Circular orbit and centripetal force

Problem

A moon travels around a planet in a circular orbit. Explain why a force is needed to stop the moon flying off along the tangent, and state the direction and sense of that force.

The velocity is tangent to the orbit, the gravitational force points towards the planet.

Solution

By Newton’s first law (the principle of inertia), a body on which no force acts continues in a straight line at constant speed. If gravity vanished, the moon would carry on along the straight line tangent to its orbit, moving away from the planet.

For the orbit to be a circle rather than a straight line, the velocity $\vec v$ (always tangent to the trajectory) must continually change direction. This requires an acceleration, and hence a force, directed towards the inside of the curve: the centripetal force.

In orbital motion it is the planet’s gravitational force that provides the centripetal force: it is directed along the line joining moon and planet, pointing from the moon towards the planet (towards the centre of the orbit). At every instant this force deflects the moon from the tangent, curving the trajectory into a circle. Equating the gravitational force and the centripetal force: $G\frac{M m}{r^2} = \frac{m v^2}{r}$ $\ev{\vec F_\text{grav} = \vec F_\text{centripetal}, \text{ directed from the moon towards the planet (centre of the orbit)}}$

Links

Topics: Gravitation Concepts: Centripetal force · Uniform circular motion · Orbits Skills: Symbolic set-up