Prove that the length of a day becomes T' = 6 h instead of T = 24 h if the earth suddenly contracts to half its present radius (consider earth as a spherical body), without having any change in its mass.

Prove that the length of a day becomes T' = 6 h instead of T = 24 h if

1.	Prove that the length of a day becomes T' = 6 h instead of T = 24 h if the earth suddenly contracts to half its present radius (consider earth as a spherical body), without having any change in its mass.
Answer» Solution :Since the earth is a SOLID sphere its moment of inertia `I PROP R^(2)` (R= radius of the earth ) [` :.` moment of inertia of solid sphere `= (2)/(5) "MR"^(2)`] So, if the present radius is R and the CHANGED radius is `(R )/ (2)` then, `(I)/(I). =(R^(2))/(((R)/(2))^(2))=4` Again if the present angular velocity is `omega` and the changed angular velocity is `omega` , then according to the PRINCIPLE of conservation of angular momentum, `Iomega=I.omega.or,omega.=(I)/(I.)omega=4omega` `:. (2pi)/(T.)=4. (2pi)/(T)or,T.=(T)/(4)=(24)/(4)=6h`.