Question

The mass and the diameter of a planet are three times the respective values for the earth. The period of oscillation of a simple pendulum on the earth is 2 s. The period of oscillation of the same pendulum on the planet would be
1. \(\frac{2}{{\sqrt 3 }}s\)
2. \(\frac{3}{2}s\)
3. \(2\sqrt 3 {\rm{\;}}s\)
4. \(\frac{{\sqrt 3 }}{2}{\rm{\;}}s\)

Answer 1

Correct Answer - Option 3 : \(2\sqrt 3 {\rm{\;}}s\)

Concept:

A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. It is a resonant system with a single resonant frequency. For small amplitudes, the period of such a pendulum can be approximated by

\(T = 2\pi \sqrt {\frac{l}{g}} \)

Calculation:

Period of motion of a pendulum is given by

\(T = 2\pi \sqrt {\frac{l}{g}}\) ----(1)

On the surface of earth, let period of motion is T_e and acceleration due to gravity is g_e

\(\therefore \;{T_e} = 2\pi \sqrt {\frac{l}{{{g_e}}}}\) ----(2)

On another planet, let period of motion is T_P and gravitational acceleration is g_p

\(\therefore \;{T_p} = 2\pi \sqrt {\frac{l}{{{g_p}}}}\) ----(3)

(∴ Pendulum is same, so l will be same)

From Equations (2) and (3),

\(\frac{{{{\rm{T}}_{\rm{e}}}}}{{{{\rm{T}}_{\rm{p}}}}} = \frac{{2{\rm{\pi }}\sqrt {\frac{{\rm{l}}}{{{{\rm{g}}_{\rm{e}}}}}} }}{{2{\rm{\pi }}\sqrt {\frac{{\rm{l}}}{{{{\rm{g}}_{\rm{p}}}}}} }} = \sqrt {\frac{{{{\rm{g}}_{\rm{p}}}}}{{{{\rm{g}}_{\rm{e}}}}}} \) ----(4)

Now, \({{\rm{g}}_{\rm{e}}} = \frac{{{\rm{G}}{{\rm{M}}_{\rm{e}}}}}{{{\rm{R}}_{\rm{e}}^2}}\)

and \({{\rm{g}}_{\rm{p}}} = \frac{{{\rm{G}}{{\rm{M}}_{\rm{p}}}}}{{{\rm{R}}_{\rm{p}}^2}}\)

Given, M_p = 3M_e

and R_p = 3R_e

\(\therefore {{\rm{g}}_{\rm{p}}} = \frac{{{\rm{G}} \times 3{{\rm{M}}_{\rm{e}}}}}{{9{\rm{R}}_{\rm{e}}^2}} = \frac{1}{3}\cdot\frac{{{\rm{G}}{{\rm{M}}_{\rm{e}}}}}{{{\rm{R}}_{\rm{e}}^2}} = \frac{1}{3}{{\rm{g}}_{\rm{e}}}\) 

\(\Rightarrow {\rm{\;}}\frac{{{{\rm{g}}_{\rm{p}}}}}{{{{\rm{g}}_{\rm{e}}}}} = \frac{1}{3}{\rm{\;or\;}}\sqrt {\frac{{{{\rm{g}}_{\rm{B}}}}}{{{{\rm{g}}_{\rm{e}}}}}} = \frac{1}{{\sqrt 3 }}\) ----(5)

From Equations. (4) and (5) \(,{{\rm{T}}_P} = \sqrt 3 {{\rm{T}}_{\rm{e}}}\)

\({\rm{or\;}}{{\rm{T}}_{\rm{P}}} = 2\sqrt 3 {\rm{s\;}}\left( {{{\rm{T}}_{\rm{e}}} = 2{\rm{s}}} \right)\)