For an object orbiting a planet, such as an artificial satellite orbiting the Earth, gravity provides the centripetal force that keeps it in orbit (Figure 17.9). This is a simple situation as there is only one force acting on the satellite–the gravitational attraction of the Earth. The satellite follows a circular path because the gravitational force is at right angles to its velocity.

From Chapter 16, you know that the centripetal force F on a body is given by:

$F = \frac{{m{v^2}}}{r}$

Consider a satellite of mass m orbiting the Earth at a distance r from the Earth’s centre at a constant speed v. Since it is the gravitational force between the Earth and the satellite that provides this centripetal force, we can write:

$\frac{{GMm}}{{{r^2}}} = \frac{{m{v^2}}}{r}$

where M is the mass of the Earth. (There is no need for a minus sign here as the gravitational force and the centripetal force are both directed towards the centre of the circle.)
Rearranging gives:

${v^2} = \frac{{GM}}{r}$

This equation allows us to calculate, for example, the speed at which a satellite must travel to stay in a circular orbit. Notice that the mass of the satellite m has cancelled out. The implication of this is that all satellites, whatever their masses, will travel at the same speed in a particular orbit. You would find this very reassuring if you were an astronaut on a space walk outside your spacecraft (Figure 17.10). You would travel at the same speed as your craft, despite the fact that your mass is a lot less than its mass.
The equation can be applied to the planets of our solar system – M becomes the mass of the Sun.

Now look at Worked example 3.