If the mass and the radius of the earth suddenly reduce to half their present values,
(i) how many hours will be there in a day?

Explanation

Time taken by the earth to spin once about its axis is the measure of a day. If the earth is taken as a uniform sphere , its moment of inertia about the axis of spinning, `I=2/5MR^2` (M=mass of the earth and R=radius of the earth). Let angular velocity of the earth be `omega`. When both mass and radius of the earth becomes half the present values, moment of inertia and angular velocity change to `I^. and omega^.`.
`therefore I^.=2/5(M/2)(R/2)^2=1/8xx2/5 MR^2=1/8I`
From law of conservation of angular momentum
`Iomega =I^.omega^. or, Iomega=1/8Iomega^. or, omega^.=8omega`
As angular velocity increases to 8 times the present value, length of a day will be `(24)/8` or 3h.