When some stars use up their fuel, they undergo a catastrophic explosion called

Question

When some stars use up their fuel, they undergo a catastrophic explosion called a supernova. This explosion blows much or all of a star's mass outward, in the form of a rapidly expanding spherical shell. As a simple model of the supernova process, assume that the star is a solid sphere of radius R that is initially rotating at 2.7 revolutions per day. After the star explodes, find the angular velocity, in revolutions per day, of the expanding supernova shell when its radius is 5.0R. Assume that all of the star's original mass is contained in the shell.

Explanation / Answer

First of all the two parts to this problem are to recognize that you should use conservation angular momentum, and also realize that the moment of inertia of the supernova changes from a solid sphere to a spherical shell

Now, you can use conservation of angular momentum because no external torques are acting on the star, this means:

Ib wb = Ia wa

I=moment of inertia
w=angular velocity
b= before explosion

a = after explosion

Ib=2/5MR^2
wb=2.7rev/day

Now, the moment of inertia of a solid sphere is 2/5MR^2.

And the moment of inertia of a spherical shell is 2/3 MR^2, so the moment of inertia of the supernova is

Ia=2/3M(5.0R)^2

Therefore, we have

wa = Ib wb/Ia = 2/5MR^2 (2.9rev/day)/(2/3M(5.0R)^2) = 0.4*2.9/16.67 =
wa = 0.0696 rev/day

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