Energy in SHM — KE + PE = ½kA² (constant): Question bank

1.6.5 · D5 · Physics › Oscillations & Waves › Energy in SHM — KE + PE = ½kA² (constant)

Do symbols baar baar aate hain, isliye pehle inhe lock in karo.

True or false — justify karo

True or false: Agar tum amplitude

A

ko double kar do, toh total energy

E

bhi double ho jaati hai.

False.

E = \frac{1}{2} k A^{2}

, isliye

E

scales karta hai

A^{2}

ke saath — double

A

se energy

4 \times

ho jaati hai,

2 \times

nahi.

True or false: Turning points par kinetic energy zero hoti hai.

True.

x = \pm A

par mass direction reverse karta hai, isliye ek instant ke liye

v = 0

, hence

K = \frac{1}{2} m v^{2} = 0

; saari energy

U

mein hoti hai.

True or false: Equilibrium (

x = 0

) par potential energy zero hoti hai.

True.

U = \frac{1}{2} k x^{2} = 0

jab

x = 0

, isliye poora

E

kinetic hai — yahi woh jagah hai jahan mass sabse fast hota hai.

True or false: Kinetic energy, displacement

x (t)

ke same frequency par oscillate karti hai.

False.

K \propto sin^{2} (ω t)

, aur

sin^{2}

sin

se do baar zyada repeat hota hai, isliye

K

(aur

U

)

2 ω

par oscillate karte hain —

x (t)

ki frequency se double.

True or false: Ek full cycle mein average KE, average PE ke barabar hoti hai.

True. Dono ka average

\frac{1}{4} k A^{2} = \frac{1}{2} E

hota hai; energy average par 50–50 baanti jaati hai, chahe kisi bhi instant par split alag ho.

True or false: Total energy

E

, mass

m

par depend karti hai jab

\frac{1}{2} k A^{2}

ke form mein likhi ho.

Is form mein False —

\frac{1}{2} k A^{2}

mein koi

m

nahi hai. Mass

ω

ke andar chhup jaata hai: equivalent form

E = \frac{1}{2} m ω^{2} A^{2}

dikhata hai ki

ω = k / m

ke saath

m

cancel hokar

\frac{1}{2} k A^{2}

bann jaata hai.

True or false: Same spring par ek bhaari mass, same amplitude tak khichne par same total energy store karta hai.

True.

E = \frac{1}{2} k A^{2}

sirf

k

aur

A

use karta hai; mass speed aur period change karta hai lekin stored energy nahi.

True or false:

x

ko half karne se potential energy half ho jaati hai.

False.

U = \frac{1}{2} k x^{2} \propto x^{2}

, isliye

x

ko half karne par PE

\frac{1}{4}

reh jaati hai, half nahi.

True or false: Ek ideal (undamped) oscillator mein,

E

kaafi cycles ke baad dheere dheere ghatta hai.

False. Ideal SHM mein sirf conservative restoring force hoti hai aur koi friction nahi, isliye

E

exactly constant rehta hai. Energy decay Damped Oscillations mein hota hai, yahan nahi.

Why questions

Total energy sirf amplitude se kyun fix hoti hai, mass abhi kahan hai isse kyun nahi?

Kyunki

E = \frac{1}{2} k A^{2}

turning point par evaluate hota hai jahan

K = 0

aur saari energy

\frac{1}{2} k A^{2}

hai; woh value poori motion ke liye conserved rehti hai, isliye mass jahan bhi ho, split change ho sakta hai lekin sum

\frac{1}{2} k A^{2}

rehta hai.

KE aur PE displacement ki frequency se double frequency par kyun oscillate karte hain?

Dono squares par depend karte hain (

sin^{2}, cos^{2}

), aur squaring wave ke negative half ko positive par fold kar deta hai, isliye pattern har displacement cycle mein do baar repeat hota hai —

sin^{2} θ = \frac{1 - c o s 2 θ}{2}

use karne par explicit

2 ω

dikhta hai.

Mass us point par fastest kyun chalta hai jahan force zero hai?

Force

x = 0

par zero hai, lekin mass poore inward safar mein accelerate hota raha hai, isliye woh equilibrium par pahunchta hai jab saari energy

K

mein convert ho jaati hai — maximum speed exactly wahan hoti hai jahan net force momentarily zero hoti hai.

A^{2}

scaling waves ki baat karte waqt kyun wapas aati hai?

Ek wave mein kaafi saare oscillators hote hain, har ek

\frac{1}{2} k A^{2}

carry karta hai, isliye transported energy (intensity) bhi amplitude squared ke saath scale karti hai — dekho Energy in Waves — intensity ∝ amplitude².

KE aur PE compare karte waqt hum

m ω^{2}

k

se kyun replace kar sakte hain?

Kyunki

ω = k / m

ka matlab hai

ω^{2} = k / m

, isliye

m ω^{2} = k

; ye KE ko

k

aur

A

ke terms mein rewrite karta hai, jis se ise directly

U = \frac{1}{2} k x^{2}

se compare kiya ja sakta hai same units mein.

Restoring force ka conservative hona constant energy ke liye key kyun hai?

Ek conservative force work ko PE ki tarah store karti hai aur use exactly wapas deti hai bina kisi loss ke, isliye koi energy heat ke roop mein nahi nikalta — yahi woh condition hai jo Conservation of Mechanical Energy require karta hai.

Edge cases

Exactly

x = 0

ke instant par energy split ka kya hota hai?

Saari energy kinetic hai:

K = E = \frac{1}{2} k A^{2}

aur

U = 0

. Ye maximum speed ka single point hai.

x = A

(turning point) par KE kya hai?

Zero.

K = \frac{1}{2} k (A^{2} - A^{2}) = 0

; mass instantaneously rest par hai aur saari energy

U = \frac{1}{2} k A^{2}

mein hai.

Degenerate case: agar amplitude

A = 0

ho, toh motion aur energy kya hai?

Koi motion nahi hai — mass hamesha equilibrium par baitha rehta hai.

E = \frac{1}{2} k (0)^{2} = 0

: dono banks khali hain.

v = ω A^{2} - x^{2}

mein agar tum (galti se)

x > A

plug in karo toh kya milta hai?

Square root ke andar ek negative number, jo unphysical hai — ye signal karta hai ki mass kabhi

x = \pm A

se aage nahi ja sakta, kyunki iska matlab hoga negative KE.

A

ek hard wall hai jo energy se set hoti hai.

Do symmetric points

x = + 0.5 A

aur

x = - 0.5 A

par energies kaise compare karti hain?

Dono identical hain:

U = \frac{1}{2} k (0.5 A)^{2}

aur

K = \frac{1}{2} k (A^{2} - 0.25 A^{2})

sirf

x^{2}

par depend karte hain, jo

\pm x

ke liye same hai. Energy equilibrium ke baare mein symmetric hai.

Boundary check: kya koi aisa point hai jahan KE aur PE dono simultaneously maximum hon?

Nahi.

K

peak karta hai

x = 0

par jahan

U = 0

hai, aur

U

peak karta hai

x = \pm A

par jahan

K = 0

hai. Unke maxima opposite locations par hain — ye bilkul out of step hain.

Recall Har trap ki ek-line summary

"Energy" word ek sum chhupata hai: $K + U = \frac{1}{2} k A^{2}$ fixed hai, lekin split centre (all $K$ ) aur edges (all $U$ ) ke beech flip karta hai, saab kuch squared quadratically scale karta hai, aur $sin^{2} / cos^{2}$ squaring hi woh cheez hai jo energies ko $2 ω$ par beat karaati hai.

Question bank — Energy in SHM — KE + PE = ½kA² (constant)

True or false — justify karo

Error dhundho

Why questions

Edge cases