Parallel axis theorem — I = I_CM + Md² — proof: Foundations

1.5.6 · D1 · Physics › Rotational Mechanics › Parallel axis theorem — I = I_CM + Md² — proof

Parent proof follow karne se pehle, tumhe uska alphabet already pata hona chahiye. Neeche, har symbol ke saath plain words → ek picture → yeh topic ko kyun chahiye — is order mein taaki har naya symbol sirf pehle wale pe hi lean kare.

1. Ek rigid body, aur use pieces mein kaatna

Kisi bhi shape ke saath physics karne ke liye, hum pretend karte hain ki woh tiny bricks se bani hai.

Figure — Parallel axis theorem — I = I_CM + Md² — proof

Saare chunk masses add karo toh poora recover ho jaata hai: $\sum_{i} m_{i} = M$

2. Ek chunk ko locate karna: coordinates $x_{i}, y_{i}$

Yeh batane ke liye ki chunk kahan hai, hum ek address system chahiye.

3. Perpendicular distance $r_{i}$ aur hum use square kyun karte hain

Address $(x_{i}, y_{i})$ pe Pythagoras rule se, jab axis origin se guzarti hai: $r_{i}^{2} = x_{i}^{2} + y_{i}^{2}$

4. Moment of inertia $I$ : show ka star

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

Is quantity ki poori kahani ke liye dekho Moment of inertia — definition.

5. Center of mass (CM): balance point

CM ka address saare chunk addresses ka mass-se-weighted average hai: $x_{C M} = \frac{\sum _{i} m _{i} x _{i}}{M}, y_{C M} = \frac{\sum _{i} m _{i} y _{i}}{M}$

Ise compute karne ke liye dekho Center of mass — definition and computation.

6. Parallel axes aur distance $d$

Agar doosri axis $x y$ -plane mein address $(a, b)$ par pierce karti hai jabki CM-axis origin par hai, tab Pythagoras se: $d^{2} = a^{2} + b^{2}$

7. Yeh foundations theorem ko kaise feed karti hain

Upar se neeche padho: chunks se mass aur addresses milte hain → addresses se distances milti hain → distances se $I$ milta hai → CM cross terms ko khatam karta hai → parallel-axis geometry $d$ supply karta hai → theorem nikal aata hai.

8. Ek chhota worked sanity-check

9. Yeh tools aage kahan dikhenge

Radius of gyration — $I = M k^{2}$ repackage karta hai; same $I$ aur $M$ chahiye.
Rotational kinetic energy — $I$ ko mass ke spin analogue ki tarah use karta hai.
Perpendicular axis theorem — $\sum m_{i} r_{i}^{2}$ par bana ek aur axis-shifting shortcut.
Physical pendulum — parallel-axis theorem ka ek direct customer.
Poore proof par wapas: parent (Hinglish).

Equipment checklist

Recall Self-test: kya tum har ek ko ek plain sentence mein explain kar sakte ho?

Rigid body kya hai? ::: Ek solid jinke parts kabhi ek doosre ke relative move nahi karte — yeh ek frozen shape ki tarah poori tarah spin karta hai. $m_{i}$ ka kya matlab hai? ::: Body ko jis $i$ -we tiny chunk mein kaata gaya hai uski mass — uski kalpna ki gayi hai. $\sum_{i}$ kya karta hai? ::: Har chunk $i$ ke liye ek quantity add karta hai — bahut saare plus-signs ki chain ka shorthand. $M$ kya hai? ::: Total mass, $\sum_{i} m_{i}$ — jo scale read karta hai. $x_{i}, y_{i}$ kya represent karte hain? ::: Ek chosen origin se chunk $i$ ka sideways aur upward address. $z_{i}$ ko kyun ignore kar sakte hain? ::: Ek vertical axis tak ki distance sirf sideways position par depend karti hai, axis ke along height par nahi. $r_{i}$ kya hai aur square kyun? ::: Ek chunk se axis tak ki perpendicular distance; $r_{i}^{2} = x_{i}^{2} + y_{i}^{2}$ , aur inertia square use karta hai isliye door ki mass bahut zyada count hoti hai. Moment of inertia $I$ kya hai? ::: Ek number, $\sum_{i} m_{i} r_{i}^{2}$ , jo measure karta hai ki body ko ek axis ke baare mein start ya stop karna kitna mushkil hai. $\int r^{2} d m$ kya hai? ::: Ek smooth body ke liye $\sum m_{i} r_{i}^{2}$ ka continuous version — same idea, infinitely baareek chunks. Center of mass kya hai? ::: Mass-weighted average position — woh point jahan body balance hoti hai. Proof mein $\sum m_{i} x_{i}$ aur $\sum m_{i} y_{i}$ kyun vanish ho jaate hain? ::: Kyunki origin CM par rakha jaata hai, jahan mass-weighted position average zero hoti hai. Parallel axes kya hain? ::: Do lines jo same direction mein point karti hain aur kabhi nahi miltin. $d$ kya hai? ::: Do parallel axes ke beech ki perpendicular (sabse chhoti, right-angle) distance.