System with external forces — conditions for conservation

1.4.4 · Physics › Momentum & Collisions

KISKA baar kar rahe hain hum

Key conceptual move yeh hai: conservation is baat par depend karta hai ki aap boundary kahaan kheenchte ho. Gravity "external" hai agar sirf ek ball aapka system hai, lekin "internal" hai agar Earth + ball aapka system hai.

KAISE: Newton ke laws se condition derive karo (scratch se)

Step 1 — Ek particle ke liye Newton ka 2nd law. ${\vec{F}}_i=\frac{d{\vec{p}}_i}{dt}$ Yeh step kyun? Yeh momentum form mein force ki definition hai — hamaara ek hi starting axiom.

Step 2 — Particle $i$ par force ko split karo. $i$ par har force ya toh kisi doosre member $j$ se hai (internal) ya bahar se (external): ${\vec{F}}_i=\underset{\text{internal}}{\underbrace{\sum _{j\ne i}{\vec{f}}_{ij}}}+\underset{\text{external}}{\underbrace{{\vec{F}}_i^{ext}}}$ Kyun? Har force ka ek source hota hai; woh source ya toh hamare chosen system ke andar hai ya bahar. Koi teesra option nahi.

Step 3 — Saare particles par sum karo. ${\sum }_i{\vec{F}}_i={\sum }_i{\sum }_{j\ne i}{\vec{f}}_{ij}+{\sum }_i{\vec{F}}_i^{ext}=\frac{d}{dt}{\sum }_i{\vec{p}}_i=\frac{d\vec{P}}{dt}$

Step 4 — Internal sum ko Newton ke 3rd law se khatam karo. Har pair ke liye, ${\vec{f}}_{ij}=-{\vec{f}}_{ji}$, toh pairs mein cancel ho jaate hain: ${\sum }_i{\sum }_{j\ne i}{\vec{f}}_{ij}=\vec{0}$ Kyun? Internal forces equal-and-opposite twins mein aate hain; unka vector sum zero hota hai. Yahi reason hai kyun internal forces kabhi total momentum nahi badal sakte (ek explosion soch lo: fragments bahar udte hain, lekin saare pieces ka $\vec{P}$ unchanged rehta hai).

Step 5 — Master equation. $\boxed{\frac{d\vec{P}}{dt}=\sum _i{\vec{F}}_i^{ext}={\vec{F}}_{ext}^{net}}$

KYUN har relaxation sach hai

Worked examples

Common mistakes (steel-manned)

Connections

• Newton's Third Law — kyun internal forces cancel hoti hain.
• Impulse–Momentum Theorem — $\Delta \vec{P}=\int \vec{F}dt$, impulsive approximation ki basis.
• Center of Mass Motion — ${\vec{F}}_{ext}^{net}=M{\vec{a}}_{cm}$, same equation, dobaara boli gayi.
• Elastic vs Inelastic Collisions — kinetic energy ke liye alag condition.
• Conservation of Linear Momentum — special case ${\vec{F}}_{ext}=0$.

Concept Map