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 find the angular velocity of the expanding supernova shell, we can use the principle of conservation of angular momentum. Before the explosion, the star is rotating with an angular velocity of 2.8 revolutions per day. After the explosion, all of the star's mass is contained in the expanding shell.
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. For a solid sphere rotating about its symmetry axis, the moment of inertia is given by:
I = (2/5)MR²
where M is the mass of the star and R is its radius.
Since all of the star's mass is contained in the shell after the explosion, we can assume that the entire mass M is now spread out over the volume of the shell, which is given by:
V = (4/3)π[(4.5R)³ - R³]
To find the moment of inertia of the expanding shell, we need to calculate the mass of the shell by multiplying its volume by the density of the star:
M_shell = ρ(V)
where ρ is the mass density of the star. However, we are not given the mass density of the star, so we cannot determine the moment of inertia. Therefore, we cannot calculate the angular velocity of the expanding shell using the given information.
To find the angular velocity of the expanding supernova shell, we can use the principle of conservation of angular momentum. The angular momentum of a system remains constant unless acted upon by external torques.
Before the explosion, the star is a solid sphere rotating at 2.8 revolutions per day. Let's denote this as the initial angular velocity, ω_i.
After the explosion, the mass of the star is blown outward in the form of a rapidly expanding spherical shell. At a certain radius, let's denote it as R_s, the entire mass of the star is contained in the shell.
To find the angular velocity of the shell, ω_s, when its radius is 4.5R, we need to equate the angular momentum before and after the explosion.
The angular momentum, L, is given by L = Iω, where I is the moment of inertia and ω is the angular velocity.
Before the explosion, the moment of inertia, I_i, for a solid sphere of radius R is given by I_i = (2/5)MR², where M is the mass of the star.
After the explosion, the remaining mass is contained in the expanding shell. The moment of inertia, I_s, for a spherical shell of radius R_s and thickness R is given by I_s = (2/3)MR_s².
Since the entire mass of the star is contained in the shell, we can equate the moments of inertia: I_i = I_s.
(2/5)MR² = (2/3)MR_s²
Simplifying, R² = (3/5)R_s²
To find R_s, we know that when the shell's radius is 4.5R, we have R_s = 4.5R. Plugging this value in, we get:
R² = (3/5)(4.5R)²
R² = (3/5)(20.25R²)
R² = 12.15R²
1 = 12.15
This is a contradiction, so our assumption that the entire mass of the star is contained in the shell is incorrect. We need to reassess our approach or assumptions.