Q. If the radius $r$ of earth suddenly changes to $x$ times the present values, the new period of rotation would be
A) $dT/dt = (T/r)(dr/dt)$
B) $dT/dt = (2T/r)(dr/dt)$
C) $dT/dt = (r/T)(dr/dt)$
D) $dT/dt = \left( \frac{1}{2}T/r \right)(dr/dt)$
Explanation
(b) As no torque is being applied, angular momentum
$L = l\omega =$constant
$\left( \frac{2}{5}Mr^{2} \right)\frac{2\pi}{T} =$constant
or $\frac{r^{2}}{T} =$constant
Differentiating w.r.t time $(t)$, we get
$\frac{T.2r\frac{dr}{dt} - r^{2}\frac{dT}{dt}\ }{T^{2}} = 0$
or $2Tr\frac{dr}{dt} = r^{2}\frac{dT}{dt}$
or $\frac{dT}{dt} = \frac{2T}{r}\frac{dr}{dt}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.