Moment of inertia I = Σmᵢrᵢ² — concept

1.5.5 · Physics › Rotational Mechanics

Moment of inertia HAI kya?

Lock karne wale key points:

$r_{i}$ axis tak ki perpendicular distance hai — origin tak ki distance NAHI.
$I$ kisi object ki fixed property nahi hai; axis badlo, aur $I$ badal jaata hai.
Yeh hamesha positive hota hai ( $m r^{2}$ ka sum, dono squared/positive hain).

Distance squared kyun hoti hai? (First principles se derivation)

Hum $I = \sum m_{i} r_{i}^{2}$ ko diya hua nahi maante. Chalte hain ise build karte hain.

Step 1 — Wahan se shuru karo jis par hum already trust karte hain: kinetic energy. Speed $v$ se move karta ek single particle kinetic energy rakhta hai $K E = \frac{1}{2} m v^{2} .$ Yeh step kyun? Yeh solid linear mechanics hai — hum ise apni foundation ke roop mein use karte hain.

Step 2 — Linear speed ko rotational speed mein convert karo. Jab ek body angular velocity $ω$ ke saath ek axis ke baare mein rigidly rotate karti hai, toh har particle radius $r_{i}$ ke ek circle mein jaata hai. Us particle ki speed hoti hai $v_{i} = ω r_{i} .$ Yeh step kyun? Saare particles ek same $ω$ share karte hain (rigid body), lekin jo zyada door hai (bada $r_{i}$ ) woh zyada tez move karta hai. Yahi woh core hai jahan $r$ enter hota hai.

Step 3 — Saare particles ki kinetic energy add karo. $K E_{rot} = \sum_{i} \frac{1}{2} m_{i} v_{i}^{2} = \sum_{i} \frac{1}{2} m_{i} (ω r_{i})^{2} = \frac{1}{2} (\sum_{i} m_{i} r_{i}^{2}) ω^{2} .$ Yeh step kyun? $ω$ sabke liye common hai, isliye hum $\frac{1}{2} ω^{2}$ bahar nikaal lete hain. Bacha hua bracket mass-aur-geometry ki ek pure property hai.

Step 4 — Demand karo ki rotational formula linear wale jaisa dikhe. Hum chahte hain ki $K E_{rot} = \frac{1}{2} I ω^{2}$ , $K E = \frac{1}{2} m v^{2}$ ko mirror kare. Dono expressions ko match karne par force hota hai: $I = i \sum m_{i} r_{i}^{2}$ Yeh step kyun? $r$ par square kinetic energy mein $v$ par square se inherit hua hai, $v = ω r$ ke saath combine hokar. Yahi deep reason hai ki distance squared kyun hoti hai — koi arbitrary choice nahi.

Ise compute kaise karein — worked examples

Figure — Moment of inertia I = Σmᵢrᵢ² — concept

Forecast-then-Verify

Common mistakes (Steel-man + fix)

Recall Feynman: ek 12-saal ke bache ko samjhao

Socho ki ek ball string se baandhi hai aur tum use ghuma rahe ho. Agar string chhoti hai, toh ghoomna aasaan hai. Agar lambi string nikaal do, toh ball ko ghoomaana aur rokna dono bahut mushkil ho jaate hain. Moment of inertia bas yeh hai ki "yeh ghoomaana kitna mushkil hai." Bhaari cheezein ghoomana mushkil hoti hain — aur jo cheezein centre se door spread hoti hain woh extra mushkil hoti hain, kyunki door wale parts ko bahut bade circles mein bahut tezi se swing karna padta hai. Isliye "door wale" parts double-extra count hote hain (hum distance ko square karte hain).

Active-recall flashcards

#flashcards/physics

Point masses ke liye moment of inertia define karo.

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

, jahan

r_{i}

mass

m_{i}

ki axis se perpendicular distance hai. Units

kg m^{2}

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

mein

r_{i}

exactly kya hai?

Particle ki axis of rotation tak ki perpendicular distance (origin tak ki distance nahi).

Moment of inertia mein distance squared kyun hoti hai?

Kyunki

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

mein

v^{2}

hai, aur

v = ω r

hai, isliye

v^{2} = ω^{2} r^{2}

—

r

par square kinetic energy se inherit hua hai.

K E = \frac{1}{2} I ω^{2}

mein mass ki jagah kaunsi rotational quantity leti hai?

Moment of inertia

I

Kya

I

kisi object ki fixed property hai?

Nahi — yeh chosen axis par depend karta hai. Same object ki alag-alag axes ke liye alag-alag

I

hoti hai.

Bilkul axis par baitha ek mass

I

mein kitna contribute karta hai?

Zero, kyunki

r = 0

hai isliye

m r^{2} = 0

Agar kisi particle ki axis se distance double kar do, toh

I

kitne factor se badlegi?

4× (kyunki

r^{2}

Hoop vs disk (same

M

R

): central axis ke baare mein kiski

I

zyada hai, aur kyun?

Hoop ki (

M R^{2}

); uska saara mass max radius par hai, vs disk (

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

) jiska mass centre ke zyada paas hai.

Skater arms andar kheenchne par zyada tez kyun ghoomta hai?

Andar kheenchne se

r

kam hota hai,

I

kam hoti hai; angular momentum

L = I ω

conserved hai, isliye

ω

badh jaata hai.

Connections

Kinetic energy ½mv² — woh foundation jis se humne $I$ derive kiya
Rotational kinetic energy ½Iω² — direct application
Angular velocity ω — $v = ω r$ supply karta hai
Centre of mass (first moment Σmr) — contrast: $r$ mein linear hai, squared nahi
Parallel axis theorem — axis shift hone par $I$ kaise badlti hai
Angular momentum L = Iω — skaters tez kyun spin karte hain
Torque τ = Iα — rotation ke liye Newton ka 2nd law

Linear	Rotational
mass $m$	moment of inertia $I = \sum m_{i} r_{i}^{2}$
velocity $v$	angular velocity $ω$
$K E = \frac{1}{2} m v^{2}$	$K E = \frac{1}{2} I ω^{2}$