Please help me, I don't understand how to solve this:
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
To solve this problem, we need to analyze the conservation of angular momentum. Angular momentum is conserved when there is no external torque acting on a rotating object.
First, let's consider the initial state of the star before the explosion. The star is a solid sphere of radius R and is rotating at 2.8 revolutions per day. The angular momentum (L_i) of the star is given by:
L_i = (moment of inertia) x (angular velocity)
The moment of inertia (I_i) of a solid sphere is given by:
I_i = (2/5) * (mass) * (radius^2)
Since all of the star's mass is contained in the supernova shell after the explosion, we can substitute (mass) with the mass of the shell.
Now, let's consider the final state when the supernova shell has a radius of 4.5R. The angular momentum (L_f) of the expanding supernova shell is given by:
L_f = (moment of inertia) x (angular velocity)
To find the angular velocity (omega), we can equate the initial and final angular momenta:
L_i = L_f
(2/5) * (mass) * (radius^2) * (initial angular velocity) = (moment of inertia) * (final angular velocity)
Substituting the expressions for moment of inertia, we have:
(2/5) * (mass) * (R^2) * (2.8 revolutions/day) = (2/5) * (mass) * (4.5R^2) * (final angular velocity)
Now, we can solve for the final angular velocity:
final angular velocity = (2/5) * (R^2) * (2.8 revolutions/day) / (4.5R^2)
Simplifying the expression:
final angular velocity = (2/5) * (2.8 revolutions/day) / 4.5
final angular velocity = 1.5556 revolutions/day
Therefore, the angular velocity of the expanding supernova shell when its radius is 4.5R is approximately 1.5556 revolutions per day.