# 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

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**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.