Moment of inertia of - rod (about end, centre), disk, ring, sphere (solid, hollow), cylinder: Foundations

1.5.8 · D1 · Physics › Rotational Mechanics › Moment of inertia of - rod (about end, centre), disk, ring,

Parent note padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare saamne fenkta hai. Yeh page har ek ko zero se build karta hai, us order mein jo agli cheez ko samajhne deta hai. Yahan kuch bhi assume nahi kiya gaya ki tumne yeh topic pehle dekha hai.

0. Woh picture jisme sab kuch rehta hai

the parent topic ka har formula EK hi setup ke baare mein hai: ek solid body, aur space mein ek seedhi line jise axis kehte hain, jiske around body spin karti hai. Pehle woh picture apne dimag mein fix karo.

Figure — Moment of inertia of - rod (about end, centre), disk, ring, sphere (solid, hollow), cylinder

Topic ko axis ki zaroorat kyun hai? Kyunki "spin karna kitna mushkil hai" ka koi jawab nahi jab tak tum yeh na kaho ki kiske around spin karna hai. Wahi rod ek taraf se spin karna aasaan hai aur doosri taraf se mushkil. Axis woh reference hai jisse hum har distance measure karne wale hain.

1. Mass — $m$ aur $M$

Picture: ek bhaari bowling ball vs usi size ki beach ball — bowling ball mein zyada mass hai.
Topic ko isko kyun chahiye: moment of inertia mass ka rotational cousin hai. Cousin banane ke liye, pehle mass chahiye. Bada $M$ grand total hai; chhota $m_{i}$ $i$ -ve tiny chunk ka mass hai jab hum body ko kaatke pieces mein tod dete hain.

Chhota subscript $i$ bas matlab hai " $i$ -va piece" — piece number 1, 2, 3, … Hum body ko bahut saare pieces mein kaatke unhe label karte hain.

2. Axis se doori — $r$

Yeh chapter ka sabse important symbol hai, aur jisme log galti karte hain.

Picture: ek pencil (axis) seedha khada karo. Ek marble side mein pakad lo. $r$ woh hai ki marble pencil se sideways kitna door hai — slanted distance nahi, seedha across wala distance.
Perpendicular kyun? Jab body spin karti hai, har piece ek circle travel karta hai. Us circle ka radius exactly yahi perpendicular distance hai. Jo piece axis ke directly upar hai (chhota sideways gap) woh tiny circle banata hai; jo piece bahut door hai woh huge circle banata hai.

3. Angle aur angular speed — $θ$ , $ω$

Picture: ek clock ki suyi. $θ$ = suyi abhi kahan point kar rahi hai; $ω$ = suyi kitni tezi se sweep kar rahi hai.
Key fact jo body par sabhi share karte hain ek $ω$ : ek rigid body mein, har piece same time mein same angle ghoomta hai. Paas ke pieces aur door ke pieces sabka same $ω$ hota hai — lekin door wale ordinary speed mein zyada fast move karte hain, kyunki unka circle bada hota hai.

Woh aakhri sentence is poore chapter ka beej hai. "Zyada fast move karna" ko precise banao.

4. Linear speed $v$ , aur bridge $v = ω r$

Yeh formula kyun? Ek poore chakkar mein ek piece circumference $2 π r$ travel karta hai. Agar woh $ω / (2 π)$ chakkar per second karta hai, toh uski speed hai $ω / (2 π) \times 2 π r = ω r$ . $2 π$ cancel ho jaate hain — clean.
Picture: merry-go-round par rim wala bachcha ( $r$ bada) tezi se fenkta hai; pole ke paas wala bachcha ( $r$ chhota) barely hilta hai. Same $ω$ , alag $v$ , kyunki $v = ω r$ .
Topic ko isko kyun chahiye: yeh woh link hai jo spinning problem ko kinetic-energy problem mein convert karta hai, jahaan $r^{2}$ paida hota hai (agla section).

5. Kinetic energy — $K E = \frac{1}{2} m v^{2}$ — aur $r^{2}$ KAHAN SE AATA HAI

Ab poore chapter ka janam dekho. Ek piece $m_{i}$ lo distance $r_{i}$ par. Uski speed hai $v_{i} = ω r_{i}$ . Uski energy: $K E_{i} = \frac{1}{2} m_{i} v_{i}^{2} = \frac{1}{2} m_{i} (ω r_{i})^{2} = \frac{1}{2} m_{i} ω^{2} r_{i}^{2}$

Har piece ko add karo. $\frac{1}{2}$ aur $ω^{2}$ sabhi pieces ke liye same hain, toh unhe bahar nikalo: $K E = \frac{1}{2} ω^{2} (\sum_{i} m_{i} r_{i}^{2})$

Bracket $\sum_{i} m_{i} r_{i}^{2}$ exactly wahi hai jise hum $I$ naam dete hain. Is substitution ki poori kahani ke liye dekho Rotational Kinetic Energy.

6. Sum symbol $\sum$ aur integral $\int$

Picture: $\sum$ asli bricks stack karna aur unka weight add karna hai. $\int$ sand daalna hai — tum grains count nahi kar sakte, toh continuously add karte ho.
Topic ko $\int$ kyun chahiye: ek rod ya disk kuch beads nahi hai; yeh continuous matter hai. Ek continuous body mein $r^{2}$ add karne ke liye hum $\sum m_{i} r_{i}^{2}$ ko $\int r^{2} d m$ se replace karte hain, jahaan $d m$ ek infinitesimal sliver ka mass hai.

7. Density — $λ$ , $σ$ , $ρ$ — geometry ko $d m$ mein badalna

Integral karne ke liye hume sliver ka mass $d m$ body ki shape use karke express karna hoga. Density yahi kaam karti hai.

Picture: $λ$ = ek wire ke grams per centimetre; $σ$ = ek paper sheet ke grams per square centimetre; $ρ$ = ek metal block ke grams per cubic centimetre.
Topic ko unki kyun zaroorat hai: integral $\int r^{2} d m$ tab tak nahi ho sakta jab tak $d m$ ko kisi aisi coordinate ke terms mein na likha jaaye jis par integrate kar sako ( $d x$ , $d r$ , $d θ$ ). Density converter hai, aur yeh is chapter ki har standard body ke liye uniform (har jagah same) hai.

8. Shape symbols — $L$ , $R$

$R$ (body ki fixed property) ko $r$ se confuse mat karo ( $r$ ek sliver ki moving distance-to-axis hai, jo $0$ se $R$ tak sweep karti hai jab tum body ke across scan karte ho).

9. Do theorems jis par parent rely karta hai

Parent note apne answers ko do shortcuts se check karta hai. Tumhe inhe yahan prove karne ki zaroorat nahi, bas jaanna hai ki yeh kya claim karte hain.

Ek aur idea jo topic quietly use karta hai: Radius of Gyration, ek single distance $k$ jisme $I = M k^{2}$ — "agar sara mass $k$ par baith jaata, toh tumhe same $I$ milta."

Prerequisite map

Equipment checklist

Recall Kya tum ready ho? (answers dhako)

Axis of rotation kya hai, ek line mein? ::: Woh fixed seedhi line jiske around body spin karti hai; uske upar ke points nahi hilte. $r$ exactly kya measure karta hai? ::: Ek mass piece se axis tak ki perpendicular (shortest, right-angle) doori — centre tak NAHI. $r$ $I$ mein squared kyun hai? ::: Ek power $v = ω r$ se, doosri $\frac{1}{2} m v^{2}$ mein $v$ ko square karne se. $ω$ kya hai aur ek rigid body par isko kaun share karta hai? ::: Angular speed rad/s mein; ek rigid body ka har piece same $ω$ share karta hai. $v$ aur rotation ke beech ka bridge batao. ::: $v = ω r$ . $λ$ , $σ$ , $ρ$ ka matlab kya hai aur unka $d m$ batao? ::: mass/length ( $d m = λ d x$ ), mass/area ( $d m = σ d A$ ), mass/volume ( $d m = ρ d V$ ). End mein $M$ use karke density ko replace kyun karte hain? ::: Taaki final $I$ mein sirf measurable totals $M$ , $R$ ya $L$ rahe. $R$ aur $r$ mein kya fark hai? ::: $R$ body ka fixed radius hai; $r$ ek sliver ki running distance-to-axis hai, jo $0 \to R$ sweep karti hai. Parallel Axis Theorem batao. ::: $I = I_{c m} + M d^{2}$ , jahaan $d$ parallel axes ke beech ka gap hai. Axis naam liye bina $I$ compute kyun nahi kar sakte? ::: "Spin karna kitna mushkil hai" undefined hai jab tak tum yeh na kaho ki kiske around spin karna hai — axis har $r$ set karta hai.