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A clock $S$ is based on oscillation of a spring and a 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
$S$ will run faster than $P$
$P$ will run faster than $S$
They will both run at the same rate as on the earth
None of these
Solution
Frequency of first system $S$ is $\sqrt{k / m}$ where $k$ is the spring constant and $m$ is the mass of spring system.
Frequency of second system $P$ is $\sqrt{g / l}$ where $l$ is the length of the pendulum.
Now, Acceleration due to gravity is given as $g=G M / R^{2}=G \times \rho \times 4 \pi R^{3} / 3 R^{2}=$ $4 \pi \rho G R / 3$
As the $R$ is doubled, $g$ will get doubled, so the frequency of pendulum system $P$ will become $\sqrt{2}$ times of that of earth.
Thus now, $P$ will run faster than $S$.
