Centre of mass concepts, motion of COM, variable mass systems, collisions and conservation laws (JEE Main focus).
1) Centre of Mass (COM)
- Centre of mass is a hypothetical point where the entire mass of a system can be assumed to be concentrated.
- If all external forces act at COM, the system undergoes pure translation.
- COM may lie inside or outside the body.
r_CM = (m₁r₁ + m₂r₂ + ... + mₙrₙ) / (m₁ + m₂ + ... + mₙ)
x_CM = (m₁x₁ + m₂x₂ + ... ) / (Total mass)
y_CM = (m₁y₁ + m₂y₂ + ... ) / (Total mass)
Centre of mass lies closer to the heavier mass.
2) Centre of Mass of a Rigid Body
r_CM = ( ∫ r dm ) / ( ∫ dm )
x_CM = ( ∫ x dm ) / ( ∫ dm )
y_CM = ( ∫ y dm ) / ( ∫ dm )
- If gravitational field is uniform, centre of mass = centre of gravity.
3) Motion of Centre of Mass
M r_CM = Σ mᵢ rᵢ
M v_CM = Σ mᵢ vᵢ
M a_CM = Σ F_ext
- If ΣF_ext = 0, then COM velocity is constant.
- If system is initially at rest and no external force acts, COM remains at rest.
4) Variable Mass System (Rocket Motion)
F_ext = dP/dt
M dv/dt = F_ext + u (dM/dt)
Thrust force: F_thrust = u (dM/dt)
- u = velocity of ejected mass relative to the rocket.
- In rocket motion, weight can often be neglected compared to thrust.
5) Collision
- Collision is a short-time interaction involving large mutual forces.
- Total impulse on the system during collision is zero.
ΔP₁ + ΔP₂ = 0
⇒ Total momentum conserved (if no external force)
6) Types of Collision
- Elastic collision: Momentum and kinetic energy conserved.
- Inelastic collision: Momentum conserved, KE not conserved.
- Perfectly inelastic: Bodies stick together after collision.
Coefficient of restitution:
e = (velocity of separation) / (velocity of approach)
0 ≤ e ≤ 1
- e = 1 → perfectly elastic
- e = 0 → perfectly inelastic
7) Head-on Elastic Collision (1D)
v₁f = [(m₁ − m₂)/(m₁ + m₂)] v₁i + [2m₂/(m₁ + m₂)] v₂i
v₂f = [2m₁/(m₁ + m₂)] v₁i + [(m₂ − m₁)/(m₁ + m₂)] v₂i
- If m₁ = m₂ → bodies exchange velocities.
8) Perfectly Inelastic Collision & Ballistic Pendulum
(m + M)v = mu
v = (mu)/(M + m)
Fractional loss of KE:
ΔK / K = M / (M + m)
- Used in bullet–block (ballistic pendulum) problems.
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Last modified: December 14, 2025
