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

1.6.6 · D5 · Physics › Oscillations & Waves › Simple pendulum — small angle approximation, T = 2π√(L - g)

Is page mein jo symbols use hote hain — pehle padho

Traps se pehle, har ek letter ko pin down karte hain taaki kuch bhi surprise na kare. Picture dekho: usme poora setup aur har symbol kisi cheez se juda hua dikha hai jise tum point kar sako.

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

Definition Paanch symbols

$m$ — bob ka mass, end mein woh chhoti weight (kilograms mein). "Kitna stuff" swing kar raha hai.
$L$ — pendulum ki length: pivot (jahan se hang karta hai) se bob ke centre of mass tak ki straight-line distance, bob ke top tak nahi (metres mein). Yeh woh radius hai jo bob apni circular arc sweep karte waqt banata hai.
$g$ — gravitational acceleration ki magnitude, $g \equiv ∣ g ∣ \approx 9.8 m/s^{2}$ Earth pe. Hamesha positive; yeh measure karta hai ki gravity kitni strongly pull karti hai.
$θ$ — vertical se angular displacement: string aur seedhi-neeche rest position ke beech ka angle (radians mein). $θ = 0$ matlab seedha neeche hang karna; $θ > 0$ matlab ek taraf khicha hua.
$θ_{0}$ — amplitude: $θ$ ki sabse badi value, yaani woh angle jahan se bob ko release kiya. Yeh "swing kitni badi hai" ko label karta hai.

Woh result jiske around ye traps orbit karte hain

Neeche sab kuch ek formula test karta hai, toh yahan ek hi saans mein poori kahani hai, swing geometry ki ek picture ke saath.

True ya false — justify karo

Pendulum ka period bob ke mass pe depend karta hai.

False —

m L \ddot{θ} = - m g sin θ

mein mass

m

dono sides se divide ho jaata hai, toh

T = 2 π L / g

mein koi

m

nahi hai. Gravity ek bhaari bob ko zyada pull karti hai lekin uska inertia utna hi zyada resist karta hai, exactly cancel karte hue.

String ki length double karne se period double ho jaati hai.

False —

T \propto L

, toh

L

double karne se

T

2 \approx 1.41

se multiply hota hai.

T

double karne ke liye

L

chaar guna karna padega.

Formula

T = 2 π L / g

sab swing angles ke liye exact hai.

False — yeh

sin θ \approx θ

se aata hai, jo sirf ek approximation hai. Asli period amplitude ke saath badhti hai:

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

, jahan

θ_{0}

amplitude (max angle) hai.

Pendulum Moon pe Earth se zyada tezi se swing karta hai.

False — Moon ka chhota

g

root ke neeche

L / g

mein aata hai, toh chhota

g

matlab bada

T

: slow, aalsi swings.

String mein tension bob ko arc ke along speed karne mein koi kaam nahi karta.

True — tension string ke along point karti hai, jo hamesha arc (motion ki direction) ke perpendicular hoti hai, toh motion ke along uski component zero hoti hai. Sirf gravity ka tangential component swing drive karta hai.

Swing ke lowest point pe bob pe net force zero hoti hai.

False — tangential force wahan zero hai (angular motion ka equilibrium), lekin bob circle mein move kar raha hai, toh tension

m g

se zyada hai upar ki taraf centripetal force provide karne ke liye. Net force upar pivot ki taraf point karti hai.

Isochronism ka matlab hai period same rehti hai chahe bob kitna bhi bhaari ho.

False — isochronism specifically matlab hai period amplitude se independent hai (small swings ke liye). Mass-independence alag fact hai; dono ko merge mat karo.

Agar string ko same length ki stiff rod se replace karo,

T = 2 π L / g

bina kisi change ke apply hota rehta hai.

False — ek rod ka apna mass length ke along distributed hota hai, toh yeh ek physical pendulum ban jaata hai jiska period uske moment of inertia pe depend karta hai, simple-pendulum formula pe nahi.

Error dhundho

"Ek student likhta hai

sin (10°) \approx 10

, toh small-angle rule break ho gayi hai."

Rule hai

sin θ \approx θ

jahan

θ

radians mein ho.

10°

hai

0.1745

rad, aur

sin (10°) = 0.1736

— excellent agreement. Error hai

sin

ko degree number se compare karna.

"Kyunki

T = 2 π L / g

, ek longer pendulum ka bada

ω

hoga aur woh zyada baar per second swing karega."

Ulta hai.

ω = g / L

, toh bada

L

chhota

ω

deta hai — per second kam cycles aur longer period. Lambe pendulums slow hote hain.

"Restoring force hai

F = - m g θ

, toh yeh linear hai aur yeh hamesha SHM hai."

Asli tangential force hai

F = - m g sin θ

, jo linear nahi hai. Yeh sirf small-angle approximation ke baad

- m g θ

reduce hoti hai, toh yeh SHM sirf small swings ke liye hai.

L

string ki length hai, pivot se bob ke top pe knot taka hai wahan tak measured."

L

pivot se bob ke centre of mass tak ki distance hai, uske top tak nahi. Ek bade bob ke liye yeh difference matter karta hai.

"Ek tezi se ghoomne wale merry-go-round pe, ek hanging pendulum ka period abhi bhi

2 π L / g

hai kyunki gravity unchanged hai."

Ek rotating (non-inertial) frame mein effective gravity

g

ko outward centrifugal effect ke saath combine karta hai, restoring acceleration change kar deta hai. Plain

g

nahi, effective

g_{eff}

use karo.

- m g sin θ

mein minus sign sirf ek convention hai aur drop kiya ja sakta hai."

Nahi kiya ja sakta — minus encode karta hai ki force $θ = 0$ ki taraf wapas point karti hai. Iske bina equation runaway growth describe karti, oscillation nahi. Dekho Restoring Force and Equilibrium.

"Amplitude badhane se bottom pe speed badhti hai, toh pendulum ko har cycle tezi se complete karni chahiye."

Bob bade amplitude ke liye bottom pe zyada tezi se move karta hai, lekin woh ek longer arc bhi travel karta hai. Small angles ke liye ye balance ho jaate hain (isochronism); large angles ke liye longer path jeet jaata hai aur period actually badhti hai.

Why questions

Gravity ko components mein split kyun kiya jaata hai, seedha

m g

use karne ki jagah?

Kyunki sirf gravity ka arc ke along (tangential) part bob ki speed change karta hai; radial part tension se cancel ho jaata hai. Splitting woh force isolate karti hai jo actually oscillation drive karti hai.

sin θ \approx θ

hold karne ke liye

θ

radians mein kyun hona chahiye?

Taylor series

sin θ = θ - θ^{3} /6 + \dots

radian measure assume karke derive ki jaati hai, jahan

sin

ka derivative

cos

cleanly hota hai. Degrees mein

π /180

ka extra factor aata hai aur

sin θ \approx θ

fail ho jaata hai. Dekho Taylor Series and Small-Angle Approximations.

Mass algebraically nahi, physically kyun cancel hota hai?

Gravity ki pull mass ke saath scale hoti hai (

m g

), aur acceleration ke khilaaf resistance (inertia) bhi mass ke saath scale hoti hai. Dono mass factors same

m

hain, toh jo acceleration produce hoti hai woh mass-free hai — same reason se sab objects same rate se girte hain.

T = 2 π / ω

kyun hai aur, say,

π / ω

4 π / ω

kyun nahi?

Ek complete cycle mein phase

2 π

radians advance karta hai, aur

ω

phase rate hai (radians per second). Ek cycle ka time total phase divided by rate hai:

2 π / ω

. Dekho Angular Frequency and Period.

Pendulum equation bilkul mass-spring equation jaisi kyun lagti hai?

Dono ka form

\overset{q}{¨} = - ω^{2} q

hai: ek restoring "acceleration" displacement ke proportional. Spring ke liye

ω^{2} = k / m

; pendulum ke liye

ω^{2} = g / L

. Same maths, same SHM. Dekho Mass-Spring System.

g

find karne ke liye ek swing time karne ki jagah 10 swings time karke 10 se divide karna better kyun hai?

Tumhari reaction-time error (say

0.2

s) har measurement mein fixed hai. 10 periods mein spread hone pe yeh sirf

0.02

s per period contribute karti hai — fractional error mein das guna reduction. Dekho Measuring g with a Pendulum.

Bada amplitude asli period ko chhota nahi, lamba kyun karta hai?

Large

θ

ke liye,

sin θ < θ

, toh asli restoring force linear approximation se kamzor hai. Kamzor restoring pull matlab slower return aur longer period.

Edge cases

Jab amplitude

180°

(bob almost seedha upar balanced) ke paas jaata hai toh period ka kya hota hai?

Period infinity ki taraf jaati hai — top ke paas restoring force

sin θ \to 0

ho jaati hai, toh bob arbitrarily long time tak lingering karta hai. Simple formula yahan bilkul galat hai.

Agar bob ko exactly

θ = 0

se zero speed ke saath release karo, kya motion hogi?

Koi nahi — yeh already equilibrium mein hai jahan koi restoring force nahi aur koi velocity nahi, toh yeh rest mein rehta hai. Yeh degenerate zero-amplitude "oscillation" hai.

Free fall mein pendulum ka period kya hai (ek elevator jiska cable cut ho gaya)?

Effectively infinite / undefined — free fall mein effective

g

zero hai, koi restoring force nahi hai, aur bob sirf float karta hai. Formula deta hai

T = 2 π L /0 \to \infty

L \to 0

ka limit lo: formula kya predict karta hai, aur kya yeh physical hai?

T \to 0

: infinitely fast swinging. Mathematically consistent, lekin physically point-mass idealization bahut pehle break ho jaati hai, toh ise limiting trend samjho, real device nahi.

Agar

g

double ho jaaye (ek hypothetical stronger-gravity planet), period kaise change hoti hai?

T \propto 1/ g

, toh

g

double karne se period

2 \approx 1.41

se divide ho jaati hai — pendulum noticeably tezi se swing karta hai.

Ek ideal (frictionless) pendulum ki energy many swings ke baad kya hoti hai?

Yeh conserved rahti hai — kinetic aur potential energy aapas mein trade hoti hain lekin unka sum constant rehta hai, toh amplitude kabhi decay nahi hoti. Real pendulums air aur friction se energy lose karte hain. Dekho Energy in Oscillations.

Active recall

Recall Period ki kaunsi do "independences" ko log sabse zyada confuse karte hain?

Mass-independence (koi bhi bob, same $T$ ) versus amplitude-independence / isochronism (kisi bhi size ki small swings, same $T$ ). Ye alag facts hain alag reasons ke saath.

Recall Woh single approximation batao jo pendulum ko SHM banata hai.

$sin θ \approx θ$ (radians), sirf small angles ke liye valid — sab kuch downstream isi pe depend karta hai.

Connections

Simple Harmonic Motion — woh pattern jis par har trap wapas aata hai
Restoring Force and Equilibrium — minus sign ka source
Taylor Series and Small-Angle Approximations — radians kyun, approximate kyun
Angular Frequency and Period — $T = 2 π / ω$ relation
Energy in Oscillations — energy edge case
Mass-Spring System — analogous SHM
Measuring g with a Pendulum — many swings time karne ka trick