Question

What will be the duration of the day if the earth suddenly shrinks to 1/64 of its volume mass remaining unchanged?

Answer 1

Whenever the mass is constant and body is a rotating body then we can apply the law of conservation of angular momemtum i.e L=I×w(1)

Where I=moment of inertia of the body

w=angular speed

Now V=volume i.e

Vintial=Vfinal

4/3×pi× r1^3=1/64 ×4/3×pi×r2^3

solving the above equation we get r2=1/4

Using r2 in equation 1 we get

I=2/5mr^2

L=2/5m(1/4)^2×w

It means that the momemt of inertia is decreased by a factor 16 from its orginal value.

It means the angular speed have to increase by a factor of 16.

And angular speed (w)=distance/time1 (2)

The earth distance is constant so equation 2 will be converted to 6400km/time1

Now equating it we get

6400km/time1=1/16×6400km/time2

Time 1=24hrs

So time 2 will be will be 24/16=1.5hrs.

The duration will be 1.5 hrs

Hope this is satisfactory