The centre of mass or is the mean or average location of all the mass in a system. In the case of a rigid body the position of the centre of mass is fixed in relation to the body. For example the centre of mass of a pool ball is exactly in the middle of it. In the case of a loose distribution of masses in free space such as pellets scattered from a shotgun or the planets of the solar system the position of the centre of mass is a point in space among them that may or may not correspond to the position of any individual mass.
The concept is useful since sometimes we want to know where exactly to apply a force on a body.
The centre of gravity of a body corresponds to the point where all the gravitational force acts, in other words the single point through which the resultant of the gravitational forces on the component particles of the body acts.
Homogeneous objects. Think about where the c.g of objects like these might lie
- Centre of gravity of a lamina or flat sheet – drop two plumblines from two separate fixed points – where they cross is the c.g
- Centre of gravity of an L shaped body- as above but the lines may cross outside of the body
- Centre of gravity of a snooker cue – the sum of all the mass elements and their associated clockwise moments equals the sum of all the mass elements on the other side of the pivot and their associated moments. Result – it balances nearer the thicker end.
- Centre of gravity of a lab stool – will lie outside the body of the material. Where’s the c.g of a doughnut?
Non- homogeneous objects.
- Bowls – eccentric balls – the ball has an eccentric c.g, thus when bowled, there’ll be a turning moment curling the ball into a curved path.
- London buses – have a very low c.g, so they can be tilted to over 40 degrees and the weight force still acts between the wheels so they won’t topple over.
- Racing cars – are subject to very large turning forces, the c.g needs to be very close to the ground otherwise they’d flip over.
For practical purposes, the centre of gravity corresponds exactly to the centre of mass because the gravitational field of the Earth (the gravitational force exerted on every kilogram of mass) is the same at 9.81N kg-1 close to the Earth’s surface
- Centre of gravity of the Burj Khalifa in Dubai. g = F/m, but is g constant? assuming the structure is homogeneous (it isn’t) there will be an imperceptibly small reduction in g at the c.g, since the centre of the building is so far away from the ground. The c.m will be less than 0.1mm away from the c.g
Three kinds of equilibrium.
Imagine a thick candle. Try balancing it on its point. It topples over, of course. This is an example of UNSTABLE equilibrium – the slightest movement moves the c.g away from and outside of the base – in this case a point – and a turning moment will cause it to topple.
If the candle is stood upright on its base, it will stay where it is, even if another (very) small lateral force is applied. As long as the c.g stays within the base, it will not topple over, but it will tend to return to its original position. This is a STABLE equilibrium.
Finally, lay the candle on its side. It can roll but its c.g simply moves laterally with the rolling motion. This is called NEUTRAL equilibrium.
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