The Sun rotates around its axis once in 27 days. What will be the new period of revolution if the Su

Question

The Sun rotates around its axis once in 27 days. What will be the new period of revolution if the Sun were to suddenly collapse to one-third ($1/3$) of its present radius without the influence of any external torque? Assume the Sun to be a uniform sphere.

Professor PC Thomas Classes · Answer

(c) : By the law of conservation of angular momentum, the initial and final angular momenta are equal: $I_1 \omega_1 = I_2 \omega_2$. For a uniform sphere, $I = \frac{2}{5}MR^2$. Therefore, $\frac{2}{5} M R_1^2 \left(\frac{2\pi}{T_1}ight) = \frac{2}{5} M R_2^2 \left(\frac{2\pi}{T_2}ight)$. This simplifies to $\frac{R_1^2}{T_1} = \frac{R_2^2}{T_2}$. We are given $R_2 = \frac{R_1}{3}$. Substituting this, we get $\frac{R_1^2}{27} = \frac{(R_1/3)^2}{T_2} = \frac{R_1^2}{9 T_2}$. Solving for $T_2$ yields $T_2 = \frac{27}{9} = 3 	ext{ days}.$