Motion of centre of mass — external force determines a_CM

1.4.11 · Physics › Momentum & Collisions

Centre of Mass KYA hota hai?

Mass-weighted KYU? Kyunki ek heavy particle system ko zyada "anchor" karta hai banisnbat ek light particle ke. Agar aap sirf positions ka average lete, toh ek feather aur ek anvil apne midpoint par balance hote — jo clearly galat hai. Mass se weight karne par balance point anvil ke paas aa jaata hai.

External force $a_{CM}$ ko KAISE control karta hai — poori derivation

Hum ${\vec{F}}_{ext}=M{\vec{a}}_{CM}$ scratch se derive karte hain.

Step 1 — Definition se shuru karo aur $M$ se multiply karo: $M{\vec{R}}_{CM}={\sum }_i{m}_i{\vec{r}}_i$ Ye step kyun? Isse fraction clear ho jaata hai taaki differentiation clean rahe.

Step 2 — Ek baar differentiate karo (velocity): $M{\vec{V}}_{CM}={\sum }_i{m}_i{\vec{v}}_i={\sum }_i{\vec{p}}_i={\vec{P}}_{total}$ Ye step kyun? Isse ek gem reveal hoti hai: total momentum equals total mass times CM velocity, $\vec{P}=M{\vec{V}}_{CM}$.

Step 3 — Dobara differentiate karo (acceleration): $M{\vec{a}}_{CM}={\sum }_i{m}_i{\vec{a}}_i={\sum }_i{\vec{F}}_i$ jahan ${\vec{F}}_i$ particle $i$ par total force hai (Newton's 2nd law har particle ke liye).

Step 4 — Har force ko internal + external mein split karo: ${\sum }_i{\vec{F}}_i=\underset{{\vec{F}}_{ext}}{\underbrace{\sum _i{\vec{F}}_i^{ext}}}+\underset{\text{internal pairs}}{\underbrace{\sum _i\sum _{j\ne i}{\vec{f}}_{ij}}}$

Step 5 — Newton's 3rd law use karke internal sum ko khatam karo: Har pair ke liye, ${\vec{f}}_{ij}=-{\vec{f}}_{ji}$. To ye pairs mein cancel ho jaate hain: ${\sum }_i{\sum }_{j\ne i}{\vec{f}}_{ij}=0$ Ye step kyun? — proof ka dil. Internal forces hamesha equal-and-opposite couples mein aate hain, isliye ye sum to zero ho jaate hain aur CM ko accelerate nahi kar sakte.

Corollary: Agar ${\vec{F}}_{ext}=0\Rightarrow {\vec{a}}_{CM}=0\Rightarrow {\vec{V}}_{CM}=\text{constant}$ → conservation of momentum.

WORKED EXAMPLES

COMMON MISTAKES (Steel-man + fix)

Active Recall

Flashcards

Connections

• Conservation of Linear Momentum — ${\vec{F}}_{ext}=0$ ka special case.
• Newton's Third Law — reason ki internal forces cancel kyun hoti hain.
• Collisions (Elastic & Inelastic) — CM frame unhe simplify karta hai.
• Rocket Propulsion — external thrust vs internal mass ejection.
• Rigid Body Dynamics — CM wo point hai jahan extended bodies ke liye $\vec{F}=M\vec{a}$ apply hota hai.

Concept Map