formula: Effective% = d₁ + d₂ − (d₁ × d₂)/100
= 20 + 10 − (20×10)/100 = 30 − 2 = 28% ✓

⚠️ Exam trap: 20% + 10% ≠ 30%. Yeh formula yaad karo: d₁ + d₂ − d₁d₂/100. Agar teen discounts hain toh pehle pehle do ka effective nikalo, phir teesre ke saath apply karo.

📌 Three discounts 10%, 20%, 25%: First two → 10+20−2 = 28%. Now 28% and 25% → 28+25−7 = 46% effective discount.

🔗 Step 4 — Advanced: Profit aur Discount dono saath (CP→MP→SP chain)

Real shopkeeper aise karta hai:
1️⃣ Cheez kharidta hai CP pe (Cost Price)
2️⃣ Tag lagata hai MP pe (Marked Price) — CP se zyada, jisme profit hidden hai
3️⃣ Discount deta hai → SP milta hai (actually jo milta hai)
4️⃣ Profit/Loss = SP vs CP compare karo

Example: CP = ₹500, 40% markup → MP = ₹700. Phir 20% discount → SP = 700×0.8 = ₹560.
Profit on CP = 560 − 500 = ₹60 = 12% profit even after giving 20% discount!

Key Formula: Net Effect
SP = CP × (1 + markup/100) × (1 − discount/100)
Profit% = [(SP − CP)/CP] × 100

💡 Dukandaar trick: "30% markup, 10% discount" sounds generous — but: SP = CP × 1.30 × 0.90 = CP × 1.17 → still 17% profit!

🎬 Live Animation — MP se SP tak ka safar dekho

Marked Price (₹) 1200

Discount % 20

🧮 Try It — Successive Discounts Calculator

Marked Price MP (₹)

First Discount d₁ (%)

Second Discount d₂ (%)

🤝

Partnership — Profit Share Formula

Arithmetic · Capital × Time · Share Ratio · Sleeping vs Working Partner

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A : B = (C_A × T_A) : (C_B × T_B)
A's Profit = Total Profit × [C_A×T_A / (C_A×T_A + C_B×T_B)]

Profit share is proportional to Capital × Time invested

C = Capital (₹) T = Time (months) Ratio = Profit sharing ratio

🧒 Step 1 — Partnership kya hai? Dum se simple example

Socho tum aur tumhara dost milke ek dukaan kholta hai.

🟣 Tum dete ho ₹3000
🟠 Dost deta hai ₹2000

Saal ke baad dukaan ne ₹5000 profit kamaya.

Ab yeh profit dono mein equally nahi bategaa — balki jisne zyada lagaya, usse zyada milega!

Ratio = 3000 : 2000 = 3 : 2
Tumhara share = 5000 × 3/5 = ₹3000
Dost ka share = 5000 × 2/5 = ₹2000

Simple tha na? Yahi hai Simple Partnership!

📌 Rule: Profit share = Capital share. Jitna lagao, utna pao!

⏱️ Step 2 — Time ka factor: Compound Partnership (Ye tricky hai!)

Alag alag time ke liye invest karo toh sirf capital se kaam nahi chalta — Capital × Time consider karo.

Kyun? Socho ek banda ₹1000 deta hai 12 mahine ke liye, aur doosra ₹2000 deta hai sirf 3 mahine ke liye. Doosra 2x paisa lagaya lekin sirf 1/4 time ke liye. Toh pehle wale ka contribution zyada effective hai!

A invests ₹5000 for 12 months → effective = 5000 × 12 = 60,000
B invests ₹6000 for 8 months → effective = 6000 × 8 = 48,000
Ratio = 60,000 : 48,000 = 5 : 4

Total profit ₹18,000 mein:
A = 18000 × 5/9 = ₹10,000
B = 18000 × 4/9 = ₹8,000

💡 Shortcut: Simplify ratio before calculating. 60000:48000 → divide both by 12000 → 5:4. Always simplify!

📌 Equal share trap: A=₹4000×6mo=24000, B=₹6000×4mo=24000 → Ratio = 1:1 → Equal profit even though B invested more money!

😴 Step 3 — Sleeping Partner vs Working Partner (Advanced)

Sleeping Partner = Sirf paisa lagata hai, kaam nahi karta
Working Partner = Paisa bhi lagata hai + kaam bhi karta hai

Working partner ko pehle ek salary/commission milti hai profit se, phir baaki profit capital ratio mein baatta hai.

Example:
A (sleeping) = ₹10,000 invest
B (working) = ₹5,000 invest + ₹2,000/year working salary
Total profit = ₹11,000

Step 1: B ko ₹2,000 salary pehle → Remaining = ₹9,000
Step 2: Capital ratio = 10,000:5,000 = 2:1
Step 3: A gets 9,000 × 2/3 = ₹6,000, B gets 9,000 × 1/3 = ₹3,000
B's total = 3,000 + 2,000 = ₹5,000, A gets ₹6,000

⚠️ Exam me: "Working partner gets ₹X per month" → Pehle wo salary poore year se nikalo, profit mein se hatao, PHIR ratio mein baanto.

📅 Step 4 — Bich mein join karna (Common exam type!)

Question: A starts business on Jan 1 with ₹8,000. B joins on Apr 1 with ₹12,000. Year end pe profit ₹14,400. Find each share.

A's time = 12 months, B's time = 9 months (Apr to Dec)
A's equivalent = 8,000 × 12 = 96,000
B's equivalent = 12,000 × 9 = 108,000
Ratio = 96,000 : 108,000 = 96 : 108 = 8 : 9

A's share = 14,400 × 8/17 = ₹6,776
B's share = 14,400 × 9/17 = ₹7,624

💡 Always convert months from join date to end of year. Jan=12mo, Feb=11mo, Apr=9mo, Jul=6mo, Oct=3mo

🎬 Capital aur Time badlao → Profit share LIVE dekho

A's Capital (₹00) 50

A's Time (months) 12

B's Capital (₹00) 60

B's Time (months) 8

🧮 Try It — Profit Split Calculator

A's Capital (₹)

A's Time (months)

B's Capital (₹)

B's Time (months)

Total Profit (₹)

📈 Sequences & Probability

Arithmetic & Geometric Progression

AP: aₙ = a+(n-1)d · Sₙ = n/2(2a+(n-1)d) · GP: aₙ = arⁿ⁻¹

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AP nth term: aₙ = a + (n−1)d
AP Sum: Sₙ = n/2 × [2a + (n−1)d] = n × (a + l)/2
GP nth term: aₙ = a × rⁿ⁻¹
GP Sum: Sₙ = a(rⁿ−1)/(r−1) [r≠1]

a = first term · d = common difference (AP) · r = common ratio (GP) · l = last term · n = term number

🧒 Step 1 — Sequence kya hoti hai? Bilkul basic se shuru

Socho tum ek seedhi line mein numbers likh rahe ho, ek pattern follow karke. Jaise:

🎵 2, 4, 6, 8, 10... — Har baar 2 add ho raha hai
🎵 3, 9, 27, 81... — Har baar 3 se multiply ho raha hai

In patterns ko hi Progression kehte hain!

Pehli wali mein hamesha same amount ADD hota hai → Arithmetic Progression (AP)
Doosri mein hamesha same number MULTIPLY hota hai → Geometric Progression (GP)

📌 Real life AP: EMI payments (same amount every month), steps on a staircase, 5, 10, 15, 20... (table of 5)

📌 Real life GP: Bacteria doubling every hour (1, 2, 4, 8...), compound interest growth, viral video shares

➕ Step 2 — Arithmetic Progression (AP) — Poora samjho

AP kya hai: Ek sequence jisme consecutive terms ka difference always same hota hai. Isko common difference (d) kehte hain.

AP: a, a+d, a+2d, a+3d, ...

Example: 5, 8, 11, 14, 17... → a=5, d=3

Formula 1 — nth Term: aₙ = a + (n−1)d
→ 10th term = 5 + (10−1)×3 = 5 + 27 = 32

Formula 2 — Sum of n terms:
Sₙ = n/2 × [2a + (n−1)d]
→ S₁₀ = 10/2 × [2×5 + 9×3] = 5 × [10+27] = 5 × 37 = 185

Shortcut Sum: Sₙ = n × (first + last) / 2
→ S₁₀ = 10 × (5 + 32) / 2 = 10 × 37/2 = 185 ✓ Same answer!

Special AP sums to remember:
1+2+3+...+n = n(n+1)/2
1²+2²+3²+...+n² = n(n+1)(2n+1)/6
Sum of first n odd numbers = n²

💡 Gauss ka trick: 1+2+3+...+100 = 100×101/2 = 5050. Pair banao: 1+100=101, 2+99=101... 50 pairs → 50×101=5050!

📌 Exam: Find sum of all multiples of 7 from 7 to 98 → AP: a=7, d=7, l=98 → n=(98-7)/7+1=14 → S=14×(7+98)/2=14×52.5=735

✖️ Step 3 — Geometric Progression (GP) — Exponential growth samjho

GP kya hai: Ek sequence jisme consecutive terms ka ratio always same hota hai. Isko common ratio (r) kehte hain.

GP: a, ar, ar², ar³, ...

Example: 2, 6, 18, 54, 162... → a=2, r=3

Formula 1 — nth Term: aₙ = a × rⁿ⁻¹
→ 5th term = 2 × 3⁴ = 2 × 81 = 162 ✓

Formula 2 — Sum (r ≠ 1):
Sₙ = a(rⁿ − 1) / (r − 1) [when r > 1]
→ S₅ = 2×(3⁵−1)/(3−1) = 2×242/2 = 242

r = 1 case: Sₙ = n×a (all terms same!)

Infinite GP Sum (|r| < 1): S∞ = a / (1−r)
Example: 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1−0.5) = 2

⚠️ GP grows EXPLOSIVELY! Wheat on chessboard: start with 1 grain, double each square = 2⁶³ ≈ 9.2 × 10¹⁸ grains (more than all wheat ever grown!)

📌 Bacteria example: 100 bacteria, doubles every hour. After 10 hours = 100 × 2¹⁰ = 100 × 1024 = 1,02,400 bacteria

⚖️ Step 4 — AP vs GP: Side-by-side comparison (Exam shortcut)

Property	AP	GP
Pattern	Add same d	Multiply same r
nth Term	a + (n−1)d	a × rⁿ⁻¹
Sum Sₙ	n/2[2a+(n-1)d]	a(rⁿ−1)/(r−1)
Growth type	Linear (line)	Exponential (curve)
Check karo if	aₙ₊₁ − aₙ = const	aₙ₊₁ / aₙ = const

💡 Arithmetic Mean of AP = (first + last) / 2 = middle term. Geometric Mean of GP = √(first × last) for 3-term GP.

🎬 AP (bars) vs GP (exponential) — dono simultaneously dekho

First term a 2

AP diff d / GP ratio r 3

🧮 Try It — AP nth Term & Sum Calculator

First term a

Common diff d

n (term number)

🎲

Probability — P(A) = n(A)/n(S)

Probability · 0 to 1 · P(A∪B) = P(A)+P(B)-P(A∩B)

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P(A) = n(A) / n(S)
P(A') = 1 − P(A)

Probability = "kitna chance hai ki yeh ho jaaye"

P(A) = event A hone ki probability n(A) = favorable outcomes (jo chahiye) n(S) = total possible outcomes (Sample Space) A' = "A nahi hona" (complement)

🧒 Step 1 — Probability kya hai? Roz ki life se samjho

Tumne kabhi suna hoga "aaj baarish hone ke 70% chances hain." Yeh hi Probability hai — "kitna chance hai ki koi cheez ho jaaye" ka number!

Probability hamesha 0 aur 1 ke beech ek number hota hai (ya 0% se 100%):
• P = 0 → bilkul impossible (kabhi nahi hoga) — jaise "sooraj kal nahi uthega"
• P = 1 → bilkul certain (zaroor hoga) — jaise "kal sooraj uthega"
• P = 0.5 → 50-50 chance — jaise coin toss mein heads aana

Formula yaad rakhne ka simple tareeka:
Probability = Jo chahiye (Favorable) / Jo ho sakta hai (Total)

Yeh "Sample Space" (n(S)) — yani saare possible outcomes ki list — aur "Favorable outcomes" (n(A)) — yani jo humein chahiye unki list — dono ka ratio hai!

💡 Pehla step hamesha yeh hai: poocho "TOTAL kitne outcomes ho sakte hain?" phir poocho "Mujhe JO chahiye, woh kitne hain?" — phir simple divide karo!

📌 Weather forecast, cricket match jeetne ka chance, exam mein pass hone ka chance, lottery — sab jagah probability ka use hota hai!

📐 Step 2 — Classic examples se formula gehraai se samjho

① Coin Toss: Ek coin mein 2 sides hain — Heads, Tails. Total outcomes n(S) = 2. Heads aane ka favorable n(A) = 1.
P(Heads) = 1/2 = 0.5 = 50%

② Dice Throw: Ek dice mein 6 faces hain (1,2,3,4,5,6). Total n(S) = 6. "6 aana" ka favorable n(A) = 1.
P(6 aana) = 1/6 ≈ 0.167 = 16.7%

③ Playing Cards: Deck mein total 52 cards hain. Ace sirf 4 hote hain (1 har suit mein).
P(Ace) = 4/52 = 1/13 ≈ 0.077 = 7.7%

④ Coloured Balls (multi-favorable example): Bag mein 3 red, 5 blue balls hain — total 8 balls.
P(red) = 3/8 = 0.375 = 37.5%
P(blue) = 5/8 = 0.625 = 62.5%
(Notice: P(red) + P(blue) = 3/8+5/8 = 8/8 = 1 — kyunki ball red ya blue hi hoga, koi teesra option nahi!)

💡 Trick: Sab possible outcomes ki probability ka SUM hamesha 1 (ya 100%) hota hai — yeh check karke apna answer verify kar sakte ho!

⚠️ Total outcomes count karte waqt sabse common mistake: kuch outcomes miss kar dena ya repeat count kar dena. Hamesha pehle poori list banao.

📌 2 dice ek saath throw karo, total sum=7 ka probability: outcomes jo sum 7 dete hain = (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) = 6 ways. Total outcomes = 6×6=36. P = 6/36 = 1/6 ≈ 16.7%

🔍 Step 3 — Complement aur "OR" (Union) rule — gehraai se samjho

① Complement Rule — P(A') = 1 − P(A):
"A' " ka matlab hai "A NAHI hona". Socho: agar koi event A hone ka chance hai, toh "A na hone" ka chance bacha hua sab hissa hai!
Example: Dice se "6 NAHI aana" ka chance? P(6 aana)=1/6, toh P(6 nahi aana) = 1 − 1/6 = 5/6 ≈ 83.3%
Yeh kab use hota hai? Jab "kam se kam ek baar hoga" ya "kabhi nahi hoga" jaisa sawaal ho — complement se solve karna aasaan ho jaata hai!

② Union Rule — P(A∪B) = P(A) + P(B) − P(A∩B):
Yeh formula tab use hota hai jab poochna ho "A YA B mein se kam se kam ek ho jaaye" ka chance.
"− P(A∩B)" kyun ghataya jaata hai? Kyunki agar A aur B dono ek saath ho sakte hain (overlap), toh unhe double count na ho jaaye, isliye ek baar wapas ghata dete hain!

Worked example: Card deck se "King YA Heart" nikalne ka chance?
P(King) = 4/52, P(Heart) = 13/52, P(King AND Heart) = 1/52 (sirf "King of Hearts")
P(King∪Heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 30.8%

💡 Yaad rakho: "AND" (∩) matlab dono ek saath chahiye, "OR" (∪) matlab koi ek bhi chalega (ya dono bhi). "OR" wala question aksar zyada bada answer deta hai!

⚠️ Agar A aur B kabhi overlap nahi karte (jaise "coin pe Heads" aur "coin pe Tails" — dono kabhi ek saath nahi ho sakte), toh P(A∩B)=0, aur formula seedha P(A)+P(B) ban jaata hai. Yeh "Mutually Exclusive" events kehlaate hain.

📌 Dice se "2 ya 5" aane ka chance (mutually exclusive, overlap nahi): P(2)+P(5) = 1/6+1/6 = 2/6 = 1/3 ≈ 33.3%

🌍 Step 4 — Real life mein Probability kahan kahan dikhti hai?

Roz ki life mein probability:
• Weather forecast: "70% chance of rain" — meteorologists past data se calculate karte hain
• Insurance companies: Tumhe accident hone ka chance kitna hai, usi se premium decide karte hain
• Medical tests: Ek dawai kaam karne ka chance kitna hai — clinical trials se nikalta hai
• Games & Gambling: Casino, lottery, card games — sab probability pe based hain
• Sports: Team ke jeetne ka chance, player ke score karne ka chance
• Stock Market: Kisi share ka price upar ya neeche jaane ka chance

Common mistakes jo students karte hain:
1. Total outcomes galat count karna (sample space poori nahi banana)
2. "AND" aur "OR" confuse kar dena (multiply vs add karna ulta)
3. Mutually exclusive na hone par bhi P(A∩B) ko zero maan lena
4. Probability ko 1 se zyada ya negative likh dena (yeh kabhi possible nahi!)

💡 Final answer dene se pehle hamesha check karo: kya yeh number 0 aur 1 ke beech hai? Agar 1 se zyada ya negative aaya, toh kahin calculation mein galti hui hai!

📌 Class mein 30 students, 18 ladke, 12 ladkiyan. Randomly 1 student choose kiya jaaye, ladki hone ka chance? P(ladki) = 12/30 = 2/5 = 40%

🎬 Sliders move karo — favorable (green) vs total outcomes dekho

Favorable outcomes3

Total outcomes8

👀 Har ball ek "possible outcome" hai. Hare balls = favorable (jo chahiye), grey balls = baaki sab. Probability = hare balls ÷ total balls!

🧮 Try It

Favorable

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