Look up the contraction ratio Rsun/Rearth. It is about 100
For an exact answe, you will need the radii or diameters of the sun and the earth, to calculate the contraction ratio Rinitial/Rfinal. It is approximatley 100, but you need a more precise vlaue
Remember that angular momentum, I*w, is conserved. That means that M*R^2/P stays the same, where P is the rotation period.
I*w = Constant*M*R^2/P
You can forget about the Mass m and the constant. They stay the same.
Since R decreases by about 100, P must decrease by a factor of about 10^4. That will make the new period about 29.2 days/10^4 = 4 minutes
For the ratio of kinetic energies, remember that for a rotating sphere, KE = (1/2)I*w^2
The KE ratio will be about
(Iw)final*wfinal/[(Iw)initial*winitial]
wfinal/winitial = 10^4
The increase in KE comes from work done by gravity during contraction.
A star with the same mass and diameter as the sun rotates about a central axis with a period of about 29.2 days. Suppose that the sun runs out of nuclear fuel and collapses to form a white dwarf star with a diameter equal to that of the earth. Assume the star acts like a solid sphere and that there is no loss of mass in the process. You will need some data from the inside front cover of you text. (a) What would be the new rotation period (s) of the star? (b) What is the ratio of final to inital kinetic energies
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