Moment of inertia I = Σmᵢrᵢ² — concept: Foundations

1.5.5 · D1 · Physics › Rotational Mechanics › Moment of inertia I = Σmᵢrᵢ² — concept

Is page pe assume kiya gaya hai ki tumne kuch bhi nahi dekha. Isse pehle ki hum $I = \sum_{i} m_{i} r_{i}^{2}$ ko zor se padh sakein, humein iske har symbol ko earn karna hoga: summation sign, chhota subscript $i$ , $m_{i}$ aur $r_{i}$ ka matlab, "perpendicular distance to an axis" dikhta kaisa hai, aur woh do tools ( $v = ω r$ aur $K E = \frac{1}{2} m v^{2}$ ) jinpar derivation tikka hua hai. Hum inhe ek ek karke build karte hain.

1. "Axis of rotation" kya hota hai? (woh picture jis par sab kuch tika hai)

Distances se pehle, mass se pehle, humein woh cheez chahiye jisse hum distance lete hain.

Figure — Moment of inertia I = Σmᵢrᵢ² — concept

Figure dekho. Violet vertical line axis hai. Blue dot seedha uske upar hai — jab disk spin karta hai, woh dot kabhi nahi hilta. Magenta dot side mein hai — woh ek circle mein ghoomta hai. Orange dot aur bhi door hai, toh uska circle bada hai.

Topic ko yeh kyun chahiye: moment of inertia hamesha ek chosen axis ke baare mein hota hai. Line badlo, aur har distance badal jaata hai, toh $I$ bhi badal jaata hai. Pehle axis point kiye bina "moment of inertia" hota hi nahi.

2. Mass $m$ — "kitna stuff hai" ka number

Picture: $m$ ko ek bag of marbles ke weight ki tarah socho — isse koi fark nahi padhta bag kahan hai, bas kitna bhara hua hai.

Topic ko yeh kyun chahiye: mass woh raw "amount of stuff" hai jo motion ko resist karta hai. Spinning-resistance ko ordinary mass se shuru karna padta hai, phir geometry ke zariye modify karna padta hai.

3. Subscript $i$ aur bahut saare particles $m_{i}$

Real objects ek lump nahi hote — socho ki body ko chhote chhote tukdon mein kaato aur unhe $1, 2, 3, \dots$ label karo

Picture: ek body chhote chhote colored balls mein dotted, har ek ka apna tag.

Topic ko yeh kyun chahiye: ek object ke alag alag tukde axis se alag alag distances par hote hain, toh humein unhe alag alag treat karna padta hai — isliye hum har ek ko $i$ se tag karte hain.

4. Summation sign $\sum$

Topic ko yeh kyun chahiye: ek object mein hazaron tukde hote hain. Har " $+$ " likhna ek poora page bhar dega. $\sum$ humein "sab tukdon par add karo" ek hi stroke mein kehne deta hai.

5. Perpendicular distance $r_{i}$ — sabse subtle wala

Yeh woh symbol hai jo students aksar galat samajhte hain, isliye hum ise apna figure dete hain.

Figure dhyan se padho:

Violet line axis hai.
Orange dashed segment mass se perpendicular (right angle par, chhota square se dikhaya gaya) axis par girta hai. Iski length $r_{i}$ hai — woh circle ka radius jo woh mass sweep karega.
Grey dotted line woh (lambi) distance hai origin tak. Woh use karne ki galat cheez hai.

Topic ko yeh kyun chahiye: ek tukde ka circle ka size — kitna door use swing karna padta hai — exactly yahi perpendicular distance hai. Yahi karta hai door wale mass ko spin karna itna mushkil.

6. Angular velocity $ω$ aur link $v = ω r$

Spinning-speed ko ordinary speed se connect karne ke liye, humein ek aur idea chahiye.

Ab key bridge. Perpendicular distance $r$ par ka ek tukda $r$ radius ke circle par ghoomta hai. Ek second mein poori body angle $ω$ ghoomti hai, toh tukda $ω r$ length ka arc travel karta hai. Woh arc length per second uski ordinary straight-line speed $v$ hai:

Figure usi spinning arm par do dots dikhata hai: andar wala (magenta) chhota circle dheere trace karta hai, bahar wala (orange) bada circle tezi se — lekin dono ek turn usi time mein complete karte hain, toh unka $ω$ same hai jabki unka $v$ alag hai.

Topic ko yeh kyun chahiye: yahi exact jagah hai jahan distance $r$ physics mein ghus jaata hai. Poori kahani ke liye dekho Angular velocity ω.

7. Kinetic energy $K E = \frac{1}{2} m v^{2}$ — woh foundation jis par hum trust karte hain

Picture: ek fast heavy truck mein bahut zyada $K E$ hoti hai; ek slow marble mein almost kuch nahi. " $\frac{1}{2}$ " ek bookkeeping constant hai, aur $v^{2}$ kehta hai energy speed ke square ke saath badhti hai.

Topic ko yeh kyun chahiye: yeh woh solid linear fact hai jis par poori derivation khadi hai. Jab hum $v = ω r$ ko is $v^{2}$ mein plug karte hain, square $r$ par padta hai — aur exactly isliye $I$ mein $r^{2}$ hota hai naa ki $r$ . Dekho Kinetic energy ½mv².

Recall Kyun

$r$ par square (sirf $r$ nahi) inevitable hai Kyunki $K E$ mein $v^{2}$ hai, aur $v = ω r$ matlab $v^{2} = ω^{2} r^{2}$ . Square inherited hai — yeh kabhi free choice nahi tha. Isse compare karo Centre of mass (first moment Σmr) se, jo $r$ ko first power mein use karta hai kyunki woh ek alag sawaal se aata hai (balance point, energy nahi).

8. Symbols ko saath jodhna

Ab — aur sirf ab — parent formula ka har symbol earn ho chuka hai:

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

Ise zor se padhna, sab kuch jo humne build kiya uske saath:

Har part kahan se aaya:

$\sum_{i}$ — sab tukdon par add karo (§4)
$m_{i}$ — har tukde ki mass (§3)
$r_{i}$ — chosen axis tak perpendicular distance (§5)
square — $K E$ ke $v^{2}$ se $v = ω r$ ke zariye inherit kiya (§6, §7)

Parent note phir $I$ ko Rotational kinetic energy ½Iω², Angular momentum L = Iω, Torque τ = Iα mein feed karta hai, aur ise Parallel axis theorem ke saath refine karta hai.

Prerequisite map

Equipment checklist

Test karo khud ko — daayein side cover karo aur reveal karne se pehle har ek ka jawab do.

Axis of rotation kya hai, ek sentence mein?

Woh fixed straight line jiske around ek body spin karta hai; uske upar ke points nahi hilte, baaki sab uske baare mein circles sweep karte hain.

m

ka matlab kya hai aur uski unit kya hai?

Mass — matter ki miqdar — kilograms (

kg

) mein measure ki jaati hai.

m_{i}

mein subscript

i

kya hai?

Ek counter jo body ke har tukde ko label karta hai: piece 1, piece 2, aur aage isi tarah.

\sum_{i} x_{i}

ko plain words mein expand karo.

x_{1} + x_{2} + x_{3} + \dots

add karo, har piece

i

ke liye ek term.

r_{i}

exactly kya hai?

Piece

i

se axis line tak perpendicular (right-angle) distance — origin se distance NAHI.

Ek mass axis par seedha baith ke far ho lekin line par hi ho — uska

r_{i}

kya hai?

Zero, toh woh

I

mein kuch contribute nahi karta.

ω

kya hai aur ek rigid body ke sab tukde kya share karte hain?

Angular velocity

rad/s

mein; har tukda same

ω

share karta hai.

v

aur

ω

ko link karne wala bridge equation batao.

v = ω r

— bada perpendicular distance matlab bada straight-line speed.

Kinetic energy likho aur uska crucial feature batao.

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

; crucial feature

v^{2}

hai (energy speed ke square ke saath badhti hai).

I

mein

r

square kyun aata hai?

Kyunki

K E

mein

v^{2}

hai aur

v = ω r

, toh

v^{2} = ω^{2} r^{2}

— square

r

par padta hai.