Centre of mass

The centre of mass (center of mass in the US) of an object is the point where its whole mass can be thought of as concentrated. AQA's GCSE Physics specification puts it this way: the weight of an object may be considered to act at a single point, its centre of mass. For a uniform regular object, such as a ruler or a cube, it is at the centre.

What is the centre of mass?

Every part of an object has weight. Rather than add up thousands of tiny forces, physics treats the total weight as one force acting through one point. That point is the centre of mass. If you support an object directly below (or above) its centre of mass, it balances.

You will also see centre of gravity. It is the point where the weight acts. In a uniform gravitational field, such as near the Earth's surface for any everyday object, the centre of gravity and the centre of mass are in the same place, so the terms are used interchangeably.

Where is the centre of mass?

• Uniform regular shapes: at the centre. AQA's A-level specification expects you to know that the centre of mass of a uniform regular solid is at its centre: the middle of a rod, the centre of a sphere or cube.
• Symmetrical shapes: on every line of symmetry, so where two lines of symmetry cross.
• Irregular or uneven objects: nearer the heavier part. A hammer's centre of mass is close to its head.
• It can be outside the object. A ring's centre of mass is in the empty middle, and the L shape below has its centre of mass in the gap.

How to find the centre of mass of a flat shape

For a thin, flat object (a lamina) such as a piece of card, use a plumb line: a weight on a string, which always hangs straight down.

1. Pierce the card in three places close to its outline, spread out around it.
2. Push a pin through the first hole into a clamp so the card can turn freely, then let the plumb line hang from that pin too.
3. Once the card and string stop moving, draw along the string on the card.
4. Repeat from the other two holes.
5. The centre of mass is where the lines cross. The third line is a check: all three should meet at one point.

It works because a hanging object always settles with its centre of mass straight under the pin, so every line you draw passes through it.

Calculating the centre of mass

For several masses in a line, the position of the centre of mass is the mass-weighted average of their positions:

Example 1: two masses on a light rod

A 2.0 kg mass is at one end of a light rod (x = 0) and a 3.0 kg mass is 1.5 m away. Where is the centre of mass?

x̄ = (2.0 × 0 + 3.0 × 1.5) ÷ (2.0 + 3.0) = 4.5 ÷ 5.0 = 0.90 m from the 2.0 kg mass, nearer the heavier mass as expected.

Example 2: a rod with a load

A uniform rod of mass 4.0 kg and length 2.0 m has a 1.0 kg mass fixed to one end. Treat the rod's mass as acting at its centre, 1.0 m from the other end. Then x̄ = (4.0 × 1.0 + 1.0 × 2.0) ÷ 5.0 = 6.0 ÷ 5.0 = 1.2 m from the unloaded end.

Example 3: an L-shaped lamina

A uniform L shape is made from two rectangles of card: A, 4 cm by 1 cm, and B, 1 cm by 3 cm, as in the figure. For a uniform card, mass is proportional to area, so use areas as the "masses".

• x̄ = (4 × 2 + 3 × 0.5) ÷ 7 = 9.5 ÷ 7 = 1.36 cm
• ȳ = (4 × 0.5 + 3 × 2.5) ÷ 7 = 9.5 ÷ 7 = 1.36 cm

G is at (1.36, 1.36), on the line joining the centres of A and B, but in the empty corner of the L. If you hung this shape from a corner, the plumb lines would cross there, off the card.

Centre of mass and stability

An object standing on a base stays upright as long as the vertical line through its centre of mass (the line of action of its weight) falls inside the base. Tilt it until that line passes the edge, and its weight makes a moment that turns it further over: it topples.

For a box tilted about one bottom edge, it is on the point of toppling when its centre of mass is directly above that edge. If the base has width w and the centre of mass is at height h, the angle of tilt at that point is given by:

A box 0.40 m wide with its centre of mass 0.50 m high topples at tan θ = 0.20 ÷ 0.50 = 0.40, so θ = 21.8°. Lower the centre of mass to 0.30 m and θ rises to 33.7°.

So objects are more stable with a low centre of mass and a wide base, which is why racing cars are built low and wide.

Centre of mass in moments problems

In a moments question, the weight of a beam, ladder or plank acts at its centre of mass. For a uniform beam that is the midpoint, so the weight's moment about a pivot is weight × distance from the pivot to the midpoint. The principle of moments guide works through beams like this, and the moments calculator checks your answers.

Common mistakes

• Assuming it is always in the middle. That is only true for uniform, symmetrical objects.
• Thinking it must be inside the object. Rings, L shapes and boomerangs have it outside.
• Forgetting the masses in the average. Use Σmx ÷ Σm, not just the average of the positions.
• Using height instead of half the width. In tan θ = (w/2)/h, the top is half the base width.
• Mixing up centre of mass and pivot. The pivot is where the object is supported; the weight acts at the centre of mass.

To practise forces and moments in a short timed paper, try today's A-level Physics: Year 1 or GCSE Physics Higher paper.

Read next

Maths · GraphsHow to find the centre of a circle: 3 methodsHow to find the centre of a circle: by construction with two chords, from the equation by completing the square, and from three points, with examples.