Moment of inertia I = Σmᵢrᵢ² — concept: Question bank

1.5.5 · D5 · Physics › Rotational Mechanics › Moment of inertia I = Σmᵢrᵢ² — concept

True or false — justify

True/false batao AUR ek-sentence ka reason do. Sahi answer ke saath galat reason matlab galat answer hai.

Moment of inertia kisi object ki ek fixed property hai, bilkul uski mass ki tarah.

False. Mass fixed hoti hai, lekin

I

chosen axis par depend karta hai — same object ke alag-alag axes ke baare mein alag-alag

I

hote hain kyunki har particle ki perpendicular distance

r_{i}

badal jaati hai.

I

negative ho sakta hai agar tum koi strange axis choose karo.

False.

I = \sum m_{i} r_{i}^{2}

positive masses aur squared distances ka sum hai, aur square kabhi negative nahi hota, isliye

I \geq 0

hamesha.

Rotation ke axis par bilkul baitha ek particle phir bhi

I

mein thoda contribute karta hai kyunki uski mass hai.

False. Uski perpendicular distance

r = 0

hai, isliye uska term

m r^{2} = 0

hota hai — axis par mass moment of inertia mein kuch nahi contribute karta.

Equal mass wale do objects ka unke centres ke baare mein hamesha equal moment of inertia hota hai.

False.

I

is baat par depend karta hai ki mass kahan hai: same mass ka ek hoop (

M R^{2}

) ek disk (

\frac{1}{2} M R^{2}

) se zyada hota hai kyunki hoop apni saari mass maximum radius par rakhta hai.

Agar

ω = 0

(body rest mein hai) toh uska moment of inertia bhi zero hai.

False.

I = \sum m_{i} r_{i}^{2}

mein

ω

bilkul nahi hai —

I

ek geometry-aur-mass property hai jo chahe body spin kare ya na kare, exist karti hai.

Har particle ki axis se distance double karne par

I

double ho jaata hai.

False. Kyunki distance squared hai,

r

double karne par har term

2^{2} = 4

se multiply hoti hai, isliye

I

4× ho jaata hai, 2× nahi.

Kisi object mein mass add karna uska moment of inertia sirf badha sakta hai, kabhi ghata nahi sakta.

True. Har added term

m r^{2} \geq 0

hai, isliye jo bhi mass tum chipkao wo sum mein sirf add hi kar sakti hai — mass add karke

I

kabhi kam nahi ho sakta.

Total mass fixed rakhte hue mass ko axis ke paas laana

I

ko ghata deta hai.

True. Chhota

r

matlab chhota

r^{2}

har term mein, isliye

I

kam ho jaata hai — yahi woh skater karta hai jo apni arms andar khichti hai.

Spot the error

Har line mein ek mistake chhipi hai. Use naam do aur correct karo.

I

compute karne ke liye har particle ki origin se distance use karo."

Error hai origin — tumhe axis line tak ki perpendicular distance use karni chahiye. Axis ke upar door baitha mass ka bada position vector hota hai lekin

r = 0

hota hai.

I = \sum m_{i} r_{i}

, mass times distance, sab sum kar lo."

Square missing hai.

r

mein linear woh formula first moment (centre of mass) hai, inertia nahi;

I

r^{2}

chahiye kyunki woh kinetic energy mein

v^{2}

se aata hai.

"Kyunki

K E_{rot} = \frac{1}{2} I ω^{2}

hai, moment of inertia sirf tab exist karta hai jab object spin kare."

I

sirf geometry se define hota hai (

\sum m_{i} r_{i}^{2}

); kinetic energy ka formula

I

use karta hai lekin use create nahi karta. Ek stationary object ka bhi ek well-defined

I

hota hai.

"Skater speed up hoti hai kyunki arms andar karne se use zyada energy milti hai."

Energy yahan naive sense mein conserve nahi hoti — woh arms khinchte waqt work karti hai. Speed-up angular momentum

L = I ω

constant rehne se hota hai: chhota

I

zyada bada

ω

force karta hai.

I

ke units

kg m

hain kyunki yeh mass times a distance hai."

Distance squared hai, isliye units ==

kg m^{2}

== hain, jo definition mein

r^{2}

se match karte hain.

"Kyunki

I

ek scalar hai, object ko chahe kaise bhi ghuma lo, yeh same rehna chahiye."

Scalar (fixed axis ke liye) matlab "ek single number, koi direction nahi," lekin woh number phir bhi chosen axis ke saath badalta hai — scalar ka matlab axis-independent nahi hota.

"Same mass aur radius ki ek ring aur ek solid disk same

ω

par same rotational KE store karti hain."

Alag

I

matlab alag KE. Ring ka

I = M R^{2}

hai, disk ka

I = \frac{1}{2} M R^{2}

, isliye equal

ω

par ring double rotational energy store karti hai.

Why questions

"Kya" nahi, "kyun" ka answer do.

Distance squared kyun hai, linearly kyun nahi aata?

Yeh inherited hai: kinetic energy

\frac{1}{2} m v^{2}

hai (ek

v^{2}

), aur rotation ke liye

v = ω r

hai, isliye

v^{2} = ω^{2} r^{2}

—

r

par square seedha

v

par square se aata hai.

r_{i}

kisi point tak straight-line distance ki jagah axis tak perpendicular distance kyun honi chahiye?

Har particle ek circle mein travel karta hai jiska radius axis tak uski perpendicular reach hai; us circle ka radius uski speed

v = ω r

set karta hai, isliye sirf perpendicular distance kinetic energy control karti hai.

Door wali mass "double-extra" kyun "count" hoti hai?

r

distance par ek particle

ω r

speed se move karta hai, aur uska KE

(ω r)^{2}

ke hisaab se jaata hai — isliye do guna door hona matlab do guna fast move karna aur use square karna, jo spin ke resistance ko chaar guna de deta hai.

I

ko "mass ka rotational analogue" kyun kaha jaata hai?

K E = \frac{1}{2} I ω^{2}

mein yeh bilkul wahi jagah baithta hai jahan

m

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

mein baithta hai; yeh measure karta hai ki spin change karna kitna mushkil hai, jaise mass measure karta hai ki straight-line motion change karna kitna mushkil hai.

Same object ke kai alag-alag moments of inertia kyun ho sakte hain?

Kyunki

I

ek axis ke relative define hota hai, aur axis badalne par har

r_{i}

badal jaati hai; mass distribution same rehta hai lekin naye axis ke relative uska spread alag hota hai.

Derivation mein

\frac{1}{2} ω^{2}

ko sum se bahar kyun kheench sakte hain?

"Rigid" ka matlab hai ki har particle same

ω

share karta hai, isliye

ω

saare terms mein common factor hai aur bahar factor ho sakta hai, chhod ke pure geometric sum

\sum m_{i} r_{i}^{2}

Edge cases

Definition ko uski limits tak push karo.

Saari mass axis par hai.

I

kya hai?

Har

r_{i} = 0

hai, isliye

I = \sum m_{i} \cdot 0 = 0

— axis se kuch bhi off nahi hai swing karne ke liye, isliye body us particular rotation ke liye koi resistance nahi offer karti.

Ek single point mass hai, aur axis seedha uske through jaati hai.

I

kya hai?

I = 0

r = 0

ke saath akela term vanish ho jaata hai, jo intuition se match karta hai ki axis par ek point bas apni jagah spin karta hai.

Do axes parallel hain lekin alag hain; kya same mass distribution same

I

de sakti hai?

Sirf tab agar do axes mass se sahi tarike se equidistant hoon — generally nahi. Parallel axis theorem dikhata hai ki

I

centre of mass ke through axis ke baare mein sabse chhota hota hai.

Jab axis ek fixed mass lump se infinitely door push hoti hai,

I

ka kya hota hai?

Har

r_{i} \to \infty

, isliye

I \to \infty

— door axis ek tiny mass ko bhi ek enormous circle sweep karaata hai, isliye spin karna infinitely mushkil ho jaata hai.

Do point masses ko ek massless rigid rod connect karta hai. Kya rod

I

mein contribute karta hai?

Nahi — zero mass ke saath, har rod element

0 \cdot r^{2} = 0

deta hai; sirf do point masses (

m_{1} r_{1}^{2} + m_{2} r_{2}^{2}

) count hoti hain.

Kya

ω

I

ki value mein kahin appear hota hai?

Nahi.

I = \sum m_{i} r_{i}^{2}

mein sirf masses aur distances hain;

ω

kinetic energy aur angular momentum mein aata hai, lekin

I

khud spin-independent hai.

Kya do bahut alag dikhne wale objects ka kisi axis ke baare mein identical

I

ho sakta hai?

Haan —

I

saari mass distribution ko ek number mein collapse kar deta hai, isliye ek compact heavy object aur ek spread-out lighter object same

\sum m_{i} r_{i}^{2}

produce kar sakte hain.

Connections

Kinetic energy ½mv² — woh $v^{2}$ jo $r^{2}$ force karta hai
Rotational kinetic energy ½Iω² — jahan $I$ mass ka role play karta hai
Angular velocity ω — ek rigid body mein common factor
Centre of mass (first moment Σmr) — $r$ mein linear wala impostor
Parallel axis theorem — $I$ axis ke saath kaise shift hota hai
Angular momentum L = Iω — skater kyun speed up hoti hai
Torque τ = Iα — rotation ke liye Newton ka 2nd law