Applications — mass, centre of mass, moments of inertia: Foundations

4.4.22 · D1 · Maths › Multivariable Calculus › Applications — mass, centre of mass, moments of inertia

Koi bhi moment of inertia compute karne se pehle, tumhe notation bina atke padhna aana chahiye. Ye page har symbol ko ek ek karke introduce karta hai, ek aisi order mein jahan har naya concept pichle concept pe build hota hai — kabhi bhi kisi cheez ko define karne se pehle use nahi kiya jaata.

0. Stage: ek flat plate plane mein rakhi hai

Sab kuch ek plate pe hota hai — ek patla flat shape (ek "lamina"). Hum ise ek kaagaz pe flat rakhte hain aur uske through do number lines kheenchte hain: ek horizontal aur ek vertical.

Figure — Applications — mass, centre of mass, moments of inertia

Ye kyun chahiye. Baad mein aane wala har symbol ( $ρ$ , $d A$ , $M_{x}$ , $I_{y}$ …) ek aisa rule hai jo kisi point ka $x$ aur $y$ padhta hai, dono origin se measure kiye gaye. Agar tum picture mein origin, $x$ aur $y$ ko point nahi kar sakte, toh koi bhi formula kaam ka nahi. Tum in axes se phir miloge jab Double Integrals over General Regions mein integration limits set up karoge.

1. Region $R$ — wo shape jiske upar hum sum karte hain

Neeche diye gaye triangle ko dekho: $R$ poora shaded interior hai. Jab hum $\iint_{R}$ likhte hain, neeche ka chota $R$ hume bata raha hai "sirf un tiles ko add karo jo is shape ke andar hain."

Ye kyun chahiye. Ek real plate poora infinite plane nahi hai — ye ek bounded shape hai. $R$ us shape ko naam deta hai taaki integral ko pata ho kahan se shuru aur band karna hai. $R$ ko inequalities ke roop mein describe karna (jaise $0 \leq x \leq 1, 0 \leq y \leq 1 - x$ ) exactly wo skill hai jo Double Integrals over General Regions mein hai.

2. Tiny tile aur uska area $d A$

Ye poore subject ka dil hai. Hum ek spread-out plate ko ek saath handle nahi kar sakte, toh hum ise tiny rectangular tiles ki ek grid mein kaat dete hain.

"Element" word kyun? $d A$ building element hai — iint. Poori plate inhi lakhon bricks se bani hai. Integral wo machine hai jo sab bricks ko wapas add karti hai jab wo zero size mein shrink hoti hain.

3. Integral signs $\int$ aur $\iint$

$\iint_{R} (something) d A = lim_{tiles \to 0} \sum_{all tiles in R} (something at that tile) \times (tile area)$

Do signs kyun, ek kyun nahi? Ek plate two-dimensional hai: usse cover karne ke liye tum $x$ direction aur $y$ direction dono mein sweep karte ho. Har direction mein ek $\int$ aata hai. Unhe ek ke baad ek karna (inner phir outer) Double Integrals over General Regions ki poori method hai; ek solid body ke liye teen signs chahiye, yaani Triple Integrals.

4. Density $ρ$ — har tile kitni heavy hai

Greek letter $ρ$ (bolo "roh", curly-p letter) density ka hamaara naam hai.

Tab EK tile ka mass hai

$d m = ρ (x, y) d A ⟸ ρ (mass per area) \times d A (area of tile) .$

$ρ$ kyun chahiye. Iske bina, ek badi tile aur ek choti tile equally count hoti, aur ek heavy region ek light region ke barabar count hoti. Density hi hai jo geometry (area) ko physics (mass) mein convert karti hai.

5. Axis se distance — "lever arm"

Aakhri raw ingredient distance hai. Teen distances matter karti hain, aur har ek bas wo coordinate hai jo tumhare paas already hai.

$x^{2} + y^{2}$ kyun? Ye Pythagoras theorem hai: seedha hop us right triangle ka hypotenuse hai jiske do legs $x$ (across) aur $y$ (up) hain. Har leg ko square karo, add karo, square-root lo — diagonal milta hai. Upar ki picture exactly yahi triangle dikhati hai. (Squaring minus sign bhi hata deta hai, isliye $r$ hamesha $\geq 0$ hota hai.)

Sign kyun matter karta hai. Ek aisi plate socho jo $y$ -axis ke dono taraf hai, equal aur opposite. Agar hum har jagah $∣ x ∣$ use karte, toh dono halves add ho jaate aur balance point off-centre chala jaata — galat. Signed $x$ use karke, left half ke negative contributions right half ke positive ones se subtract hote hain, aur balance point correctly middle mein aata hai. Jab bhi koi region ek axis cross kare, ye yaad rakhna.

6. First power vs second power — ek line mein poora plot

Ab har symbol define ho chuka hai, teen recipes clearly padhi jaati hain. Sirf coordinate ki power dekho:

Kyunki do axes hain, moment aur inertia dono do labelled versions mein aate hain — ek per axis. Ye parent topic jo notation use karta hai, wo ab define hai taaki kabhi bhi bina announce ke na aaye:

Power $0$ ( $1$ se multiply karo) → mass $m$ . Tum bas cheezein count kar rahe ho.
Power $1$ (signed coordinate se multiply karo) → moment $M_{x}$ ya $M_{y}$ ; $m$ se divide karo balance point paane ke liye. Pehli power isliye kyunki balance leverage ke baare mein hai, jo linear hai — aur signed hai taaki opposite sides cancel ho sakein.
Power $2$ (coordinate² se multiply karo) → moment of inertia $I_{x}, I_{y}, I_{0}$ . Doosri power isliye kyunki spin energy speed² ke saath badhti hai, aur speed radius ke saath badhti hai. Us $\frac{1}{2} I ω^{2}$ ki kahani ke liye Rotational Kinetic Energy and Angular Momentum dekho.

Balance point ke liye $m$ se divide kyun? Moment $M_{y}$ "total sideways leverage" hai. Balance point $(\overset{x}{ˉ}, \overset{y}{ˉ})$ pe rakha gaya ek single point mass $m$ wahi leverage reproduce karna chahiye, toh $m \overset{x}{ˉ} = M_{y}$ , jisse $\overset{x}{ˉ} = M_{y} / m$ milta hai. Jab $ρ$ constant ho toh ye cancel ho jaata hai aur balance point pure-geometry centroid ban jaata hai — jo Centroids and the Pappus Theorems ka subject hai.

7. Bar notation $\overset{x}{ˉ}, \overset{y}{ˉ}$

8. Polar tiles ke baare mein ek baat (preview)

Kabhi kabhi ek round region polar coordinates mein bahut aasaan hoti hai, jahan ek point ko $(x, y)$ se nahi balki origin se uski distance aur ek direction se name kiya jaata hai.

In coordinates mein tile rectangle nahi balki ek curved wedge hoti hai, aur uska area hai

$d A = r d r d θ (extra r mandatory hai),$

jahan $d r$ bahar ki taraf ek tiny step hai aur $d θ$ ek tiny turn. Woh extra $r$ Jacobian factor hai; poori kahani Polar Coordinates and the Jacobian mein hai aur uska general version Change of Variables and Jacobians mein. Abhi bas $r$ aur $θ$ symbols note kar lo taaki baad mein shock na lage.

Prerequisite map

Ise bottom-up padho: origin axes ko anchor karta hai, axes tumhe points deti hain, points region aur lever arms banate hain, region tiles mein kata jaata hai, density tile-area ko tile-mass mein convert karti hai, integral unhe add karta hai, aur coordinate ko jo power raise karte ho woh select karta hai — mass vs moment vs inertia.

Equipment checklist

Khud test karo — right side cover karo aur har cheez zor se jawab do.

Origin kya hai, aur ise kisliye use kiya jaata hai?

Point

(0, 0)

jahan axes cross karti hain; har coordinate aur distance

r

isi se measure hoti hai.

Symbol

ρ (x, y)

ka plain words mein kya matlab hai?

Point

(x, y)

pe mass per unit area; ye point se point pe alag ho sakta hai.

d A

kya hai, aur ye zero mein kyun shrink karna chahiye?

Ek tiny tile ka area; zero mein shrink karne se summed answer blocky/approximate ki jagah exact ban jaata hai.

d m

ρ

aur

d A

ke terms mein likho.

d m = ρ (x, y) d A

— ek tile ka mass.

Double integral mein do integral signs kyun use hote hain?

Ek plate 2-D hai; tum

x

aur

y

dono directions mein tiles sweep karte ho, ek direction mein ek

\int

(x, y)

pe tile ki

x

-axis se perpendicular distance kya hai?

Size

∣ y ∣

, kyunki tum seedha horizontal axis tak jaate ho.

(x, y)

se origin tak distance kya hai, aur kyun?

r = x^{2} + y^{2}

, Pythagoras se — ye

x

aur

y

legs wale right triangle ka hypotenuse hai.

Moments mein SIGNED coordinate

x

kyun use hota hai lekin inertias mein

x^{2}

Signed

x

se left (

x < 0

) aur right (

x > 0

) mass balance ke liye cancel ho sakta hai; squaring sign mita deta hai kyunki spin-resistance dono sides mein same hoti hai.

M_{y}

aur

M_{x}

define karo.

M_{y} = \iint_{R} x d m

(

y

-axis ke baare mein moment),

M_{x} = \iint_{R} y d m

(

x

-axis ke baare mein moment).

In moments se balance point kaise nikaalte hain?

\overset{x}{ˉ} = M_{y} / m

aur

\overset{y}{ˉ} = M_{x} / m

: ek point mass

m

same moments reproduce karna chahiye.

I_{x}

I_{y}

, aur

I_{0}

define karo.

I_{x} = \iint y^{2} d m

I_{y} = \iint x^{2} d m

I_{0} = \iint (x^{2} + y^{2}) d m

—

x

-axis,

y

-axis, aur origin ke baare mein spinning ke against resistance.

Mass, moment, aur moment of inertia paane ke liye

d m

ko kisse multiply karte hain?

1

se, signed coordinate se, aur coordinate squared se.

\overset{x}{ˉ}

mein overbar kya signify karta hai?

Woh ek special solved value — balance-point coordinate, running variable nahi.

Polar coordinates mein

r

aur

θ

kya hain, aur

d A

kya hai?

r

= origin se distance,

θ

= positive

x

-axis se anticlockwise angle (radians);

d A = r d r d θ

, extra

r

Jacobian factor hai.

Ye the parent topic ke liye mana gaya poora toolkit hai. Jab checklist ki har line automatic ho jaaye, toh wahan ke formulas sentences jaisi padhte hain.