Circular motion

What you'll learn

• Why an object moving at constant speed in a circle is still accelerating.
• How period, frequency and angular velocity describe circular motion.
• How to use $a=\frac{v^2}{r}$ and $F=\frac{m{v}^2}{r}$.
• How to identify the real force providing the centripetal force in different situations.

Before circular motion: velocity and acceleration

You already know that speed is a scalar quantity: it only has a magnitude. Velocity is a vector quantity: it has magnitude and direction.

This matters because an object can have constant speed but changing velocity if its direction is changing.

In circular motion, direction changes continuously. So even if the object moves around the circle at constant speed, it is still accelerating.

Uniform circular motion

Uniform circular motion means motion in a circle at constant speed.

The velocity is always tangential to the circle, meaning it points along the tangent to the path at that instant. The acceleration points towards the centre of the circle.

Period, frequency and speed

For circular motion, it is useful to describe how quickly the object completes revolutions.

They are linked by:

$$f=\frac{1}{T}$$

For one complete revolution, the distance travelled is the circumference of the circle:

$$2\pi r$$

So the speed is:

$$v=\frac{2\pi r}{T}$$

Since $f=\frac{1}{T}$, you can also write:

$$v=2\pi rf$$

Angular velocity

Sometimes circular motion is described using the angle swept out per second.

One full revolution is an angle of $2\pi$ radians, so:

$$\omega =\frac{2\pi }{T}$$

Since $f=\frac{1}{T}$:

$$\omega =2\pi f$$

Linear speed and angular velocity are linked by:

$$v=\omega r$$

Centripetal acceleration

The inward acceleration in circular motion is called centripetal acceleration.

For uniform circular motion:

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

Using $v=\omega r$, you can also write:

$$a={\omega }^{2}r$$

Both are useful. Use the one that matches the information given.

Centripetal force

From Newton’s second law:

$$F=ma$$

If the acceleration is centripetal, then the resultant force must also be towards the centre. This inward resultant force is called the centripetal force.

Combining $F=ma$ with $a=\frac{v^2}{r}$ gives:

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

You may also use:

$$F=m{\omega }^{2}r$$

For example:

• For a mass on a string, tension provides the centripetal force.
• For a car turning on a flat road, friction provides the centripetal force.
• For a satellite in orbit, gravity provides the centripetal force.

Resultant force: the free-body diagram approach

Circular motion questions often become much easier if you draw a free-body diagram, meaning a diagram showing the forces acting on the object.

The key question is:

Which forces have components towards or away from the centre of the circle?

Then apply Newton’s second law towards the centre:

$$F_{resultant\; towards\; centre}=\frac{m{v}^2}{r}$$

For a horizontal circle, weight often acts vertically downwards and may be balanced by an upward force. The horizontal inward force is then the centripetal force.

Cars going round bends

For a car on a flat circular bend, the centripetal force is usually provided by friction between the tyres and the road.

If the required centripetal force is too large, the tyres cannot provide enough friction and the car skids outwards. The car does not skid because there is an outward force; it skids because there is not enough inward force to keep it on the circular path.

Vertical circular motion

In vertical circular motion, such as a rollercoaster loop or a ball on a string moving in a vertical circle, weight changes how the forces combine at different points.

At the top of a vertical circle, weight acts towards the centre. At the bottom, weight acts away from the centre.

For a mass on a string:

• At the top, tension and weight may both act towards the centre.
• At the bottom, tension acts towards the centre but weight acts away from the centre.

So the resultant force towards the centre changes around the circle.

Practical skills: investigating circular motion

A common practical setup uses a rubber bung attached to a string and whirled in a horizontal circle. The radius can be measured, and the period can be found by timing several revolutions and dividing by the number of revolutions.

Good practical habits include:

• Time many revolutions to reduce percentage uncertainty in the period.
• Keep the radius as constant as possible.
• Repeat readings and calculate a mean period.
• Use $v=\frac{2\pi r}{T}$, then test whether the centripetal force matches $\frac{m{v}^2}{r}$.

If a graph is used, choose axes that give a straight line. For example, if mass and radius are constant, $F$ is proportional to $v^2$, so a graph of $F$ against $v^2$ should be a straight line through the origin.

Choosing the right equation

The main circular motion equations are:

$$v=\frac{2\pi r}{T}$$ $$\omega =\frac{2\pi }{T}=2\pi f$$ $$v=\omega r$$ $$a=\frac{v^2}{r}={\omega }^{2}r$$ $$F=\frac{m{v}^2}{r}=m{\omega }^{2}r$$

A reliable method is:

1. Identify the radius of the circular path.
2. Find speed or angular velocity.
3. Find centripetal acceleration if needed.
4. Use Newton’s second law to connect the required inward resultant force to the real forces acting.