The centre of mass of a system of particles

One point that stands for the whole

A spinning hammer thrown across a field tumbles in a complicated way, yet one point of it follows a smooth parabola, just like a thrown stone. That point is the centre of mass.

Particles

For particles of mass $m_i$ at positions $x_i$, the centre of mass is the mass-weighted average position:

and the same for $y$ and $z$. Heavier particles pull it toward themselves.

Move the masses and watch it.

You should find the centre of mass always inside the triangle of the three masses, pulled toward the heaviest. With equal masses it sits at the triangle's centre.

Uniform bodies

For a uniform body with a centre of symmetry (a rod, a ring, a disc, a sphere, a rectangle), the centre of mass is at that centre. A body made of simple pieces can be split: treat each piece as its whole mass at its own centre.

A body with a hole can be treated as the full body plus a negative mass where the hole is.

Watch out. The centre of mass need not lie inside the material: a ring's is at its empty centre, and a boomerang's lies between its arms.

A worked example

Three children sit on a square raft: $30\ \text{kg}$ at $(0, 0)$, $40\ \text{kg}$ at $(3.0, 0)$ and $50\ \text{kg}$ at $(3.0, 4.0)$, in metres.

1. Total mass. $M = 120\ \text{kg}$.
2. $x$. $\sum m_ix_i = 30\ \text{kg} \times 0\ \text{m} + 40\ \text{kg} \times 3.0\ \text{m} + 50\ \text{kg} \times 3.0\ \text{m} = 270\ \text{kg m}$, so $x_{cm} = \dfrac{270\ \text{kg m}}{120\ \text{kg}} = 2.25\ \text{m}$.
3. $y$. $y_{cm} = \dfrac{50\ \text{kg} \times 4.0\ \text{m}}{120\ \text{kg}} = 1.67\ \text{m}$.

Think first. Put the origin on one of the masses: every term with a zero drops out.

Practise with new numbers

Try it in Python

The program finds the centre of mass of the children on the raft.

Remember these

• $x_{cm} = \dfrac{\sum m_ix_i}{\sum m_i}$, and the same for $y$ and $z$.
• Uniform symmetric bodies: the centre of mass is at the centre.
• Split complex shapes into pieces; a hole is a negative mass.

Check yourself

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

1. A $2.0\ \text{kg}$ ball and a $6.0\ \text{kg}$ ball are $1.6\ \text{m}$ apart. How far is the centre of mass from the lighter ball?

2. Where is the centre of mass of a uniform metre rule?

3. The Earth is about $81$ times as heavy as the Moon, and they are $3.84 \times 10^{5}\ \text{km}$ apart. How far from the Earth's centre is their centre of mass?

4. Can a body balance on a point that is not under its centre of mass?