Question

How to find total energy of a partical moving in circular motion

Answer 1

Suppose you have a satellite of mass mm at a distance rr. If we assume the satellite is small enough to behave as a point mass the moment of inertia of the satellite is:

I=mr2I=mr2

so its kinetic energy is:

E=12Iw2=12mr2ω2(1)(1)E=12Iw2=12mr2ω2

But for a body moving in a circle of radius rr at an angular velocity ωω we have the identity;

v=rωv=rω

where vv is the tangential velocity. If we substitute for rωrω in equation (1) we get:

E=12mv2E=12mv2

which is of course just the usual equation for kinetic energy. So you can treat the energy of the satellite either as rotational kinetic energy or translational kinetic energy as you find convenient.

This only works because the satellite is small enough compared to the Earth that we can assume every point is moving at the same tangential velocity. If you have some large object pivoting around a point within it, e.g. a disk rotating about its centre, the tangential velocity of all the points in the object will vary with distance from the pivot. In that case it's simpler to assume the angular form for the kinetic energy.