Centre of mass

Finding the centre of mass

Narrated video. Transcript:

Throw a cricket bat spinning into the air. Every point on it follows a messy, wobbly path.

But one special point moves along a smooth parabola, just like a thrown ball. That point is the centre of mass.

Start simple. Two particles sit on the x-axis: mass m₁ at x₁ and mass m₂ at x₂.

The centre of mass is at X = (m₁x₁ + m₂x₂) / (m₁ + m₂). It is a weighted average of the two positions.

If the masses are equal, X is exactly halfway between them. Watch the marker sit at the midpoint.

Now make m₂ three times heavier. The marker slides towards it. The centre of mass is always closer to the heavier mass.

For many particles in a plane, do the same for x and for y: X = Σmᵢxᵢ / M and Y = Σmᵢyᵢ / M, where M is the total mass.

For a uniform ring, disc or sphere, the centre of mass is at the geometric centre. It need not be on the material: for a ring it is in the empty middle.

How the centre of mass moves

Narrated video. Transcript:

Now let the particles move. Each one feels forces from outside the system, and forces from the other particles.

Internal forces come in Newton’s third-law pairs: equal and opposite. When we add up all the forces, every pair cancels.

Only the external force is left. So total mass times the acceleration of the centre of mass equals the total external force: M A = F_ext.

The centre of mass moves as if all the mass were packed into one point, with all the outside forces acting there.

Watch a shell fired from a gun. At the top of its path it explodes into two pieces. The explosion is an internal force.

Only gravity acts from outside, so the centre of mass carries on along the same parabola, as if nothing had happened.

The total momentum of the system is P = MV, where V is the velocity of the centre of mass. If the external force is zero, P stays constant.

Then the centre of mass moves with constant velocity. This is conservation of total linear momentum for a system of particles.

Try it: find the centre of mass

Interactive simulation. Things to try:

• Drag the coloured masses. Click one to change its mass.
• Make all masses equal and see where the centre of mass goes.
• Make one mass much heavier. The centre of mass moves towards it.

Try it: an exploding shell

Interactive simulation. Things to try:

• Press Launch. The shell bursts at the top of its path.
• Change the masses and the burst speed. The pieces land in different places.
• Watch the amber cross: the centre of mass always lands at the same point.

Quick check

A 1 kg mass and a 3 kg mass are 4 m apart. How far is their centre of mass from the 3 kg mass?

1. 1 m
2. 2 m
3. 3 m
4. 0.75 m

Exploding shell

A shell would land at distance R. At the top of its path it bursts into two equal halves. One half drops straight down. Where does the other half land?

1. At R
2. At 2R
3. At 3R/2
4. At R/2

Summary

• Two particles: X = (m₁x₁ + m₂x₂) / (m₁ + m₂)
• Many particles: X = Σmᵢxᵢ / M, Y = Σmᵢyᵢ / M
• Motion of the centre of mass: M A = F_ext
• Total momentum: P = M V

• The centre of mass is a mass-weighted average of positions. It lies closer to the heavier masses and need not be on the body.
• Internal forces cancel in pairs, so only external forces change the motion of the centre of mass.
• If an exploding shell’s pieces are in flight, their centre of mass keeps to the original parabola.
• With zero external force, total momentum is conserved and the centre of mass moves at constant velocity.