Quick Revision Notes: Center of Mass (CM)

⚡

1. The Basic Definition The Center of Mass is the point where the entire mass of the system is assumed to be concentrated.

Position: R_CM = (m₁r₁ + m₂r₂ + ...) / Total Mass

Velocity: v_CM = (m₁v₁ + m₂v₂ + ...) / Total Mass

2. The Golden Rule of Forces

Internal Forces: (Explosions, collisions, tension) NEVER change the acceleration of the CM. They cancel out.

External Forces: (Gravity, friction) are the ONLY forces that move the CM.

Formula: F_ext = M_total × a_CM

3. Projectile Motion & Explosions If a projectile explodes in mid-air:

The fragments fly in different directions.

The CM continues on the original parabolic path as if nothing happened.

Tip: To find where a fragment lands, calculate the original range (R) of the CM first.

4. Collisions In free space (no external force):

Momentum is Conserved: p_initial = p_final

CM Velocity is Constant: v_CM does not change before, during, or after the collision.

5. Man on a Log (Floating Systems) If a man walks on a floating boat/log (no friction from water):

There is no external horizontal force.

Therefore, the CM position remains fixed.

As the man moves one way, the log moves the other.

Calculation: m₁Δx₁ + m₂Δx₂ = 0

6. Pulley Systems When masses accelerate in opposite directions:

The CM does accelerate (it is not zero).

It accelerates in the direction of the heavier mass.

a_CM = (m₁a₁ + m₂a₂) / (m₁ + m₂) (Watch your +/- signs!)

7. Vector Analysis (2D Problems) For explosions or collisions in 2D:

Never add velocities directly.

Resolve everything into X and Y components:

Σ p_x (initial) = Σ p_x (final)

Σ p_y (initial) = Σ p_y (final)