Motion of the centre of mass and the momentum of a system

The motion of the centre of mass

Add up Newton's second law for every particle of a system. The internal forces come in third-law pairs and cancel, so only the external forces are left:

The centre of mass moves like a single particle of mass $M$ pushed by the external forces alone.

Momentum of a system

The total momentum is the sum of the parts' momenta, and it equals

With no external force, $\vec{P}$ is conserved, and the centre of mass moves at a steady velocity (or stays at rest), whatever happens inside.

A shell that bursts

A shell bursts at the top of its flight. The burst is internal, and gravity is unchanged, so the centre of mass of the pieces carries on along the same parabola.

Try different bursts.

You should find that the other piece lands farther away the bigger the share that drops, but the centre of mass always lands at the same place, the shell's own range.

Watch out. This holds only while all the pieces are in the air. Once one piece lands, the ground pushes on it, and that is an external force.

A worked example

A $60\ \text{kg}$ diver stands on a $240\ \text{kg}$ boat at rest, and dives off the back at $2.5\ \text{m/s}$.

1. Momentum before. Zero: everything is at rest.
2. After. $60\ \text{kg} \times 2.5\ \text{m/s} + 240\ \text{kg}\,v = 0$, so the boat moves forward at $v = 0.625\ \text{m/s}$.
3. The centre of mass. The momentum is $60\ \text{kg} \times 2.5\ \text{m/s} + 240\ \text{kg} \times (-0.625\ \text{m/s}) = 0$, so $v_{cm} = 0$: it stays still, as it must.

Think first. Ask whether any external force acts along the direction you care about. If none does, the centre of mass does not move along it.

Practise with new numbers

Try it in Python

The program finds where the second piece lands for bursts of different shares.

Remember these

• $M\vec{a}_{cm} = \vec{F}_{\text{ext}}$: internal forces cannot move the centre of mass.
• $\vec{P} = M\vec{v}_{cm}$; with no external force, $\vec{v}_{cm}$ stays the same.
• A burst shell's centre of mass keeps to the old path while all the pieces fly.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A $3.0\ \text{kg}$ cart moves east at $4.0\ \text{m/s}$ and a $1.0\ \text{kg}$ cart is at rest. What is the velocity of their centre of mass?

2. Can you move your own centre of mass by pulling on your belt?

3. A cat and a dog sit at the two ends of a trolley on smooth ice, and walk toward each other. What does the trolley do?

4. Why does a gun recoil, but the centre of mass of gun and bullet does not move when it fires?