Physics Problem A rotating star collapses under the influence of gravitational forces to form a pulsar?

Question

I cannot figure out this problem, I figured out the answers for part (a) and (d) but I cannot figure out parts (b) and (c). I have tried every equation and plugged and chugged and still no correct answer...Help! Please!

A rotating star collapses under the influence of gravitational forces to form a pulsar. The radius of the star after collapse is 3.00 10-4 times the radius before collapse. There is no change in mass. In both cases the mass of the star is uniformly distributed in a spherical shape.

(a) What is the ratio of the angular momentum of the star after collapse to before collapse?

The answer to part (a) is 1

(b) What is the ratio of the angular velocity of the star after collapse to before collapse?

(c) What is the ratio of the rotational kinetic energy of the star after collapse to before collapse?

(d) If the period of the star's rotation before collapse is 2.00 107 s, what is its period after collapse?

The answer to (d) is 1.8 seconds

Answer 1

The angular momentum is Iw,

where the moment of inertia I α MR^2 and w is the angular velocity.

The rotational K.E. = 1/2 I w^2.

So:

Part (d) and (a) together tell you that

wf/wi = 2 x 10^7 / 1.8 = 1.11... x 10^7.

and thus

Rf / Ri = (wf / wi)(- 1/2) = 0.0003.

(This is actually confirming what you found for the angular speed, GIVEN the ratio of the final to initial radius.)

Thus the desired answers to parts (b) and (c) are:

(b) wi/wf = 1.8 / (2 x 10^7) = 9.00 x 10^(- 8);

(c) K.E.f /K.E.i = (Rf/Ri)^2 * (wf/wi)^2

= (0.0003)^2 * (1.11... x 10^7)^2 = 1.11... x 10^7.

QED

You might ask "Where does that HUGE increase in the kinetic energy come from?" The answer, of course, is from the gravitational energy released in the collapse. The large reduction in the radius of the object turning into a pulsar only happens because gravity has completely overcome the pressures that were earlier responsible for the support of the pre-pulsar.

Live long and prosper.

Answer 2

> The answer to part (a) is 1

Correct.

> (b) What is the ratio of the angular velocity of the star after collapse to before collapse?

Use the equation that relates angular velocity to angular momentum:

L = Iω

where:
L = angular momentum;
I = moment of inertia;
ω = angular velocity.

We know (from part (a)) that the angular momentum is the same before and after the collapse:

L_before = L_after
(I_before)(ω_before) = (I_after)(ω_after)

Now rearrange that 2nd equation to find the ratio of the angular velocities:

(ω_after) ⁄ (ω_before) = (I_before) ⁄ (I_after)

Now you just need to know I_before and I_after. Look up the formula for the moment of inertia of a uniform sphere (there's probably a table in your book that shows this). You will see that it is given in terms of the sphere's mass and radius; and you'll find that the mass will cancel out, and you can determine (I_before) ⁄ (I_after) by just knowing (R_before) ⁄ (R_after), which is given in the problem.

> (c) What is the ratio of the rotational kinetic energy of the star after collapse to before collapse?

Rotational KE is given by this formula:

KE = Iω²/2

Notice this is the same as:
(Iω)ω/2
= Lω/2

So:
KE_after ⁄ KE_before = [(L_after)(ω_after)/2] ⁄ [(L_before)(ω_before)/2]

You already know that L_after = L_before; and you figured out (ω_after) ⁄ (ω_before) already: so this quotient should be easy.

Answer 3

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