Centre of Gravity

Centre of Gravity

Have you ever tried to balance a ruler horizontally on one finger? If you stretch out one finger and carefully place the centre of the ruler on it, the ruler could be balanced. The point of the ruler that is in contact with your finger is called the centre of gravity.

Lets look at the forces present. The ruler is a long object. Think of it as being made up of many small parts. Each part has some weight pulling it down. Over the length of the ruler, there are many small forces of gravity acting. What does it mean if you can balance the ruler at a single point at the centre?

At the centre, a single upward force balances all of the small forces. This means that this collection of small forces behave like a single force acting downwards at the centre of gravity. This is why a single upward force at can balance the ruler at that point. With this understanding, we get the following definition:

The centre of gravity of a body is the point at with the weight of the body appears to act.

What if the body is more complicated than the uniform ruler? Suppose I stick a lump of platicine at one end of the ruler. You would no longer be able to balance the ruler at the centre any more. However, if you shift your finger slowly towards the lump, there would be a point where the ruler is balanced. At this point, a single force can again balance all of the weights on different parts of the ruler, including the weight on the lump. So at this new point, all the weights again act on a single point. This is the new centre of gravity.

Consider what happens if your finger moves away from the centre of gravity of the ruler. The ruler would fall to one side. This is a rotational motion. It happens because the weights on one side has a larger turning effect than the weights on the other side.

For complex solid shapes, the idea of a centre of gravity is less easy to understand. There would be a point where the weights on all parts of the solid appear to act. This point is usually inside the solid. Can we still use the idea of balancing at at point to understand this?

If the body has an irregular shape, with the centre of gravity, then the idea of balancing has to be changed. We cannot think of it like balancing a ruler horizontally, because the body is not long and narrow. We cannot put our finger just below the centre of gravity, because the centre of gravity is inside the body. So there are two problems: the idea of balancing horizontally cannot be used, and we cannot reach inside to support the centre of gravity.

Imagine, for now, that you can somehow magically reach inside and put your finger just under the centre of gravity to support the body. How can we tell whether the body is balanced in any sense? The answer is: it is balanced if it does not rotate. Recall what happens if the ruler is not balanced on your finger - as it falls, it is in fact rotating about your finger tip. So if the irregular body does not rotate, then it is balanced.

Of course, we cannot reach in and touch the centre of gravity. However, we can do something that has the same effect. We can attach a string to the body and support it from above. The body would swing for a bit and then stop. It is supported by a force at a single point from the string. This point on the surface is not the centre of gravity, which is inside the body. However, if we following the straight line of the string down through the body, we would hit the centre of gravity.

The reason why the centre of gravity must lie along this line is because the weight of the whole body appears to act downwards at this centre. The upward force from the string and the downward weight at the centre of gravity must line along the same line. Otherwise, the body would rotate. In this way, the force from the string is transmitted through the body and reaches the centre of gravity to support the weight. This is our magic, and the body is balanced, since it does not rotate. If we attach the string to a different point B and allow the body to come to rest, the straight line from the string would again go through the same centre of gravity. This means that no matter how you turn the body, the weight would always appear to act at the same point.

The Physics Notes