📐 For discrete particles

R_cm = (Σ m_i r_i) / Σ m_i

For 2 particles: x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂)

📏 For continuous body

r_cm = (∫ r dm) / M

COM of common shapes:
Uniform rod: at center
Ring: at center (empty)
Disc: at center
Triangle: at centroid (1/3 height from base)

Conservation of momentum: If no external force, total momentum = constant → V_cm = constant.

Impulse = change in momentum = ∫ F dt = mΔv

Oblique collisions: Use components. For smooth surfaces, impulse is along common normal.

💡 NEET TIPS & SHORTCUTS

• COM of a system moves as if all external forces act at it and total mass is concentrated there.
• For two-body system, COM lies on line joining them, and m₁r₁ = m₂r₂ from COM.
• In perfectly inelastic collision, maximum KE loss occurs.
• For oblique collisions, resolve velocities along normal and tangential. Tangential component remains unchanged if no friction.

⚠️ COMMON MISTAKES

• Applying conservation of momentum when external forces are present.
• Assuming energy is conserved in inelastic collisions.
• Forgetting that COM can accelerate if external forces act.
• Using wrong sign for velocities in e formula.

COM position: R_cm = Σ m_i r_i / M

Velocity of COM: V_cm = Σ m_i v_i / M

Momentum: P = M V_cm

Elastic 1D final velocities: given above

Perfectly inelastic common velocity: v_f = (m₁v₁ + m₂v₂)/(m₁+m₂)

Coefficient of restitution: e = (v₂' – v₁')/(v₁ – v₂)