CENTRE OF GRAVITY:
Centre of gravity of a body is defined as the point through which the whole weight of the body acts. A body can have only one centre of gravity for all positions of the body. It is represented or denoted by C.G. or G.
Consider the following lamina. Let’s assume that it has been exposed to gravitational field. Obviously every single element will experience a gravitational force towards the centre of earth. Further let’s assume the body has practical dimensions, then we can easily conclude that all elementary forces will be unidirectional and parallel.
Consider G to be the centroid of the irregular lamina. As shown in first figure we can easily represent the net force passing through the single point G. We can also divide the entire region into let’s say n small elements.
Let’s say the coordinates to be (x1,y1), (x2,y2), (x3,y3)………. (xn,yn) as shown in figure . Let ΔW1, ΔW2, ΔW3,……., ΔWn be the elementary forces acting on the elementary elements. Clearly,
W = ΔW1+ ΔW2+ ΔW3 +…………..+ ΔWn
When n tends to infinity ΔW becomes infinitesimally small and can be replaced as dW.
Centre of gravity :
And we have seen that
W.For some type of surfaces of bodies there lies a probability that the centre of gravity may lie outside the body. Secondly centre of gravity represents the entire lamina, therefore we can replace the entire body by the single point with a force acting on it when needed. There is a major difference between centre of mass and centre of gravity of a body.
For centre of gravity we integrate with respect to dW whereas for centre of mass we integrate with respect to dm. Mass is a scalar quantity and force a vector quantity. For general practical size objects both of them turn out to be the same as both of them are proportional and the force is unidirected (dW = dm*g) .
But when we consider large size objects such as a continent, results would turn out to be different because here the vector nature of dW comes into play.
