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 4.5R. Assume that all of the star's original mass is contained in the shell.

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 4.5R. Assume that all of the star's original mass is contained in the shell.

Answer 1

To find the angular velocity of the expanding supernova shell, we can use the conservation of angular momentum.

Angular momentum is given by the equation: L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Initially, the star is a solid sphere rotating at 2.8 revolutions per day. To find the initial angular momentum, we need to determine the moment of inertia of the solid sphere.

The moment of inertia of a solid sphere rotating around its axis can be calculated using the formula: I = (2/5) * MR^2, where M is the mass of the sphere and R is its radius.

Since all of the star's original mass is contained in the shell after the explosion, we can assume that the mass of the shell is equal to the mass of the star. Therefore, the mass of the shell is M.

The initial angular momentum (L_initial) of the star is then given by L_initial = I_initial * ω_initial.

Now let's calculate the initial angular momentum:

I_initial = (2/5) * MR^2
ω_initial = 2.8 revolutions per day

L_initial = (2/5) * MR^2 * 2.8

Next, we need to find the final angular velocity when the radius of the expanding shell is 4.5R.

According to the law of conservation of angular momentum, the initial angular momentum is equal to the final angular momentum.

L_initial = L_final

L_final = I_final * ω_final

Since all of the star's original mass is contained in the shell, we can assume that the moment of inertia of the shell is I_final = (2/5) * M(4.5R)^2.

Now, let's solve for ω_final:

L_initial = L_final
(2/5) * MR^2 * 2.8 = (2/5) * M(4.5R)^2 * ω_final

Simplifying the equation further:

2.8R^2 = 4.5^2 * ω_final

Now we can solve for ω_final:

ω_final = (2.8R^2) / (4.5^2)

Finally, we can calculate the angular velocity of the expanding supernova shell when its radius is 4.5R by substituting the given value for R:

ω_final = (2.8 * R^2) / (4.5^2)

Note: Make sure to convert the result to revolutions per day, as specified in the question.