Why does an unhinged body rotate about its centre of mass?

Question

If a force is applied to a body which does not act through its centre of mass, it rotates about its centre of mass and not any other point. Why?

Answer 1

Actually, it does rotate about other points and with the same angular velocity as that of the rotation about the center of mass.

It is just that the simplest description of the motion, after the application of the force, is a translation of the center of mass together with rotation about the center of mass.

For example, consider a uniform rod, aligned along the x axis, of mass M and length L initially at rest. whose center of mass is located at the origin. If an impulsive force of magnitude F acts on the end of the rod in the y direction with short time duration Δt the center of mass will move with velocity

v = F Δt / M and there will be a rotation about the center of mass with angular velocity

ω = (F Δt L/2 ) / ( M L² /12 )

These formulas can be derived using impulse equals change in momentum.

After a time T, the coordinates of the center of mass will be at

( 0, v T) .

We can think about the motion after the application of the force in an alternate inertial frame of reference - namely one moving parallel to the the y direction with velocity v. In this frame, the center of mass stays at (0,0).

In this frame, an ant stuck on the right hand end will be at [ ( L/2) cos(ω T), ( L/2) sin(ω T) ].

This is rotation about the origin.

The ant's velocity vector is time derivative of this coordinate pair.

This ant is going to get dizzy because if it starts out looking along the y direction, at a later time it will be looking along the -x direction and still later along the -y direction.

That is to say the ant is rotating as well as translating.

This is an intuitive way of seeing that there is rotation about every point on the rod.