Question

A clock $S$ is based on oscillation of a spring and another clock $P$ is based on pendulum motion. Both clocks run at the same rate on earth. On a planet having the same density as earth but twice the radius:-

• A. $S$ will run faster than $P$
• B. $P$ will run faster than $S$
  
• C. They will both run at the same rate as on the earth
  
• D. None of these

Answer 1

The correct option is B $P$ will run faster than $S$


Time period of clock $S$ is, $T_S=2\pi \sqrt{\frac{m}{K}}$
Time period of clock $S$ only depends on spring constant and mass of the block. So, its time period will be same on new planet as that of on the earth.

Time period of clock $P$ is, $T_P=2\pi \sqrt{\frac{l}{g}}$
Acceleration due to gravity,
$g=\frac{GM}{R^2}=\frac{4}{3}\pi \rho GR$
As given, density $\rho$ is same and $R=2R_e$. Hence $g=2g_e$
So, time perod of clock $P$ will reduce on the new planet.

Given, $T_S=T_P$ on earth.
But on the new planet, $T_P<T_S$. So $P$ will run faster than $S$.