question_answer

A massive star is spinning about its diameter with an angular speed \[{{\omega }_{0}}=\frac{\pi }{1000}rad/day.\] After its fuel is exhausted, the star collapses under its own gravity to form a neutron star. Assume that the volume of the star decreases to \[{{10}^{-12}}\] times the original volume and its shape remains spherical. Assuming that the density of the star is uniform, find the angular speed of the neutron star.

A) \[2\pi \times {{10}^{5}}\,rad/s\]  

B) \[\pi \times {{10}^{5}}\,rad/s\]

C) \[\frac{5\pi }{3}\times {{10}^{5}}\,rad/s\]  

D) \[\frac{5\pi }{2}\times {{10}^{5}}\,rad/s\]

Correct Answer: B

Solution :

[b] \[\frac{V}{{{V}_{0}}}={{10}^{-12}}\Rightarrow \frac{\frac{4}{3}\pi {{r}^{3}}}{\frac{4}{3}\pi {{R}^{3}}}={{10}^{-12}}\Rightarrow \frac{r}{R}={{10}^{-4}}\] Angular momentum conservation gives \[\frac{2}{5}M{{R}^{2}}{{\omega }_{0}}=\frac{2}{5}M{{r}^{2}}\omega \] \[\Rightarrow \,\,\,\,\omega ={{\omega }_{0}}{{\left( \frac{R}{r} \right)}^{2}}=\frac{\pi }{{{10}^{3}}}\times {{({{10}^{4}})}^{2}}=\pi \times {{10}^{5}}rad/s\]