How would be the duration of a day change if the size of earth decreased without a change in mass?

Answer 1

Simple Answer:

If the Earth shrinks, the day gets shorter and the Earth rotates faster. But, that involves certain assumptions. See below.

Angular Momentum:

This is a question about angular momentum.

If the change in size of the Earth is a result of an internal process and no external forces are involved, then total angular momentum is conserved. Angular momentum is the sum (integral) of all mass times velocity times the distance from the axis of rotation. That sum will stay the same as you redistribute mass with no external influences. If mass gets closer to the axis of rotation, then some mass has to go faster.

The moment of inertia of a uniform sphere is I=2MR^2/5. Angular momentum is L=I omega. Omega is 2 pi/T and the period, T, is one day. So, L=2 pi I T. (The Earth is not uniform, but the result is the same.) If you keep M fixed and the density uniformly increases, then omega decreases. We can't get too precise without discussing more details, but the simple analysis suggests, T/R^2 is constant, so the period is proportional to square of the radius. Halve the radius and the day decreases to one fourth.

Answer 2

If the size of Earth decreased without a change in mass, the conservation of angular momentum would cause the Earth to rotate faster. This would result in shorter days, as the rotation speed increases to compensate for the decreased radius.

Answer 3

NO