Centre of gravity

A body's weight can be represented as a single force acting at its centre of gravity, G.

Gravity acts throughout the body's matter. The centre of gravity lets us replace those distributed effects with one resultant weight arrow. Draw that arrow from G, vertically downwards near Earth.

A weight produces a moment about a pivot unless its line of action passes through the pivot. Use the body's total weight in newtons in the moment calculation.

The centre of gravity need not be the middle of the outline

For a uniform, symmetrical object in a uniform gravitational field, G is at its geometric centre. A uniform metre rule has G at the 50 cm mark. If more mass is concentrated towards one end, G shifts towards that end.

Some shapes have G outside their material: for a uniform ring, it is at the centre of the empty opening. G represents where the combined weight acts; it does not have to be a point containing material.

A horizontal rule can balance on a narrow support directly below its G. The weight's line of action then passes through the support, so weight has no moment about it. The upward support also balances the downward weight.

Locate G by suspending a lamina

A plane lamina is a thin, flat sheet, such as a rigid card cut into an irregular outline. Its centre of gravity can be found without assuming the outline is symmetrical.

When it hangs freely and settles, G lies vertically below the suspension point. Otherwise the weight would have a moment about that point and turn the lamina. A plumb line, made from a freely hanging thread and small weight, shows the vertical direction.

1. Make a small hole near the edge of the lamina. Suspend it from a pin through that hole, allowing it to rotate freely.
2. Hang a plumb line from the same suspension point. Keep the thread free and wait until both the lamina and line are still.
3. Mark the plumb-line position on the lamina using a thin line or two well-separated marks that can be joined. G lies somewhere on this traced line.
4. Suspend the same lamina from a different hole. Let it settle into its new orientation, then mark the new plumb-line position.
5. Find the intersection of the two traces on the card. This estimates G. A third suspension from another hole provides a check.

Changing the suspension hole changes how the lamina hangs, not where G lies relative to the card. One suspension gives a line of possible positions. A second, differently directed trace locates their common point.

Make the intersection useful

• Allow free rotation: do not clamp the lamina against the pin. Friction could hold it before G is vertically below the suspension point.
• Use thin marks and a still plumb line: thick or moving marks leave a broad range of possible lines.
• View perpendicular to the card: reduce parallax between the thread and the surface being marked.
• Choose useful suspension positions: traces with distinctly different directions locate an intersection more clearly than nearly parallel traces.

Real traces may meet within a small region rather than at one exact point. Use that region to judge the uncertainty in G, and check the result with another suspension. Do not choose the geometric centre simply because the lines are imperfect.

Draw one weight at G. G is not an extra force. When replacing distributed weight with its resultant, do not also keep a second full-weight arrow elsewhere on the same body.