Centre of Gravity (and Centre of Mass)

Centre of Gravity (and Centre of Mass)

What is it?

Close your eyes. Get someone to balance a ruler horizontally on your outstretched finger. You can't tell anything about the shape of the ruler. All you can tell is how heavy it is. All the weight of the ruler passes through (and is supported by) your finger.

It seems as if the weight (and the mass) is concentrated in a spot in the ruler above your finger.

This spot is the centre of gravity of the ruler. Of course, the weight isn't all concentrated in the spot above your finger; it is spread out over the whole of the ruler. But it is a useful idea to use when calculating the effect of forces.

Why is it useful?

When doing moment calculations you can say that all the weight of an object acts through the Centre of Gravity.

Look at this beam balanced at a point other than its centre of gravity:

Centre of Gravity

Note: the pivot is not below the centre of gravity.

Question:

Where must I put the 20N weight to balance the beam?

Answer:

We must satisfy the equilibrium condition that states the moments turning the beam clockwise about the pivot must equal the moments turning the beam anticlockwise about the pivot.

Consider moments about the pivot (clockwise first).

The turning force, M due to the beam around the pivot is:

M = weight of the beam x distance from pivot to the Centre of Gravity.

M = 100N x 1m = 100 Nm trying to turn the beam clockwise.

For balance, the 20N weight must produce a 100Nm turning force in the opposite direction. So the anticlockwise moment due to the 20N weight is:

M = 100Nm = 20N x distance from the centre of the weight to the pivot

M = 20 x d = 100Nm So, d = 5m

In a uniform gravitational field (like the field close to the surface of the Earth) the Centres of Gravity and Mass are in exactly the same place - and at A-level, that's as complicated as it will get.

Centre of Gravity = the point where all the weight seems to be concentrated.

Centre of Mass = the point where all the mass seems to be concentrated.

For regular shapes it is the geometrical centre of the object - for example, the centre of a cube or a sphere.

For irregular shapes, hang the object from a point on its edge and the Centre of Gravity will end up vertically below the point you are hanging it from.

Repeat this for two or more points, each time drawing a line vertically down through the point you are hanging it from (and therefore through the Centre of Gravity).

Where the lines cross is the Centre of Gravity.

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