Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions: Foundations

1.5.7 · D1 · Physics › Rotational Mechanics › Perpendicular axis theorem — I_z = I_x + I_y — proof, restri

Kuch bhi prove karne se pehle, tumhe sab kuch padh paana chahiye. Neeche har symbol, picture, aur idea hai jis par parent note tika hua hai, is tarah order kiya gaya hai ki har ek apne pehle waale se bana ho.

1. Space mein ek point: coordinates $(x, y, z)$

Picture: teen arrows ek corner par milte hue, jaise ek room ka corner jahan do deewarein floor se milti hain.

Topic ko yeh kyun chahiye: poora theorem is baare mein hai ki mass ka har tukda kahan baithta hai. Positions ko name karne ka tarika na ho, toh hum koi bhi distance nahi likh sakte, aur distance hi yahan sab kuch ka dil hai.

Figure — Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions

2. Mass ka ek tiny tukda: $d m$

Picture: ek solid plate dots ke ek cloud mein dissolve ho gayi, har dot ek $d m$ .

Topic ko yeh kyun chahiye: real objects spread out hote hain, ek point par concentrated nahi. Pure body ko handle karne ke liye hum har crumb ko alag treat karte hain, phir sab ko add kar dete hain.

3. Crumbs ko add karna: integral $\int \dots d m$

Picture: dots ka cloud, har ek thodi si value carry karta hua, sab ek running total mein funnel ho rahe hain.

Topic ko yeh kyun chahiye: proof ki aakhri line mein do integrals ek mein add hote hain. Woh step sirf is splitting rule ki wajah se allowed hai.

4. Ek point se ek axis tak distance (Pythagoras)

Yeh page par sabse important picture hai. Yahan time lagao.

Iske liye Pythagoras kyun chahiye? Kyunki woh shortest distance find karne ke liye, hum point ke coordinates se ek right triangle banate hain aur hypotenuse padh lete hain.

Yeh tool kyun aur koi doosra kyun nahi: hum ek distance chahte hain jo horizontal aur vertical offsets se bani ho jo right angle par milte hain. Pythagoras exactly woh machine hai jo do perpendicular legs ko ek straight distance mein convert karta hai — koi doosra rule yeh kaam nahi karta.

Ab ise har axis par apply karo. Kisi point ki axis se kitni door hai yeh measure karne ke liye, tum doosre do coordinates use karte ho (woh jo us axis ke across hain):

Topic ko yeh kyun chahiye: yeh teen expressions proof ka poora skeleton hain. Step 2 ke baad sab kuch inhe plug in karna hai.

5. Moment of inertia: $I$

Plain words mein padho: "har crumb ke liye, uski axis tak distance lo, use square karo, crumb ke mass se multiply karo, aur yeh sab add kar do."

Distance square kyun hoti hai, aur perpendicular kyun: axis se door ek crumb jab spin karta hai toh ek bada circle sweep karta hai, isliye woh zyada resist karta hai — aur resistance radius ke square ke saath badhti hai, sirf radius ke saath nahi. Hum perpendicular distance use karte hain kyunki sirf spin-line se sideways reach matter karti hai; axis ke along slide karna us circle ko nahi badalta jo ek crumb travel karta hai.

Topic ko yeh kyun chahiye: theorem teen moments of inertia ke baare mein ek statement hai — $I_{z}$ , $I_{x}$ , $I_{y}$ . Yahi woh object hai jo hum compute karte hain. Poori kahani ke liye Moment of inertia — definition dekho.

6. Teen characters: $I_{z}$ , $I_{x}$ , $I_{y}$

Picture: coin analogy — $I_{z}$ ek fidget-top spin hai, $I_{x}$ aur $I_{y}$ do end-over-end flips hain jo ek dusre se right angles par hain.

Topic ko yeh kyun chahiye: yeh exact teen symbols hain $I_{z} = I_{x} + I_{y}$ mein. Ek baar jab tum jaante ho ki har ek "us axis ke around spin karne ki hardness" hai, toh boxed formula ek plain English sentence ki tarah padha jaata hai.

7. Symmetry — free shortcut

Picture: ek disc — ise rotate karo aur har diameter baaki sab ke jaisi dikhti hai, isliye koi bhi diameter doosre se "mushkil" nahi ho sakti.

Topic ko yeh kyun chahiye: lagbhag har worked example mein theorem ko $I_{x} = I_{y}$ ke saath combine kiya jaata hai ek line mein answer paane ke liye, bina kisi integration ke. Symmetry arguments in MOI dekho. Lekin dhyan raho — symmetry ek convenience hai, kabhi requirement nahi; theorem tab bhi hold karta hai jab rectangle ke liye $I_{x} \neq = I_{y}$ ho.

Foundations topic ko kaise feed karti hain

Upar se neeche padho: positions hume crumbs ko name karne deti hain; Pythagoras positions ko distances mein convert karta hai; integral crumbs add karta hai; saath mein yeh $I$ define karte hain; $I$ ki teen copies plus flat-body fact ( $z = 0$ ) plus summing rule theorem deti hai.

Equipment checklist

Right-hand side cover karo aur reveal karne se pehle har ek ka jawab do.

Teen numbers

(x, y, z)

tumhe kya batate hain?

Har ek teen mutually perpendicular axes ke along kitna chalna hai ek point tak pahunchne ke liye.

d m

ka kya matlab hai?

Mass ka ek tiny crumb, itna chhota ki ek single point par baith sake.

\int f d m

ka plain words mein kya matlab hai?

Body mein har mass crumb par value

f

ko add karo.

Woh linearity rule batao jo hume integrals combine karne deta hai.

\int (a + b) d m = \int a d m + \int b d m

Pythagoras' theorem batao.

Legs

p, q

aur hypotenuse

h

ke liye:

h^{2} = p^{2} + q^{2}

Point

(x, y, z)

z

-axis se perpendicular distance²?

x^{2} + y^{2}

Point

(x, y, z)

x

-axis se perpendicular distance²?

y^{2} + z^{2}

Moment of inertia ko integral ke roop mein define karo.

I = \int r_{⊥}^{2} d m

, jahan

r_{⊥}

axis tak perpendicular distance hai.

I

mein distance square kyun hoti hai?

Spinning ki resistance us radius ke square ke saath badhti hai jo ek crumb sweep karta hai.

"Planar body" ka coordinates mein kya matlab hai?

Har mass point ka

z = 0

hai; body

x y

-plane mein flat padi hai.

I_{x} = I_{y}

kab claim kar sakte ho?

Jab body dono in-plane axes ke baare mein identical dikhti ho (ek symmetry argument) — kabhi automatic assumption ke roop mein nahi.

Connections

1.5.07 Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions (Hinglish) — woh parent topic jiske yeh foundations serve karte hain.
Moment of inertia — definition — woh $\int r^{2} d m$ jo yahan poori tarah se bana hai.
Symmetry arguments in MOI — $I_{x} = I_{y}$ shortcut.
Parallel axis theorem — sister theorem; wohi distance-and-integral toolkit chahiye.
Radius of gyration — $I$ milne ke baad use repackage karta hai.