The rotational inertia of a collapsing star changes to one third its initial value while conserving angular momentum. What is the ratio of the new rotational kinetic energy to the initial kinetic energy?

Question

The rotational inertia of a collapsing star changes to one third its initial value while conserving angular momentum. What is the ratio of the new rotational kinetic energy to the initial kinetic energy?

Answer 1

I guess the moment or inertia is 1/3 of original

If I w is constant then w' = 3 w
then if original Ke = (1/2) I w^2
the final is (1/2)(I/3)(9w^2)
= 3 times original

has to do work to pull mass in toward the axis of rotation against centripetal force.

Answer 2

To solve this problem, we can use the conservation of angular momentum and the equation for rotational kinetic energy.

Initial conditions:
Let the initial rotational inertia be represented by I_initial.
Let the initial rotational kinetic energy be represented by KE_initial.

Final conditions:
The rotational inertia changes to one third its initial value. Therefore, the final rotational inertia is represented by (1/3)I_initial.
The question asks for the ratio of the new rotational kinetic energy (KE_final) to the initial kinetic energy (KE_initial).

Conservation of angular momentum states that the initial angular momentum (L_initial) equals the final angular momentum (L_final). Mathematically, this can be expressed as:

L_initial = L_final

Since angular momentum (L) is given by the equation L = I * ω, where ω represents the angular velocity, we can write the conservation of angular momentum equation as:

I_initial * ω_initial = I_final * ω_final

Now, let's simplify the equation by replacing the values of I and ω:

I_initial * ω_initial = (1/3)I_initial * ω_final

We can simplify this expression by canceling out the I_initial term and rewriting it as:

ω_final = 3 * ω_initial

Now, let's move on to the equation for rotational kinetic energy, which is given by:

KE = (1/2) * I * ω^2

Where KE is the rotational kinetic energy, I is the rotational inertia, and ω is the angular velocity.

Using this equation, we can calculate the new rotational kinetic energy (KE_final):

KE_final = (1/2) * I_final * ω_final^2

Substituting the known values into this equation:

KE_final = (1/2) * (1/3)I_initial * (3 * ω_initial)^2

Simplifying this expression, we get:

KE_final = (1/2) * (1/3)I_initial * 9 * ω_initial^2

KE_final = (3/6) * I_initial * 9 * ω_initial^2

KE_final = (3/2) * I_initial * ω_initial^2

So, the ratio of the new rotational kinetic energy (KE_final) to the initial kinetic energy (KE_initial) is 3/2.