The centre of mass is the point where the whole weight of an object seems to act. An object supported at that point balances. This idea lets you treat a complicated object as one weight at one position when using moments.
It is the third lesson in moments and stability and links the moments work to the stability ideas that follow.
Where is the centre of mass for simple objects?
For a uniform object with a regular shape, it lies at the geometric centre. A uniform metre rule has it at the 50 cm mark.
A uniform rectangular sheet has it where the diagonals cross. A uniform sphere has it at its centre.
When an object is not uniform, the centre of mass shifts towards the heavier part.
A hammer balances nearer the head than the middle of the handle.
How do you find it for an irregular sheet?
This is a practical idea you should be able to explain in words, even if you do not perform it in an exam.
- Make three small holes near the edge of a cardboard sheet.
- Hang the sheet freely from a pin through one hole, so it can swing.
- Hang a plumb line (a weight on a string) from the same pin. When everything is still, mark where the string passes across the sheet and draw that line.
- Repeat from a second hole, and draw a second line.
- The centre of mass is where the lines cross. A third line, from the third hole, is a check: it should pass through the same point.
The reason is that a hanging object comes to rest with its weight directly below the pin. The centre of mass therefore lies somewhere on the vertical line under the pin. Two different lines pin it down to one point.
Worked example: two weights on a light rod
A light rod is 1.0 m long. An 8 N weight is fixed at one end and a 2 N weight at the other end. Where is the balance point, measured from the 8 N end?
Let the balance point be x metres from the 8 N end. Take moments about it.
Step 1, distances: the 8 N weight is x m away. The 2 N weight is (1.0 − x) m away.
Step 2, balance: 8 × x = 2 × (1.0 − x).
Step 3, solve: 8x = 2 − 2x, so 10x = 2, giving x = 0.20 m.
Answer: 0.20 m from the 8 N end.
Check: 8 × 0.20 = 1.6 N m, and 2 × 0.80 = 1.6 N m. The balance point is closer to the heavier weight, as expected.
The mistake to watch for
A common slip is to think the centre of mass is always in the middle of the object.
Mistaken answer: “The balance point is at 0.50 m because the rod is 1.0 m long.”
That is correct only if the weights are equal, or if the rod is uniform with no added weights.
The correction is to ask how the weight is spread. The centre of mass moves towards the larger weight.
Check yourself
1. Where is the centre of mass of a uniform rectangular sheet 0.40 m by 0.20 m?
Show answer
At the centre of the rectangle: 0.20 m from each short edge and 0.10 m from each long edge. It is where the diagonals cross.
2. A light rod 0.60 m long has a 12 N weight at one end and a 4 N weight at the other. How far from the 12 N end is the balance point?
Show answer
12 × x = 4 × (0.60 − x), so 12x = 2.4 − 4x, so 16x = 2.4 and x = 0.15 m. Check: 12 × 0.15 = 1.8 N m and 4 × 0.45 = 1.8 N m.
3. Why must the plumb line hang from the same pin as the sheet?
Show answer
The centre of mass lies vertically below the pin when the sheet is at rest. The plumb line marks that vertical. If it hung from a different point, the line would not show the vertical through the pin.
Where this leads next
Now use the centre of mass to explain toppling in explaining stability using a base and centre of mass. The moments and stability practice set also includes balance-point questions.
Explaining the method clearly in words is a skill in itself, and it is one of the things a teacher in online one-to-one Physics tuition can check with you through your written answers.