Foundations — Inertia tensor — principal axes, principal moments

2.1.22 · D1 · Physics › Analytical Mechanics › Inertia tensor — principal axes, principal moments

Yeh ek prerequisites page hai. Pehle parent note padh sako, uske liye tumhe har woh symbol khud se samajhna hoga jo woh tumhare samne rakhta hai. Hum ek-ek karke jaate hain, har ek kisi picture se anchor hai, aur har ek pichle par build karta hai.

0. Bilkul shuruat: ek rigid body

Socho ek eent jo lakhon chhote mass-dots se glue karke bani hai. Agar eent $10°$ ghoomti hai, toh har dot $10°$ ghoomta hai. Yahi ek fact hai jo poori theory ko possible banata hai — dekho Rigid body rotation.

Neeche ki figure yeh dikhati hai: faint dots "pehle" wali eent hain, bright dots "baad" wali. Yellow aur green arrows dekho — ek corner dot aur ek inner dot usi angle se fixed red origin ke around ghoomte hain. Kuch stretch nahi hota; sirf orientation badlti hai.

1. Position vector $\vec{r}$, basis vectors, aur length $r$

Upar wala chhota arrow $\stackrel{⃗}{}$ hamesha matlab hai "yeh ek vector hai — iska ek direction hai, sirf size nahi."

Yeh topic ko kyun chahiye: inertia tensor build hota hai is se ki har particle kahan baitha hai, yaani uske components $x,y,z$ se. Koi positions nahi, toh koi tensor nahi.

2. Mass: $m_a$, sum $\sum$, density $\rho$, aur integral $\int dm$

${\sum }_a{m}_a=m_1+m_2+m_3+\cdots =\text{total mass.}$

Ek smooth solid body mein infinitely many infinitely-small chunks hote hain. Infinitely many chhote pieces jodna exactly wahi kaam hai jo ek integral karta hai. Mass ka ek "infinitely small chunk" samajhne ke liye, hume ek aur idea chahiye: density.

Yeh topic ko kyun chahiye: $\mathbf{I}$ ki entries mass-weighted sums hain coordinate products ki. Ek solid body ke liye inhe $\rho dV$ par volume integrals ke roop mein compute karte hain — exactly aise hi parent ke cube example mein $\frac{1}{6}M{L}^2$ aata hai.

3. Angular velocity $\vec{\omega }$ — spin arrow

Figure mein disc ghoomti dikhti hai (green circulation arrows) jabki single yellow $\vec{\omega }$ arrow axis ke upar seedha khada hai — uski direction hai axis, uski length hai rate.

Yeh topic ko kyun chahiye: $\vec{\omega }$ inertia tensor ka input hai. $\vec{L}=\mathbf{I}\vec{\omega }$ padhte hain "spin arrow daalo andar, momentum arrow niklo bahar."

4. Cross product $\vec{A}\times \vec{B}$ — geometry aur components

Do arrows ko aisa multiply kiya ja sakta hai ki ek teesra arrow milta hai. Hume yeh do baar chahiye: velocity ke liye ($\vec{v}=\vec{\omega }\times \vec{r}$) aur angular momentum ke liye ($\vec{L}=\vec{r}\times m\vec{v}$).

Figure mein, blue $\vec{A}$ aur green $\vec{B}$ page mein hain; yellow shaded parallelogram unka $|\vec{A}||\vec{B}|\sin \theta$ area hai; red dot-in-a-circle hai $\vec{A}\times \vec{B}$ jo seedha page ke bahar point kar raha hai — dono ke perpendicular.

5. Dot product $\vec{A}\cdot \vec{B}$

Yeh topic ko kyun chahiye: $\mathbf{I}$ ki derivation mein term $\vec{r}(\vec{r}\cdot \vec{\omega })$ aata hai. Dot $\vec{r}\cdot \vec{\omega }$ ek number mein collapse hota hai jo phir $\vec{r}$ ko re-scale karta hai.

6. Angular momentum $\vec{L}$ — spin ka "ghoomne ki matra"

Yeh topic ko kyun chahiye: $\vec{L}$ inertia tensor ka output hai. Poora topic is sawaal ka jawab dene ke liye hai: spin $\vec{\omega }$ diya hua hai, toh $\vec{L}$ kya hai?

7. Tensor components $I_{ij}$ — general formula, pehle se

Parent note ek $3\times 3$ grid $I_{xx},I_{xy},I_{yz}$, wagera entries se bharta hai. Yahan exactly hai ki har label ka kya matlab hai, derivation mein milne se pehle bhi. Do subscripts $i$ aur $j$ mein se har ek axes $x,y,z$ mein se ek ko represent karta hai; $I_{ij}$ entry hai row $i$, column $j$ mein.

Us ek formula ko unpack karne se har woh entry milti hai jo parent use karta hai:

8. Matrix aur matrix–vector multiply

9. Diagonal, symmetric, identity — teen matrix shapes

10. Eigenvalue, eigenvector — special arrows

11. Prerequisite map

Equipment checklist

Self-test: kya tum parent note kholne se pehle har sawaal ka jawab de sakte ho?

Jab har line easy ho jaaye → jao parent note par.