Answer to Question #82038 in Classical Mechanics for Krishna

Question

If the earth suddenly shrinks to (1/27)th of its original volume with no change in mass, what is the new duration of a day ?

Answer 1

V^'/V=1/27=1/3^3 =(r^'/r)^3

From the conservation of angular momentum:

I_1 ω_1=I_2 ω_2

The moment of inertia of the sphere is

I=2/5 mr^2.

Thus,

2/5 mr^2 ω_1=2/5 m(r/3)^2 ω_2

ω_2=9ω_1.

We know that:

T~1/ω.

Thus,

T_2=T_1/4=24/9=2 hours 40 minutes.