4.4.22
Ek double (ya triple) integral bas "chote-chote pieces ko add karo" hai. Kisi flat plate ya solid body ki har physical quantity ek hi recipe se banti hai:
Total quantity = ∬ R ( quantity per tiny piece ) d A tiny piece of mass ko add karne se pehle usme kya multiply karte hain :
1 mass .axis se distance se multiply karo → moment (pehli power) → centre of mass.axis se distance² se multiply karo → moment of inertia (doosri power).
Ek point mass easy hota hai: uski ek position hoti hai aur ek mass. Lekin ek real plate ka mass ek region R density ρ ( x , y ) spread out hota hai. Alag-alag points ki density alag hoti hai aur kisi bhi axis se distance bhi alag hota hai. Isliye hum ek akele number se kaam nahi chala sakte — humein plate ko infinitesimal tiles mein kaatna hoga, har tile ko ek point mass maanna hoga, aur sum (integrate) karna hoga.
Definition Surface density (areal density)ρ ( x , y ) ( x , y ) mass per unit area . Area d A d m = ρ ( x , y ) d A . ρ homogeneous (uniform) kehte hain.
KYA: total mass = saari tiny masses ka sum.
KAISE (derivation): R i ρ ( x i , y i ) Δ A i m = lim Δ A → 0 ∑ i ρ ( x i , y i ) Δ A i = ∬ R ρ ( x , y ) d A
Intuition "Moment" actually hota kya haiEk moment measure karta hai ki kitna mass kitni door ek line se hai — yeh leverage hai. Ek chhota mass door se ek bade mass ko paas mein balance kar sakta hai. Har tile contribute karta hai (uska mass) × (uska lever arm = axis tak ki doori).
Mass se divide KYU karte hain? Centre of mass balance point hota hai — woh akela location jahan tum poora mass rakh sako aur har axis ke baare mein same leverage mile.
Coordinate axes ke baare mein moments define karo. x y y x
Centre of mass KAISE milta hai: Balance condition — body ko ( x ˉ , y ˉ ) m m x ˉ = M y , m y ˉ = M x .
Moment of inertia rotation ke against resistance measure karta hai (rotational "mass"). Ek spinning tile ki kinetic energy 2 1 ( d m ) v 2 v = ω r = 2 1 ω 2 ( r 2 d m ) r 2 speed radius ke saath badhti hai aur energy speed² ke saath badhti hai . r 2 d m
Radius of gyration R g = I / m I
Worked example (A) Triangle ka Mass & CoM, variable densityPlate: triangle with vertices ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) ρ ( x , y ) = x + y
Mass. Region: 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 − x m = ∫ 0 1 ∫ 0 1 − x ( x + y ) d y d x Inner pehle KYU? Fixed x y ∫ 0 1 − x ( x + y ) d y = x ( 1 − x ) + 2 ( 1 − x ) 2 = 2 1 − x 2 Yeh step KYU? Bas y ∫ 0 1 2 1 − x 2 d x = 2 1 ( 1 − 3 1 ) = 3 1 . m = 3 1
M y = ∬ x ρ d A ∫ 0 1 − x x ( x + y ) d y = x ⋅ 2 1 − x 2 ∫ 0 1 2 x − x 3 d x = 2 1 ( 2 1 − 4 1 ) = 8 1 . x ˉ = m M y = 1/3 1/8 = 8 3 . ρ x , y y ˉ = 8 3
Worked example (B) Uniform rectangle ka Moment of InertiaRectangle 0 ≤ x ≤ a , 0 ≤ y ≤ b ρ I x = ρ ∫ 0 a ∫ 0 b y 2 d y d x = ρ ∫ 0 a 3 b 3 d x = 3 ρ a b 3 . Yeh step KYU? ∫ 0 b y 2 d y = b 3 /3 x m = ρ ab I x = 3 m b 2 . I y = 3 m a 2 I 0 = 3 m ( a 2 + b 2 ) .
Worked example (C) Polar ki power — quarter discFirst quadrant mein radius a ρ = d A = r d r d θ x 2 + y 2 = r 2 I 0 = ρ ∫ 0 π /2 ∫ 0 a r 2 ⋅ r d r d θ = ρ ⋅ 2 π ⋅ 4 a 4 = 8 π ρ a 4 . Extra r Polar coordinates mein ek tile ka area r d r d θ na ki d r d θ
Recall Compute karne se pehle predict karoRadius a guess karo : kya I 0 a 2 a 4
Forecast: I 0 = ∬ r 2 ρ d A d A = a 4 Verify: I 0 = ρ ∫ 0 2 π ∫ 0 a r 3 d r d θ = ρ 2 π 4 a 4 = 2 π ρ a 4 = 2 m a 2 m = ρ π a 2 ∝ a 4 ∝ a 2
Common mistake "Centre of mass =x ρ Kyun sahi lagta hai: uniform plate ke liye density cancel ho jaati hai, toh x ˉ Fix: jab ρ x ˉ = ∬ ρ d A ∬ x ρ d A
r d r d θ d r d θ Kyun sahi lagta hai: d x d y Fix: ek polar tile ek curved wedge hoti hai jiska area hota hai (radial length)× = d r × r d θ r
M x I x Kyun sahi lagta hai: dono "x-axis ke baare mein hain." Fix: M x y 1 I x y 2
I 0 = I x + I y Kyun sahi lagta hai: yeh ek neat clean formula hai. Fix: yeh perpendicular-axis theorem hai, sirf x y planar lamina ke liye valid hai. 3-D solids ke liye yeh fail karta hai.
"1, r r 2 — d m
1 → massr m r 2
Aur polar mein, extra r : "polar tiles ek r
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek pizza jisme cheese kahin moti hai aur kahin patli. Mass hai pizza ki total quantity — tum pizza ke chhote-chhote squares add karte ho. Balance point (centre of mass) woh jagah hai jahan tum pizza ko ek ungli par rakh sako bina girae — yeh heavy side ki taraf jhukta hai, isliye tum har square ko uski doori aur cheese ki moti jagah se weight karte ho. Moment of inertia hai ki pizza ko spin karna kitna mushkil hai: centre se door ki cheese spin ke against zyada ladhti hai nazdeek ki cheese se — aur "door" double-hard count hoti hai kyunki hum distance ko square karte hain. Har baar same recipe: squares mein kaato, har ek ko 1
Density aur area ke terms mein d m d m = ρ ( x , y ) d A
Ek lamina ke total mass ka formula? m = ∬ R ρ d A
Moment M x M x = ∬ R y ρ d A x = y
M y M y = ∬ R x ρ d A = x
Centre of mass coordinates? x ˉ = M y / m y ˉ = M x / m
Centroid aur centre of mass mein kya fark hai? Centroid = CoM jab density constant ho (purely geometric).
x ˉ = M y / m M x / m x ˉ y x M y
x I x = ∬ R y 2 ρ d A
Origin ke baare mein moment of inertia? I 0 = ∬ R ( x 2 + y 2 ) ρ d A = I x + I y
I Rotational KE = 2 1 ω 2 ∫ r 2 d m ∝ r ∝ v 2
Perpendicular axis theorem aur uski limitation? I 0 = I x + I y
Polar coordinates mein area element? d A = r d r d θ r
Radius of gyration ki definition? R g = I / m — woh distance jo same
I deti agar poora mass wahan baith jaaye.
Ek uniform rectangle (a × b I x I x = m b 2 /3 = ρ a b 3 /3
Radius a I 0 I 0 = 2 1 m a 2 = 2 π ρ a 4
Disc ka I 0 ∝ a 4 ∝ a 2
M x I x M x y 1 I x y 2
treat each tile as point mass
multiply by distance squared
Moment Mx = integral y rho dA
Moment My = integral x rho dA