Simple pendulum — small angle approximation, T = 2π√(L - g) derivation

1.6.6 · Physics › Oscillations & Waves

Simple pendulum KYA hota hai?

YEH idealizations KYU? Ye complications ko hata deti hain taaki physics pure rahe: string ka koi mass nahi jo hilega, koi stretching nahi jo energy store karegi, koi friction nahi jo swing ko damp karegi.

$T = 2 π L / g$ ko scratch se derive KAISE karte hain

Step 1 — Forces identify karo aur restoring direction dhundho

Jab string vertical ke saath angle $θ$ banati hai, bob par do forces act karti hain: gravity $m g$ (neeche) aur tension $T_{string}$ (string ke along).

Gravity ko components mein kyun todte hain? Tension string ke along hoti hai, isliye yeh bob ko arc ke along speed up ya slow down nahi kar sakti. Sirf gravity ka component arc ke along (tangential) hi motion drive karta hai.

$F_{tangential} = - m g sin θ$

Minus sign kyun? Kyunki force $θ = 0$ ki taraf waapas point karti hai. Agar $θ > 0$ hai, toh force negative hai (restoring). Yahi minus sign saari oscillation ka dil hai.

Step 2 — Arc ke along Newton ka law likhna

Bob radius $L$ ke circular arc par move karta hai. Arc length (displacement) $s = L θ$ hai, isliye tangential acceleration hai:

$a = \frac{d ^{2} s}{d t ^{2}} = L \frac{d ^{2} θ}{d t ^{2}}$

Tangentially Newton's second law ( $ma = F_{tangential}$ ):

$m L \frac{d ^{2} θ}{d t ^{2}} = - m g sin θ$

Yeh step kyun? Humne forces ko equation of motion mein badal diya. $m$ cancel karo (mass nikal jaata hai — pendulums ko bob ke weight ki parwah nahi!):

$\frac{d ^{2} θ}{d t ^{2}} = - \frac{g}{L} sin θ$

Step 3 — Small-angle approximation

Is equation mein $sin θ$ hai — yeh abhi SHM nahi hai (SHM ke liye restoring term $θ$ mein linear chahiye).

Key trick: $θ = 0$ ke aas-paas $sin θ$ ka Taylor-expand karo:

$sin θ = θ - \frac{θ ^{3}}{6} + \frac{θ ^{5}}{120} - \dots$

Chhote $θ$ ke liye (radians mein), $θ^{3}$ term bahut chhota hai, isliye:

$sin θ \approx θ$

Yeh allow kyun hai? $θ = 0.1$ rad ( $\approx 5.7°$ ) par, $sin θ = 0.09983$ , error $\approx 0.17%$ . Negligible!

$\frac{d ^{2} θ}{d t ^{2}} = - \frac{g}{L} θ$

Step 4 — SHM recognize karo aur period read off karo

Standard SHM equation se compare karo:

$\frac{d ^{2} x}{d t ^{2}} = - ω^{2} x ⟹ ω^{2} = \frac{g}{L}$

Figure — Simple pendulum — small angle approximation, T = 2π√(L - g) derivation

Formula padhna (80/20 insight)

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Reveal karne se pehle try karo

$F = - m g sin θ$ mein minus sign kahan se aata hai?
Kis step mein mass cancel hota hai, aur iska kya matlab hai?
$θ$ radians mein kyun hona chahiye?
$T \propto L$ physically kya matlab hai?

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek park ke swing ki imagine karo. Gravity hamesha swing ko lowest point par waapas kheenchne ki koshish karti hai — jitna zyada upar kheencho, utna hi zyada gravity usse waapas kheenchti hai. Yeh "hamesha waapas kheenchna, jitna door utna zyada" wala rule hi ise steady rhythm mein aage-peechhe karta hai. Achha surprise yeh hai: koi bhaari baccha ho ya halka — dono ek same time mein ek full swing lete hain, jab tak chain ki length same ho! Lambi chains = slow aaram se jhoolna; chhoti chains = quick swings. Ek clockmaker timekeeping ke liye exactly isi steady rhythm ka use karta hai.

Flashcards

Kaunsi approximation pendulum equation ko linear (SHM) banati hai?

sin θ \approx θ

radians mein chhote

θ

ke liye.

Bob ka mass period ko affect kyun nahi karta?

m

dono sides par aata hai (

m L \ddot{θ} = - m g sin θ

) aur cancel ho jaata hai.

Simple pendulum ka period bolo.

T = 2 π L / g

Bob par tangential restoring force kya hai?

F = - m g sin θ

(minus = equilibrium ki taraf waapas point karta hai).

Agar

L

chaar guna ho jaaye toh

T

kaise change hoga?

T

double ho jaata hai, kyunki

T \propto L

Simple pendulum ke liye

ω^{2}

express karo.

ω^{2} = g / L

Pendulum data se

g

kaise dhundhen?

g = 4 π^{2} L / T^{2}

sin θ \approx θ

ke liye

θ

radians mein kyun hona chahiye?

Taylor series

sin θ = θ - θ^{3} /6 + \dots

sirf radians mein hold karta hai.

Bade amplitude

θ_{0}

ke liye period ka pehla correction term kya hai?

T \approx 2 π L / g (1 + θ_{0}^{2} /16)

"Isochronism" kya hai?

Period amplitude se independent hona (chhote swings ke liye) — Galileo ka observation.

Connections

Simple Harmonic Motion — pendulum ek special case hai
Restoring Force and Equilibrium
Taylor Series and Small-Angle Approximations
Angular Frequency and Period
Energy in Oscillations — swing mein KE/PE exchange
Mass-Spring System — analogous $ω = k / m$
Measuring g with a Pendulum (lab application)

Simple pendulum — small angle approximation, T = 2π√(L - g) derivation

Simple pendulum KYA hota hai?

$T = 2 π L / g$ ko scratch se derive KAISE karte hain

Step 1 — Forces identify karo aur restoring direction dhundho

Step 2 — Arc ke along Newton ka law likhna

Step 3 — Small-angle approximation

Step 4 — SHM recognize karo aur period read off karo

Formula padhna (80/20 insight)

Worked examples

Common mistakes (steel-manned)

Active recall

Flashcards

Connections

Concept Map

Deep Dive