What will be the duration of the day of earth if the earth suddenly shrinks to `1/27` of its original volume, mass remains unchanged ?

Explanation

We are given:
Final volume of earth = `1/27 xx` initial volume of earth
i.e., `4/3 pi R_2^2 = 1/27 xx 4/3 pi R_1^2 " or " (R_2)/(R_1) = 1/3 " " .....(i)`
Using law of conservation of angular momentum
`I_1omega_1 = I_2omega_2`
`2/5 MR_1^2 xx (2pi)/(T_1) = 2/5 MR_2^2 xx(2pi)/(T_2) " " [because I = 2/5 MR^2]`
`T_2 = ((R_2)/(R_1))^2 T_1 " " ......(ii)`
Substituting the value of from equation (i) in equation (ii), we get
`T_2 = (1/3)^2 T_1 = (T_1)/(9)`
Since `T_1 = 24 h `So `T_2 = 24/9 = 2.66 = 2.7 h`.