Question

A star of radius 1.3 × 105 km rotates about
its axis with a period of 36 days. The star
undergoes a supernova explosion, whereby its
core collapses into a neutron star of radius
11 km.
Estimate the period of the neutron star
(assume the mass remains constant).
Answer in units of s

Answer 1

You can use conservation of angular momentum and treat each body as a sphere

\[L = I\omega\]

\[L_I = L_f\]
\[ I\omega_{Big \ Star} = I \omega_{Neutron \ Star}\]
\[I_{sphere} = \frac{2}{5}mr^2\]

solve for the angular velocity of the neutron start, then convert to the period

\[\omega = 2\pi f=\frac{2 \pi}{T}\]

Answer 2

what does L and I stand for?

Answer 3

L is angular momentum and I is the moment of inertia ^_^ Have you used those before?

Can you find the angular velocity of the star given its period of rotation?

Answer 4

I've never used this before :/ maybe that's why it's so hard ha

Answer 5

^_^ You've done normal conservation of momentum before, right?

Answer 6

\[ p_i = p_f
\\ \ \\ mv_i = mv_f\]

Conservation is angular momentum is exactly like that, exact we use something called a cross product
\[\textbf r_i \times m \textbf v_i = \textbf r_f \times m \textbf v_f\]

Luckily for this problem, that simplifies down to

\[I_i\omega_i = I_f \omega_f\]

The moment of inertia, I, of an object is kinda like a term that shows "an objects tendency to resist changes in motion" - which is exactly what "mass" does in a normal momentum problem.

In a table you can look up that the moment of inertia of a sphere is

\[I_{sphere} = \frac{2}{5}mr^2\]

and the angular velocity, omega, of the original sun is given by 2 pi divided by its period of rotation (which is saying that it goes 2 pi radians, or one complete revolution, in T seconds.)

\[\omega_{sun} = \frac{2 \pi}{T_{sun}}\]

\[I_{sun}\omega_{sun} = I_{neutron} \omega_{neutron}\]

\[\Big(\cancel{\frac{2}{5}}\cancel{m}r^2_{sun} \Big)\omega_{sun} = \Big(\cancel{\frac{2}{5}}\cancel{m}r^2_{neutron} \omega_{neutron}\]
Notice that writing out both terms from the definitions of moment of inertia lets the masses cancel out ^_^ Can you see where to go from here?

Answer 7

Yes thanks for your help :)