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

1.6.6 · D1 · Physics › Oscillations & Waves › Simple pendulum — small angle approximation, T = 2π√(L - g)

Pehle aapko parent note (T = 2π√(L/g)) ki derivation padhni hai, lekin usse pehle har symbol mein fluent hona zaroori hai jo woh aap par throw karta hai. Yeh page har ek ko zero se build karta hai — seedhe words, phir ek picture, phir kyun topic ko uski zaroorat hai. Upar se neeche padho; baad mein kuch bhi aisa nahi aata jo pehle define na kiya gaya ho.

1. Pendulum ki picture (woh stage jis par sab kuch hota hai)

Neeche sab kuch EK drawing par hota hai: ek fixed point se latakti hui string, end par ek ball, aur gravity seedha neeche kheench rahi hai. Chaliye parts ko naam dete hain.

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

Yahan se kyun shuru karein? Aage aane wala har symbol is picture se liya hua ek measurement hai. Agar picture clear hai, toh symbols sirf labels hain.

2. Length $L$ — swing arc ka size

Upar figure mein amber line dekho: woh $L$ hai. Yeh set karta hai arc kitna bada hai.

Topic ko isko kyun chahiye: $L$ pendulum ke shape ki ek maatra cheez hai jo final formula $T = 2 π L / g$ mein survive karti hai. Baaki sab cancel ho jaata hai.

3. Angle $θ$ — aap kitna swing kar chuke ho

Angle kyun aur sideways distance kyun nahi? Kyunki bob string par atka hua hai — woh sirf ek curved arc par chal sakta hai, aur ek angle "curve par kitna aage" kehne ka natural tarika hai. $θ$ ka sign (positive vs negative) woh hai jo hume baad mein kehne deta hai "force hamesha $θ = 0$ ki taraf wapas point karti hai."

4. Radians — $θ$ iss tarah kyun measure karna zaroori hai

Yahan ek subtle lekin zaroori idea hai jo parent note demand karta hai: angles radians mein measured hone chahiye, degrees mein nahi.

Formula se pehle, ek aur symbol. Arc length, jise $s$ likhte hain, woh distance hai jo bob apne curved path par travel karta hai — metres mein measured, jaise circle ki rim ke around walk karna seedha cross karne ki jagah. Yeh ordinary distance ka curved-path cousin hai.

Topic ko specifically radians kyun chahiye:

Parent ka key approximation $sin θ \approx θ$ sirf radians mein true hai. Degrees mein $sin (10°) = 0.174$ hai lekin " $10$ " number uske kareeb kahin nahi hai.
Arc-length formula $s = L θ$ (derivation ke Step 2 mein use hota hai) mein sirf radians mein koi extra constant nahi hai.

5. Mass $m$ aur gravity $g$

$m$ aur $g$ dono kyun aate hain: gravity ka pull $m g$ hai, lekin bob ki reluctance to move bhi $m$ se set hoti hai. Pendulum ka magic (parent mein dikhaya gaya) yeh hai ki yeh do $m$ 's cancel ho jaate hain — isliye period mass par depend nahi karti.

6. Force, aur "restoring" ka minus sign

7. Gravity ko split karna: tangent, arc, aur tension

Bob par do forces pull karti hain: uska weight $m g$ (seedha neeche) aur string tension. Tension ko naam dete hain.

Kyunki tension string ke along hai, yeh curved path par motion drive nahi kar sakti. Toh hum weight $m g$ ko do pieces mein split karte hain: ek string ke along (jise tension balance karta hai) aur ek arc ke along (tangential, matlab "curve ke along sideways"). Sirf tangential piece hi swing drive karta hai.

$sin θ$ kyun aata hai, aur $sin$ kya hai?

Hamare split mein, weight $m g$ ek chote triangle ki hypotenuse hai, aur tangential (arc-wise) part woh side hai jo angle $θ$ ke "opposite" hai — toh yeh $m g sin θ$ hai. Minus sign section 6 se restoring rule hai. $sin$ kyun aur $cos$ kyun nahi? Kyunki neeche ( $θ = 0$ ) zero sideways push honi chahiye, aur $sin 0 = 0$ — yeh reality se match karta hai; $cos 0 = 1$ nahi karta.

8. Time $t$ , aur rate of change $\frac{d θ}{d t}$ , $\frac{d ^{2} θ}{d t ^{2}}$

Pehle hum baat kar sakein kitni fast cheezein change hoti hain, hume woh cheez chahiye jiske saath woh change hoti hain: time.

Parent note $\frac{d ^{2} θ}{d t ^{2}}$ likhta hai. Chaliye woh notation earn karte hain.

Topic ko second derivative kyun chahiye: Newton's law kehta hai force acceleration set karta hai (position nahi, speed nahi). Acceleration position ki change ki second rate hai — isliye do $d$ 's. Yeh "gravity wapas kheenchti hai" ko ek solvable equation of motion mein turn karta hai. Zyada Simple Harmonic Motion aur Angular Frequency and Period mein.

9. Angular frequency $ω$ aur period $T$

Topic ko $ω$ kyun chahiye: tidy SHM equation padhti hai $\frac{d ^{2} θ}{d t ^{2}} = - ω^{2} θ$ . Aage wala number, $ω^{2}$ , hi physics hai — pendulum ke liye yeh $g / L$ nikalta hai, jo immediately period deta hai. Dekho Angular Frequency and Period.

10. Taylor idea (sirf Step 3 padhne ke liye enough)

Topic ko isko kyun chahiye: $sin θ$ ke saath equation haath se unsolvable hai; $θ$ ke saath yeh exactly SHM equation ban jaata hai. Yeh single swap hi poori derivation ka pivot hai. Poori kahani Taylor Series and Small-Angle Approximations mein.

Prerequisite map

Equipment checklist

Khud test karo — sirf answer karne ke baad reveal karo.

L

kya measure karta hai, aur bob ke kis point tak?

Pivot se bob ke center of mass tak ki distance, metres mein.

θ

kya hai, aur yeh zero kahan hai?

String ka vertical se angle; neeche (rest) par zero.

Ek radian ko ek sentence mein define karo.

Woh angle jiska arc length circle ke radius ke barabar ho (

\approx 57.3°

Symbol

s

ka kya matlab hai, uske units kya hain, aur uska sign convention kya hai?

Signed arc displacement — bob neeche se curved path par kitna door hai, metres mein, daayein positive aur baayein negative (

θ

jaisa hi sign).

L

aur

θ

ke terms mein arc length likho, aur units condition batao.

s = L θ

, tabhi valid jab

θ

radians mein ho.

Right triangle par

sin θ

kya hota hai?

Opposite side over hypotenuse.

Tangential force

cos

ki jagah

sin

kyun use karta hai?

θ = 0

par sideways push zero honi chahiye, aur

sin 0 = 0

isse match karta hai.

- m g sin θ

mein minus sign physically kya mean karta hai?

Force hamesha equilibrium ki taraf wapas point karti hai (restoring).

T_{string}

kya hai, aur yeh swing kyun drive nahi kar sakta?

String tension, string ke along pivot ki taraf directed; string ke along hone ki wajah se uska arc ke along koi component nahi hai.

t

kya hai, aur kya uske par depend karta hai?

Seconds mein elapsed time; angle

θ (t)

uske saath change hota hai.

\frac{d ^{2} θ}{d t ^{2}}

kya represent karta hai?

Angular acceleration — angular velocity time

t

ke har second mein kitni fast change hoti hai.

T

aur

ω

ko relate karo.

T = 2 π / ω

(ek cycle

2 π

phase ka hota hai).

Small-angle approximation aur uski condition batao.

sin θ \approx θ

small

θ

ke liye radians mein measured.

m

aur

g

kya hain, units ke saath?

m

= mass (kg, inertia);

g

= gravitational field strength (

\approx 9.8 m/s^{2}

Connections

Simple Pendulum — small angle approximation, T = 2π√(L - g) derivation — woh parent jiske liye yeh page prepare karta hai
Restoring Force and Equilibrium — minus sign poori tarah explain kiya gaya
Taylor Series and Small-Angle Approximations — $sin θ \approx θ$ kahan se aata hai
Angular Frequency and Period — $ω$ aur $T$ depth mein
Simple Harmonic Motion — woh pattern jise pendulum follow karta hai
Energy in Oscillations — ek alternative viewpoint
Mass-Spring System — same maths $k$ aur $m$ ke saath
Measuring g with a Pendulum — in symbols ko kaam mein laana