Question

Two satellites of identical masses orbit the earth at different heights. The ratio of their distances from the centre of earth is $d:1$ and the ratio of acceleration due to gravity at those heights is $g:1$. Then find the ratio of their orbital velocities.
A) $\sqrt {\dfrac{g}{d}} $
B) $\sqrt {gd} $
C) $\sqrt g $
D) $\sqrt g d$

Answer 1

Hint: Orbital Velocity is the velocity of the artificial Earth's satellite for revolving around the Earth.
Mathematically, Orbital Velocity is given by :
$\sqrt {\dfrac{{GM}}{r}} $
Using the above relation we find the ratio orbital velocities of the two given satellites.

Complete step by step solution:
Let's discuss satellites and orbital velocity first and then we will find the ratio of the orbital velocities.
A satellite is a body which is continuously revolving around the bigger body. Centripetal force is responsible for the revolution of the satellite along with the Gravitational attraction between the satellite and the body around which it is revolving.
Orbital Velocity: It is the velocity which is given to an artificial Earth's satellite a few hundred kilometers above the earth surface so that it may start revolving round the Earth. It is denoted by $v_0$.
Now, we will calculate the ratio of orbital velocities of the two satellites.
Orbital Velocity is given by:
$ \Rightarrow {v_0} = \sqrt {\dfrac{{GM}}{r}} $.................(1) (orbital velocity)
From the formula for gravitational force value of gravitational acceleration can be given as:
$ \Rightarrow g = \dfrac{{GM}}{{{r^2}}}$ (gravitational acceleration)
$ \Rightarrow gr = \dfrac{{GM}}{r}$...................(2)
Now orbital velocity can be given as:
$v = \sqrt {gr} $
From equation 1 and 2 we get
$ \Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{{\sqrt {{g_1}{r_1}} }}{{\sqrt {{g_2}{r_2}} }}$
We are provided in the question that ratio of gravitational acceleration is $g:1$ and distance between the satellites is $d:1$, using this information we have
$ \Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{{\sqrt {{g_1}{d_1}} }}{{\sqrt {{g_2}{d_2}} }}$(r is replaced by d)
$ \Rightarrow v = \sqrt {gd} $

Thus option (D) is correct.

Note: We have many satellites in our solar system some are artificial and some are natural satellites. In our solar system Sun is the biggest body among all other celestial bodies present in the solar system around which planets are revolving, so planets can be said to be satellites. Similarly, the moon revolves around the Earth, and the moon is also a natural satellite.