physics help. needed please

Question

If the radius of the earth becomes half of its present value , with its mass remaining the same , the duration of the one day will become ?

1. 6 hour
2. 12 hour
3. 48 hour
4. 96 hour

 

although i think faintly think that Kepler's  formula (time)T² = (radius)R³ could be used but not so sure 

Answer 1

Moment of inertia of a sphere:

 

I=(2/5)MR²

 

Angular momentum:

 

L=Iω

 

ω is angular velocity.

 

Problem assumes M does not change, R is cut in half.  But L must remain the same, since there are no torques applied.

 

Original situation:

 

L=I₁ω₁

 

Final situation:

 

L=I₂ω₂

 

I₂ω₂=I₁ω₁

 

 

ω₂=(I₁/I₂)ω₁

 

 

ω₂=(R₁/R₂)²ω₁=(1/4)ω₁

 

Thus, length of day is reduced to one quarter of what it was, i.e., 6 hours.