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 2.8 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

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, a= before, after explosion

Ib=2/5MR^2
wb=2.8rev/day


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

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(5R)^2

so we have

wa = Ib wb/Ia = 2/5MR^2 (2.8rev/day)/(2/3M(5R)^2)
wa = 0.0672rev/day
Ib wb = Ia wa

I=moment of inertia
w=angular velocity
b, a= before, after explosion

Ib=2/5MR^2
wb=2.8rev/day


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

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(5R)^2

so we have

wa = Ib wb/Ia = 2/5MR^2 (2.8rev/day)/(2/3M(5R)^2)
wa = 0.0672rev/day
Ib wb = Ia wa

I=moment of inertia
w=angular velocity
b, a= before, after explosion

Ib=2/5MR^2
wb=2.8rev/day


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

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(5R)^2

so we have

wa = Ib wb/Ia = 2/5MR^2 (2.8rev/day)/(2/3M(5R)^2)
wa = 0.0672rev/day

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