Conservation of Angular Momentum A collapsing Star

[dsmwc8]

1. After a collapsing star decreases its radius to half its initial size, predict what will happen to its angular velocity(Assume uniform density at all times)Your answer describes the high angular speed of neutron stars. Find the change in the rotational kinetic energy of the star. Where does the energy come from?



I=2/5MR^2 I think the energy comes from the force of the stars own gravity,

I also thought that the angular velocity would increase but I don't know how to find the change in kinetic energy. Any clear explanation would be great, I think I am on the right track but need a little help.

The Attempt at a Solution

 

[Dick]

You are completely correct about the source of the energy. Find the change in angular velocity using conservation of angular momentum. Use the change in angular velocity to find the change in rotational kinetic energy.

 

[ogb p]

When a star collapses, its radius decreases while its mass remains the same. This means that its moment of inertia, or I, decreases as well, since it is directly proportional to the square of the radius. Therefore, according to the law of conservation of angular momentum, the angular velocity must increase in order to maintain the same angular momentum. This is because the product of I and angular velocity, L= Iω, must remain constant.

To find the change in rotational kinetic energy, we can use the equation E= ½Iω^2. Since the mass and density of the star are assumed to be uniform, the change in kinetic energy, ΔE, can be expressed as:

ΔE= ½I(Δω)^2

Substituting the value of I= 2/5MR^2, we get:

ΔE= ½(2/5MR^2)(Δω)^2

Since the radius decreases to half its initial size, ΔR= -R/2. Therefore, the change in angular velocity, Δω, can be calculated as:

Δω= ωf- ωi= ω(R/2)^2- ω(R)^2

= ω(R^2/4- R^2)

= ω(3R^2/4)

Substituting this value in the equation for ΔE, we get:

ΔE= ½(2/5MR^2)(ω(3R^2/4))^2

= ½(2/5MR^2)(9ω^2R^4/16)

= 9/32MR^2ω^2

Therefore, the change in rotational kinetic energy is directly proportional to the square of the initial angular velocity and the square of the initial radius. This means that as the star collapses, the energy is converted from its potential energy (due to its own gravity) to kinetic energy, resulting in an increase in angular velocity.

In conclusion, the conservation of angular momentum and the decrease in moment of inertia lead to an increase in angular velocity and rotational kinetic energy in a collapsing star. The source of this energy is the gravitational force of the star itself.