When an object’s velocity is changing, it has acceleration. In the case of uniform circular motion, the acceleration is rather unusual because, as we have seen, the object’s speed does not change but its velocity does. How can an object accelerate and at the same time have a steady speed?
One way to understand this is to think about what Newton’s laws of motion can tell us about this situation. Newton’s first law states that an object remains at rest or in a state of uniform velocity (at constant speed in a straight line) unless it is acted on by an external force. In the case of an object moving at steady speed in a circle, we have a body whose velocity is not constant; therefore, there must be a resultant (unbalanced) force acting on it.
Now we can think about different situations where objects are going round in a circle and try to find the force that is acting on them.
- Consider a rubber bung on the end of a string. Imagine whirling it in a horizontal circle above your head (Figure 16.7). To make it go round in a circle, you have to pull on the string. The pull of the string on the bung is the unbalanced force, which is constantly acting to change the bung’s velocity as it orbits your head. If you let go of the string, suddenly there is no tension in the string and the bung will fly off at a tangent to the circle.
- Similarly, as the Earth orbits the Sun, it has a constantly changing velocity. Newton’s first law suggests that there must be an unbalanced force acting on it. That force is the gravitational pull of the Sun. If the force disappeared, the Earth would travel off in a straight line.
In both of these cases, you should be able to see why the direction of the force is as shown in Figure 16.8.
The force on the object is directed towards the centre of the circle. We describe each of these resultant forces as a centripetal force – that is, directed towards the centre.

It is important to note that the word centripetal is an adjective. We use it to describe a force that is making something travel along a circular path. It does not tell us what causes this force, which might be gravitational, electrostatic, magnetic, frictional or whatever.

Vector diagrams

Figure 16.9a shows an object travelling along a circular path, at two positions in its orbit. It reaches position B a short time after A. How has its velocity changed between these two positions?
The change in the velocity of the object can be determined using a vector triangle. The vector triangle in Figure 16.9b shows the difference between the final velocity ${v_B}$ and initial velocity ${v_A}$. The change in the velocity of the object between the points B and A is shown by the smaller arrow labelled $\Delta v$. Note that the change in the velocity of the object is (more or less):
- at right angles to the velocity at A
- directed towards the centre of the circle.
The object is accelerating because its velocity changes. Since acceleration is the rate of change of velocity, it follows that the acceleration of the object must be in the same direction as the change in the velocity – towards the centre of the circle. This is not surprising because, according to $F = ma$, the acceleration a of the object is in the same direction as the centripetal force F:

Figure 16.9: Changes in the velocity vector.

Acceleration at steady speed

Now that we know that the centripetal force F and acceleration are always at right angles to the object’s velocity, we can explain why its speed remains constant. If the force is to make the object change its speed, it must have a component in the direction of the object’s velocity; it must provide a push in the direction in which the object is already travelling. However, here we have a force at ${90^ \circ }$ to the velocity, so it has no component in the required direction. (Its component in the direction of the velocity is $F\,\cos \,{90^ \circ } = 0$.) It acts to pull the object around the circle, without ever making it speed up or slow down.
You can also use the idea of work done to show that the speed of the object moving in a circle remains the same. The work done by a force is equal to the product of the force and the distance moved by the object in the direction of the force. The distance moved by the object in the direction of the centripetal force is zero; hence the work done is zero. If no work is done on the object, its kinetic energy must remain the same and hence its speed is unchanged.

Understanding circular motion

Isaac Newton devised an ingenious thought experiment that allows us to think about how an object can remain in a circular orbit around the Earth. Consider a large cannon on some high point on the Earth’s surface, capable of firing objects horizontally. Figure 16.10 shows what will happen if we fire them at different speeds.
If the object is fired too slowly, gravity will pull it down towards the ground and it will land at some distance from the cannon. A faster initial speed results in the object landing further from the cannon.
Now, if we try a bit faster than this, the object will travel all the way round the Earth. We have to get just the right speed to do this. As the object is pulled down towards the Earth, the curved surface of the Earth falls away beneath it. The object follows a circular path, constantly falling under gravity but never getting any closer to the surface.

If the object is fired too fast, it travels off into space, and fails to get into a circular orbit. So we can see that there is just one correct speed to achieve a circular orbit under gravity. (Note that we have ignored the effects of air resistance in this discussion.)