Rotational Motion (Neutron Star)

[patrickmoloney]

Homework Statement

I'm doing a question from a past paper, preparing for an upcoming exam. There is no solutions so I was wondering if my answer is correct for all parts.

Take a star to be a uniform sphere with mass $$M_{i}=3.0 \times 10^{30} Kg$$ and radius $$R_{i} = 7.0 \times 10^{8}m$$ that rotates with a period of 27.0 days.

(i) What's the star's angular speed of rotation?
(ii) What's the star's angular momentum?
(iii) At the end of it's lifetime, the star collapses to form a very compact star called a neutron star. If the star retains all it's mass when it collapses and the angular speed of the rotation of the neutron star that forms is $$\omega _{f} = 10^{4} rad/s$$. what is the radius of the neutron star in kilometres?
(iv) If the neutron star's period of rotation is observed to be increasing at a rate of $$1.2 \times 10^{-5}s/yr$$, what is the torque acting on the star?

Homework Equations

The Attempt at a Solution

(i) $$27 days = 2332800 s$$. The star rotates 360 degrees in 27 days. $$\omega = \frac{\frac{360}{2\pi}}{T} = 2.46 \times 10^{-5} rad/s$$

(ii) $$\begin{align} L & =I \omega \\ & = \frac{2}{5} M_{i}R_{i}^{2} \omega \\ & = \frac{2}{5}(3 \times 10^{30})(7 \times 10^{8})^{2} \\ & = 1.446 \times 10^{43} Kgm^{2}/s\end{align}$$

(iii) Conservation of angular momentum $$\begin{align} I_{i} \omega_{i}&= I_{f} \omega_{f} \\ \frac{2}{5}M_{i}R_{i}^{2} \omega_{i} & = \frac{2}{5}M_{f}R_{f}^{2} \omega_{f} \\ R_{f} & = \sqrt{\frac{\omega_{i}}{\omega_{f}}R_{i}^{2}} \\ & = 34.718.87 m \\ & = 34.71 km \end{align}$$

(iv) $$\begin{align} \Delta L & = I_{f} \omega_{f} - I_{i} \omega{i} \\ & = 1.2 \times 10^{34} kgm^{2}/s\end{align}$$

$$\tau = \frac{\Delta L}{\Delta T} = \frac{1.2 \times 10^{34}}{1.2 \times 10^{-5}} = 1 \times 10^{39} N$$

Thanks. Does this look okay?

 

[gneill]

Revisit your determination of the angular velocity in part (i). The method should not involve degrees at all. One "cycle" or rotation is $2 \pi ~ rad$ (just as one rotation is also 360 degrees, but your angular velocity should be specified in radians per second).

I believe that part (iv) is meant to pertain to the star once it's already in neutron star form: it's observed to be slowing at the specified rate. So it won't involve the initial state of the star at all.