Centre of mass — definition for system of particles: Foundations

1.4.9 · D1 · Physics › Momentum & Collisions › Centre of mass — definition for system of particles

Yeh parent topic ke liye toolbox page hai. Hum har woh tool uthayenge jo parent use karta hai, usse ache se dekhenge, batayenge ki iska matlab kya hai, uski picture draw karenge, aur explain karenge ki topic uske bina aage kyon nahi badh sakta. Yahan kuch bhi assume nahi kiya gaya ki aapne pehle yeh notation dekhi hai.

0. Woh scene jo hum describe kar rahe hain

Kisi bhi symbol se pehle, us physical situation ki picture banao jiske baare mein poora chapter hai.

Hamare paas kaafi chote matter ke blobs hain (inhe particles kahte hain) jo space mein float kar rahe hain. Har ek kisi location par hai aur har ek ka kuch weight hai. Yahi poora raw material hai. Neeche har symbol sirf is picture ke ek piece ka compact naam hai.

1. Ek "particle" aur uska label $i$

Picture: upar wali figure mein, har blob ek numbered tag pehne hua hai. $i$ sirf us tag se zyada kuch nahi hai.

Topic ko iske liye kyun zaroorat hai: hamare paas kaafi particles hain aur hume "us specific wale ki mass" ke baare mein baat karni hai. Label ke bina hum unhe alag nahi kar sakte. $m_{i}$ mein subscript exactly yahi tag hai jo symbol par chipka hua hai.

2. Mass $m_{i}$ aur total mass $M$

Capital $M$ chote $m_{i}$ se summing karke banta hai (agla section): $M = m_{1} + m_{2} + m_{3} + \dots$

Picture: figure mein har blob ka size uske $m_{i}$ ka hint deta hai. Saare blobs ko ek lump mein melt karo — us lump ki mass $M$ hai.

Topic ko iske liye kyun zaroorat hai: centre of mass ek mass-weighted average hai, isliye individual heavinesses $m_{i}$ decide karti hain ki har particle kitna "pull" karta hai. Ant mein $M$ se divide karna woh cheez hai jo ek raw total ko ek proper average mein convert karti hai.

3. Summation sign $\sum$

Yeh notation ka sabse important piece hai jo nail karna zaroori hai.

$n$ letter yahan sirf "total mein kitne particles hain" hai.

Yeh tool kyun aur plus likhna kyun nahi? Kyunki ek system mein $3$ particles ho sakte hain ya $3000$ . $\sum$ symbol ek aisi formula likhta hai jo kisi bhi count ke liye kaam karta hai. Yeh sawaal ka jawab deta hai: "'poori crowd par add karo' kaise express karoon bina crowd size jaane?"

4. Position, coordinates, aur vector $r_{i}$

Yeh kehne ke liye ki particle kahan hai, hume ek fixed reference point aur measure karne ka tarika chahiye.

Topic ko vectors ki kyun zaroorat hai: particles 2D ya 3D mein hote hain, isliye "position" ke liye sirf number nahi, direction bhi chahiye. Vector $r_{i}$ teeno coordinates ko ek symbol mein pack karta hai. Isliye parent ki master formula $r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}$ ek line mein likhi ja sakti hai — aur phir $x$ , $y$ , $z$ ke liye teen alag coordinate equations mein unpack ki ja sakti hai.

5. Weighted average — formula ka dil

COM literally ek positions ka weighted average hai, isliye yeh idea crystal clear hona chahiye.

COM ke liye, weights masses hain aur values positions hain. Lever ki language mein same maths ke liye Weighted Average and Moments dekho.

Yeh tool kyun? Ek plain average ek feather aur ek boulder ko equally treat karta — galat, kyunki boulder ko dominate karna chahiye "mass kahan hai". Weighted average woh ek averaging rule hai jo bhaare particles ko result ko apni taraf kheenchne deta hai, jo exactly woh physical behaviour hai jo hum chahte hain.

6. Rate of change: velocity, aur symbol $\frac{d}{d t}$

Parent ki derivation (Section 2) ko ek aur tool chahiye: position kitni tezi se change hoti hai.

Picture: ek moving blob ke do snapshots freeze karo thodi si time ke antar se; pehli jagah se doosri jagah tak chhota arrow, us choti time se divide karke, $v_{i}$ ke along point karta hai aur uski length = speed hai.

Yeh tool kyun aur sirf "distance ÷ time" kyun nahi? Kyunki velocity pal pal badal sakti hai. Derivative $\frac{d}{d t}$ instantaneous rate measure karta hai — abhi ki value, lambi trip par average nahi. Parent ise momentum $P = \sum_{i} m_{i} v_{i}$ define karne ke liye use karta hai aur phir $P = M v_{c m}$ demand karne ke liye, jo COM formula ko existence mein force karta hai.

Recall

$\frac{d r _{i}}{d t}$ tumhe kya deta hai? Particle $i$ ki instantaneous velocity $v_{i}$ — uski position ki rate of change. ::: velocity

7. Symbols ko saath mein rakhna

Ab parent ki headline formula fully readable hai: $r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}$

Ise zor se padho saath mein jo kuch hum ne build kiya:

Har piece upar ke kisi section ka tool hai — kuch bhi unexplained nahi chhoda gaya.

Prerequisite map

Equipment checklist

Test karo khud ko — reveal karne se pehle har ek ka jawab do.

m_{i}

mein subscript

i

ka kya matlab hai?

Ek label/tag jo "i-waan particle" ki taraf point karta hai; ise kabhi multiply nahi kiya jaata.

\sum_{i}

kya operation perform karta hai?

Yeh system ke har particle par ek quantity add karta hai.

m_{i}

aur

M

mein kya farq hai?

m_{i}

ek particle ki mass hai;

M = \sum_{i} m_{i}

saari masses ka total hai.

r_{i}

mein arrow kya signify karta hai, aur uske pieces kya hain?

Yeh ek vector mark karta hai (length + direction); uske pieces coordinates

(x_{i}, y_{i}, z_{i})

hain origin se measure kiye gaye.

Weighted average mein, COM ke liye weights ki role kaun play karta hai?

Masses

m_{i}

— bhaare particles ko zyada pull milta hai.

\sum_{i} m_{i} x_{i}

M

se divide kyun karna padta hai?

Sum ki units kg·m hain (ek moment);

M

(kg) se divide karne par position metres mein milti hai.

\frac{d r _{i}}{d t}

physically kya represent karta hai?

Particle

i

ki instantaneous velocity

v_{i}

Centre of mass ke liye single-line vector formula likho.

r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}

Connections

Parent: COM definition — jahan yahan ke saare tools use hote hain.
Weighted Average and Moments — Section 5 ke peeche ka pure-maths engine.
Newton's Second Law for a System of Particles — Section 6 ke derivative aur momentum tools ki zaroorat hai.
Conservation of Linear Momentum — $P = M v_{c m}$ par build karta hai.
Centre of Mass of Continuous Bodies — $\sum$ ko $\int$ se replace karta hai jab summation samajh aa jaaye.
Collisions — Elastic and Inelastic — COM frame mein in foundations ka use karke analyze kiya jaata hai.