# If earth contracts to half its radius ,what will be the length of the day in hours?

Neglecting external torques on our little sphere of dust, the angular momentum is conserved:

(I w)before = (I w)after

I = proportional to the mass and to the radius squared (note I don't have to assume a uniform sphere), so halving the radius (at the same mass) will quarter the moment of inertia I. Therefore the angular speed w must increase by a factor 4.

A day would last 6 of our current hours.

Day reduces 2² = 4 times. So new day is 24/4 = 6 hours.

If you want a full explanation...

The moment of inertia (I) of uniform solid sphere is ⅖mr². The earth is not uniform but this only changes the constant factor ⅖. So halving the radius changes I by a factor (½)=¼.

Assuming no external torques have acted, the earth's angular momentum (L) remains constant.

Since L = Iω (where ω is angular velocity) and L is constant, ω increases by a factor 4 (to compensate for ω decreasing by a factor 4).

So the earth spins 4 times as fast which means the period (a day) is reduced from 24 hours to 24/4 = 6 hours.

Infinite, since the Earth would just fly off into space, never again seeing its Sun.

Your question is too vague to proffer an answer.