Question

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:
(A) S will run faster than P
(B) P will run faster than S
(C) they will both run at same rates as on the earth
(D) they will both runs at equal rates, but not the same as on earth

Answer 1

Hint : Oxidation defines the to and fro motion of body. Pendulum is a body suspended from a fixed point so as to swing freely to and fro under the action of quality. A spring is a device that stores potential energy, specifically elastic potential energy.

Formula:
Time period of spring, ${T_{spring}}$ $= 2\pi \sqrt {\dfrac{m}{k}}$
Time period of pendulum, ${T_{pendulum}} = 2\pi \sqrt {\dfrac{l}{g}}$ .

Complete step by step answer
According to the question, clock S and clock P run at the same rate on earth.
We know that clock S is based on the oxidation of spring.
So, time period of spring which is clock S ${T_s} = 2\pi \sqrt {\dfrac{m}{k}}$
Time period of clock S depends on mass of body $\left( m \right)$ and spring constant $\left( k \right)$
Also, the time period of clock P which is based on pendulum motion is given ${T_P} = 2\pi \sqrt {\dfrac{l}{g}}$
Time period of clock P depends on length of pendulum $\left( l \right)$ and acceleration due to gravity $\left( g \right)$
We know that, $g = \dfrac{{GM}}{{{R^2}}}$ , R $=$ Radius of earth or planet
So, g $\propto$ $\dfrac{1}{{{{\left( {2R} \right)}^2}}}$ $= >$ g $\propto \dfrac{1}{{4{R^2}}}$
As radius is doubled, g decreases so the time period of clock P also decreases as it depends on R, which is different on different planets.
But, time period of clock S will remain same as ${T_s} = 2\pi \sqrt {\dfrac{m}{k}}$
Which depends on mass but not on weight, So time period of S will be the same on other planets which are double the radius of earth.
So, S will run faster than P
Option $\left( A \right)$ is correct.

Note
A clock runs slow, when the time period of its pendulum is increased and runs fast, when time period is decreased.