Question

Should the speed of two artificial satellites of the earth having the different masses but the same orbital radius be the same?

Answer 1

The task at hand in this question is to find the dependency of orbital speed on the mass of the satellite and its orbital radius. You could simply recall the expression for this quantity and thus determine the quantities on which the orbital speed is dependent and hence determine the answer.

Formula used:
Orbital speed,
$v=\sqrt{\dfrac{GM}{r}}$

Complete step by step solution:
In the question, we are asked whether the speed of two artificial satellites are the same when they have the same orbital radius and different masses. In order to answer this, we have to know on what quantities the orbital speed will depend. If we know that, it is really easy to answer this.
The orbital speed of a satellite around earth is the speed at which it orbits around earth’s centre. Mathematically, it is given by the following relation,
$v=\sqrt{\dfrac{GM}{r}}$
Where, G is the gravitational constant, M is the mass of Earth and r is the distance of the satellite from Earth’s centre. So, it is very clear that the orbital speed of the satellite is independent of its mass but depends on its orbital radius.
Thus, we could easily conclude that the given two satellites of different masses and same orbital radius will quite obviously have the same orbital speed.

Note: Orbital speed is normally given by the following expression.
$v\approx \dfrac{2\pi a}{T}\approx \sqrt{\dfrac{\mu }{a}}$
Here,$\mu \approx GM$
This formula holds true only when the orbiting body’s mass is negligible when compared to the central one (Earth, in the above case).