Centre of mass — derivation for common shapes (rod, triangle, semicircle, hemisphere): Foundations

1.4.10 · D1 · Physics › Momentum & Collisions › Centre of mass — derivation for common shapes (rod, triangle

Parent derivations padhne se pehle, tumhe un saare symbols ka poora command hona chahiye jo wahan use hote hain. Yeh page har ek ko zero se build karta hai — pehle plain words, phir ek picture, phir kyun topic ko yeh chahiye. Upar se neeche padho; har idea upar wale par lean karta hai.

0. Scale-and-average idea (woh cheez jis par sab kuch tika hai)

Picture: soch ek see-saw. Position $x_{1}$ par ek bhaari baccha rakho aur $x_{2}$ par ek halka baccha. Balance point beech mein nahi hoga — yeh bhaari bacche ki taraf khisak jaata hai. Woh balance point hi weighted average hai, aur yahi exactly centre of mass hai.

Figure — Centre of mass — derivation for common shapes (rod, triangle, semicircle, hemisphere)

Topic ko yeh kyun chahiye: har COM formula yahi see-saw hai jo laakhon tiny pieces ke saath kiya jaata hai. Agar tum samajh lo kyun bhaari side balance kheenchti hai, toh tum poora chapter samajh lo. Discrete version ke liye dekho Centre of mass — definition and system of particles.

1. Symbols $m$ , $M$ , aur $x_{i}$ (ek piece ki mass aur position)

Picture: ek wire par beads ki ek row. Har bead ek piece hai apni mass $m_{i}$ ke saath apni jagah $x_{i}$ par baitha hua. Saari bead masses jodon → $M$ .

Greek capital sigma == $\sum$ == ka bas matlab hai "uske baad jo cheezein hain unhe saari add karo." Toh $\sum_{i} m_{i} x_{i}$ padhte hain: "har piece ke liye, uski mass ko uski position se multiply karo, phir un saare products ko add karo."

Topic ko yeh kyun chahiye: master formula $x_{c m} = \frac{\sum m _{i} x _{i}}{\sum m _{i}}$ exactly step 0 ka weighted average hai, symbols mein likha hua.

2. Coordinates aur vector $r$ (koi point kahan hai, kisi bhi dimension mein)

Picture: apne page ke corner (origin) se ek dot tak ek arrow draw karo. Uska horizontal shadow $x$ hai, vertical shadow $y$ hai.

Topic ko yeh kyun chahiye: master formula ek vector ke liye ek baar likha jaata hai, $r_{c m} = \frac{1}{M} \int r d m$ , aur woh ek line secretly teen alag averages hain — ek $x$ ke liye, ek $y$ ke liye, ek $z$ ke liye. Teen formulas ki jagah $r$ likhna ink bachata hai aur dikhata hai ki idea dimension-blind hai.

3. Density: $λ$ , $σ$ , $ρ$ (mass kitni spread out hai)

Kisi chunk ko mass number mein convert karne ke liye, tumhe jaanna hoga ki mass kitni densely packed hai. Teen flavours hain depending on whether tumhari body ek line hai, ek sheet hai, ya ek solid hai.

Picture:

Ek thin wire → mass ek line ke along smeared → $λ$ use karo.
Ek flat sheet (lamina) → mass ek area par smeared → $σ$ use karo.
Ek solid lump → mass ek volume mein fill → $ρ$ use karo.

Kyun "density × size = mass": density jawaab deti hai "har unit size par kitni mass?" Kitna size actually hai usse multiply karo, toh mass milti hai. Isliye: $d m = λ d l (line), d m = σ d A (area), d m = ρ d V (volume) .$

4. Infinitesimal $d m$ , $d x$ , $d l$ , $d A$ , $d V$ (ek "tiny piece")

Picture: rod lo aur imagine karo ki use bahut saare paper-thin slices mein kaata ja raha hai. Ek slice ki width $d x$ hai; uske andar ki mass $d m$ hai. Poori rod yeh slices wapas stack karke bani hai.

Topic ko yeh kyun chahiye: ek continuous body mein koi alag beads nahi hoti jinhe $i$ se label karein. Toh hum apni khud ki beads invent karte hain — slices — har ek ki mass $d m$ hai. Ab step 1 wala row-of-beads idea phir se kaam karta hai.

5. Integral $\int$ (infinitely many tiny pieces ko add karna)

Picture: wahi paper-thin slices ka stack — ab hum unke contributions ko wapas glue karte hain. Chhote numbers $0$ aur $L$ (limits) batate hain ki slices kahan se shuru aur kahan khatam hoti hain.

Topic ko yeh kyun chahiye: master formula bead sum ko turn karta hai $x_{c m} = \frac{\sum m _{i} x _{i}}{\sum m _{i}} ⟶ x_{c m} = \frac{\int x d m}{\int d m} .$ Parent note ki har shape bas "sahi $d m$ choose karo, phir yeh integral chalao" hai. Dekho Integration as continuous summation.

Recall Uniform bodies ke liye density kyun cancel ho jaati hai?

Kyunki ek constant (jaise $λ = M / L$ ) upar wale integral aur neeche wale integral dono se bahar khicha ja sakta hai — phir woh divide ho jaata hai. Isliye ek uniform body ka COM purely ek geometric fact hai, uske centroid ke equal. ::: Ek constant numerator aur denominator dono se factor out ho jaata hai aur cancel ho jaata hai.

6. Symmetry (woh shortcut jo aadha kaam delete kar deta hai)

Picture: semicircle ko uski vertical middle line ke along fold karo — dono halves perfectly ek doosre par aa jaati hain. Har bit of mass jo left par hai, uske equal right par bhi hai, toh unki sideways pulls cancel ho jaati hain: COM ka $x$ -coordinate $0$ hona hi chahiye.

Topic ko yeh kyun chahiye: parent note semicircle aur hemisphere ke liye "by symmetry $x_{c m} = 0$ " likhta hai. Woh ek sentence ek poore integral ki jagah le leta hai. Dekho Symmetry arguments in physics.

7. Centroid vs centre of mass (ek naming clarification)

Topic ko yeh kyun chahiye: parent note ki saari shapes uniform hain, toh "COM dhundho" secretly matlab hai "centroid dhundho." Distinction tabhi matter karta hai jab density vary kare. Poori comparison mein Centroid vs centre of mass vs centre of gravity.

Yeh topic ko kaise feed karte hain

Har box ek symbol ya idea hai jo tumne abhi seekha; arrows follow karo aur tum master formula tak pahunch jaoge jo the parent topic ki har derivation power karta hai.

Equipment checklist

Answers cover karo aur khud test karo — agar koi line tumhe stump kare, derivations tackle karne se pehle woh section dobara padho.

Ek sentence mein weighted average kya hota hai? ::: Ek average jahan har value apni mass ke proportion mein count karti hai (bhaari values use apni taraf kheenchti hain).
$\sum_{i} m_{i} x_{i}$ tumhe kya karne ko kehta hai? ::: Har piece ke liye, uski mass ko uski position se multiply karo, phir saare products add karo.
$r$ mein arrow ka kya matlab hai, aur 3D mein yeh kitne numbers hold karta hai? ::: Yeh ek position vector hai (origin se ek point tak arrow); 3D mein yeh teen numbers $(x, y, z)$ hold karta hai.
$λ$ , $σ$ , $ρ$ ke units batao. ::: $kg/m$ , $kg/m^{2}$ , $kg/m^{3}$ .
Ek line, ek sheet, aur ek solid ke liye $d m$ likho. ::: $d m = λ d l$ , $d m = σ d A$ , $d m = ρ d V$ .
Symbol $d x$ physically kya represent karta hai? ::: Ek infinitely thin slice ki width, itni patli ki quantities us par constant hain.
$\int$ aur $\sum$ mein kya fark hai? ::: $\int$ infinitely many infinitesimal pieces add karta hai; $\sum$ ek finite list add karta hai.
Constant density $x_{c m}$ se kyun cancel ho jaati hai? ::: Yeh numerator aur denominator dono se factor out ho jaati hai aur divide ho jaati hai.
Symmetry axis tumhe instantly kya likhne deta hai? ::: COM us par lie karta hai, toh off-axis coordinate $0$ hai.
Uniform body ke liye, centroid vs centre of mass? ::: Dono same point hote hain.