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 1.5 revolutions per day. After the star explodes, find the angular velocity, in revolutions per day, of the expanding supernova shell when its radius is 3.7R. Assume that all of the star's original mass is contained in the shell.

To find the angular velocity of the expanding supernova shell, we can use the principle of conservation of angular momentum. Angular momentum is conserved when no external torque acts upon an object or a system.

Before the explosion, the star is rotating with an initial angular velocity of 1.5 revolutions per day. Let's denote this initial angular velocity as ω_i.

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

Since we are assuming that all of the star's original mass is contained in the shell, we can consider the mass of the shell to be the same as the initial mass of the star.

When the radius of the supernova shell is 3.7R, its mass would have spread out over a larger volume, resulting in a larger radius but with the same total mass. The moment of inertia of the expanding shell can still be calculated using the same formula, since the mass and radius of the shell both increase proportionally.

Let's denote the final radius of the supernova shell as R_f.
At this point, we need to apply the conservation of angular momentum:

Angular momentum before explosion = Angular momentum after explosion

The angular momentum (L) is given by the product of moment of inertia and angular velocity (L = I * ω).

Before the explosion: L_i = I * ω_i
After the explosion: L_f = I * ω_f

Since all of the star's original mass is contained in the shell, the moment of inertia (I) will be the same before and after the explosion.

Therefore, L_i = L_f

We can now equate the two angular momentum expressions:

I * ω_i = I * ω_f

Since the moment of inertia (I) cancels out, we get:

ω_i = ω_f

This means that the initial angular velocity (1.5 revolutions per day) is equal to the final angular velocity (ω_f) of the expanding supernova shell.

Therefore, the angular velocity of the expanding supernova shell when its radius is 3.7R is 1.5 revolutions per day.