Question

If the earth were to suddenly contract to`(1/n)th` of its present radius without any change in its mass, the duration of the new day will be nearly (in hours) ??

Answer 1

Duration of the new day`= 24/n^2 hours`

Explanation:

Conservation of angular momentum dictates that the angular momentum before and after the contraction must be the same.

`I_1omega_1=I_2omega_2`

`I = 2/5 Mr^2 larr`moment of inertia of the earth
`omega = (2pi)/T larr` angular velocity of the earth = `(2pi)/(day)`.

Let `r_2 and T_2` be the new radius and new day duration, and rewrite the conservation equation as:

`(cancel(2/5M)r_1^2)(cancel(2pi)/T_1)=(cancel(2/5M)r_2^2)(cancel(2pi)/T_2)`

`(r_1^2)/T_1=r_2^2/T_2`

`T_2=r_2^2/r_1^2 T_1`

When the earth contracts, `r_2 = 1/nr_1`

`T_2=(1/n r_1)^2/r_1^2 T_1`

`T_2= 1/n^2T_1`

`T_1 =24 hours`

`T_2= 1/n^2T_1 = 24/n^2 (hours)`

If the radius is halved, then the new day is `24/2^2 = 6` hours long.
If radius is 1/10th, the new day is `24/10^2 = ~ ¼` hours long