Question

if earth were to shrink to half its present diameter without any change in its mass the duration of the day will be
48hours
6hours
12hours
24hours
plzzz help me fast tomorrow i m having my exams

Answer 1

Lets say the speed or angular velocity that the earth is spinning with right now is
w_1 = 2pi/24
because it makes a full rotation (2pi) every 24 hours.

The law of conservation of angular momentum states that angular momentum must be preserved, so in this case, if your radius is shrinking, your angular velocity must increase to preserve angular momentum.

Angular momentum = angular velocity * moment of intertia
L = I * w

Since angular momentum is conserved, we can say that

L_1 = L_2
and therefore
I_1 * w_1 = I_2 * w_2

Moment of inertia for a solid sphere is found using the equation:
L = 2/5 * M * R^2
However in this case, we can simply use proportionality to save us some work

Mass is unchanged, 2/5 is a constant, and therefore moment of inertia is proportional to R^2 (and the constants will cancel out of our angular momentum equation)

This leaves us with:

R_1^2 * w_1 = R_2^2 * w_2

you know R_1 is one earth radius, and R_2 is one half earth radius, and w_1 is 2pi/24

Now we can simply solve for w_2

1^2 * 2pi/24 = 1/2^2 * w_2

w_2 = pi/3 = 2pi/6

That means the earth makes a full rotation (2pi) every 6 hours

Answer 2

thanks a lot