Centripetal force

What you'll learn

Why an object moving at constant speed in a circle is still accelerating.
How angular speed links to linear speed using $\omega r$ .
How to calculate centripetal acceleration and centripetal force.
How the whirling bung experiment investigates circular motion.

The key prerequisite: velocity is a vector

A vector is a quantity with both magnitude and direction. Velocity is a vector: its magnitude is speed, but it also has a direction.

So an object can have a constant speed while its velocity changes, if its direction changes. That is exactly what happens in circular motion.

Definition

Resultant force

The resultant force is the single force that has the same overall effect as all the forces acting on an object. By Newton’s second law, the resultant force is in the same direction as the acceleration.

Why a perpendicular force makes circular motion

For an object to move in a circle at constant speed, its velocity must keep changing direction. The required acceleration points towards the centre of the circle.

A resultant force that is always perpendicular to the velocity changes the direction of motion, not the speed. If this force has constant magnitude and always points towards a fixed centre, the object follows a circular path.

Circular motion showing tangent velocity and inward centripetal acceleration and force

Key Idea

The direction is the whole point

In uniform circular motion, the velocity is tangent to the circle, while the acceleration and resultant force point towards the centre.

Example

Deciding the direction of the force

A stone on a string is moving in a horizontal circle. At one instant, it is at the right-hand side of the circle and moving upwards. What is the direction of the resultant force?

The resultant force for circular motion must point towards the centre of the circle.
From the right-hand side of the circle, the centre is to the left of the stone.
Therefore, the resultant force is to the left. The velocity is upwards, so the force is perpendicular to the velocity at that instant.

Constant speed and angular speed

In circular motion, the object travels around a circumference. If the radius is $r$ , the distance for one complete revolution is $2πr2\pi r$ .

The period, $T$ , is the time for one complete revolution, measured in seconds.

Definition

Angular speed

Angular speed, $ω\omega$ , is the angle swept out per unit time. It is measured in radians per second, written as $s−1\text{rad s}^{-1}$ . One complete circle is $2π2\pi$ radians.

For one complete revolution:

ω=2πT\omega = \frac{2\pi}{T}

The linear speed around the circle is:

\frac{2\pi r}{T}

Combining these gives the key OCR equation:

\omega r

Example

Finding linear speed from angular speed

A point on a rotating platform is 0.40 m from the centre. The platform has angular speed $s−16.0\ \text{rad s}^{-1}$ . Calculate the linear speed of the point.

Use the relationship linking linear speed and angular speed:
$\omega r$
Substitute the values:
$6.0\ \text{rad s}^{-1} \times 0.40\ \text{m}$
Since radians are dimensionless, the unit becomes metres per second:
$2.4\ \text{m s}^{-1}$

Centripetal acceleration

Centripetal means “centre-seeking”. In circular motion, the acceleration is directed towards the centre of the circle, even when the speed is constant.

Definition

Centripetal acceleration

Centripetal acceleration is the acceleration of an object moving in a circular path, directed towards the centre of the circle.

The OCR equations are:

\frac{v^2}{r}

and

\omega^2 r

You can see the link between them by substituting $\omega r$ into $\frac{v^2}{r}$ :

\frac{(\omega r)^2}{r} = \omega^2 r

Example

Calculating centripetal acceleration

A model aircraft flies in a horizontal circle of radius 25 m at a constant speed of $s−118\ \text{m s}^{-1}$ . Calculate its centripetal acceleration.

The speed and radius are given, so choose:
$\frac{v^2}{r}$
Substitute the values:
$\frac{(18\ \text{m s}^{-1})^2}{25\ \text{m}}$
Calculate and include the direction:
$13\ \text{m s}^{-2}$
The acceleration is towards the centre of the circle.

Common Mistake

Thinking constant speed means no acceleration

Acceleration means change in velocity, not just change in speed. In circular motion at constant speed, the velocity is changing because its direction is changing.

Centripetal force is not a new type of force. It is the resultant inward force that causes centripetal acceleration.

Using $F = ma$ with $\frac{v^2}{r}$ :

\frac{mv^2}{r}

Using $\omega^2 r$ :

m\omega^2 r

The centripetal force is measured in newtons, N, and always acts towards the centre of the circular path.

In real situations, the inward resultant force might be provided by tension, friction, gravity, or a combination of forces.

Common Mistake

Adding a fake centripetal force

Do not draw “centripetal force” as an extra force on a free-body diagram. First draw the real forces, then identify which resultant component acts towards the centre.

Tip

Check what is being kept constant

If speed $v$ is constant, $\propto \frac{1}{r}$ . If angular speed $ω\omega$ is constant, $\propto r$ . These sound similar but describe different situations.

Example

Calculating the force on a car going round a bend

A car of mass 950 kg travels around a flat circular bend of radius 45 m at $s−112\ \text{m s}^{-1}$ . Calculate the centripetal force required.

The mass, speed and radius are known, so use:
$\frac{mv^2}{r}$
Substitute the values:
$\frac{950\ \text{kg} \times (12\ \text{m s}^{-1})^2}{45\ \text{m}}$
Calculate:
$\times 10^3\ \text{N}$
This force acts towards the centre of the bend. On a flat road, it is provided by friction between the tyres and the road.

Investigating circular motion with a whirling bung

A common practical uses a rubber bung attached to a string passing through a tube. A hanging mass provides the tension in the string. The bung is whirled in a horizontal circle.

Whirling bung experiment with hanging mass providing tension

If the hanging mass is stationary, its weight provides the tension in the string:

F = m_h g

where $m_h$ is the hanging mass. This tension acts towards the centre of the bung’s circular path, so it provides the centripetal force for the bung:

mhg=mbv2rm_h g = \frac{m_b v^2}{r}

where $m_b$ is the mass of the bung.

To measure the speed of the bung, time many revolutions rather than just one. If $N$ revolutions take time $t$ , then:

\frac{t}{N}

and

\frac{2\pi r}{T}

A useful analysis method is to plot $m_h g$ against $v2r\frac{v^2}{r}$ . The gradient should be the mass of the bung, $m_b$ , because:

m_b\left(\frac{v^2}{r}\right)

Common Mistake

Assumptions in the bung experiment

The tension is only approximately equal to $m_h g$ if the hanging mass stays stationary and friction where the string passes through the tube is small.

Tip

Improving the measurement

Time at least 10 or 20 revolutions, repeat the timing, and measure the radius from the centre of the tube to the centre of the bung. Use a marker on the string just below the tube to help keep the radius constant.

Example

Analysing whirling bung data

A bung of mass 0.0500 kg is whirled at radius 0.500 m. It completes 20 revolutions in 14.2 s. A hanging mass of 0.200 kg is used. Check whether the data are consistent with the hanging weight providing the centripetal force.

Calculate the period and then the speed:
$\frac{14.2\ \text{s}}{20} = 0.710\ \text{s}$ $\frac{2\pi \times 0.500\ \text{m}}{0.710\ \text{s}} = 4.43\ \text{m s}^{-1}$
Calculate the centripetal force needed for the bung:
$\frac{0.0500\ \text{kg} \times (4.43\ \text{m s}^{-1})^2}{0.500\ \text{m}}$ $1.96\ \text{N}$
Compare this with the hanging weight:
$Nm_h g = 0.200\ \text{kg} \times 9.81\ \text{N kg}^{-1} = 1.96\ \text{N}$
The two values match to three significant figures, so the data support the model well.

Exam technique

In the exam

Start by drawing a quick sketch: velocity tangent to the circle, acceleration and resultant force towards the centre.
Choose the equation from the quantities given: use $\omega r$ , $\frac{v^2}{r}$ , $\omega^2 r$ , $\frac{mv^2}{r}$ or $m\omega^2 r$ .
In practical questions, remember that the mass moving in the circle is the bung, while the hanging mass provides the tension through its weight.

Self review

Check yourself

Why can an object moving at constant speed in a circle still have acceleration?
If the angular speed doubles while the radius stays the same, what happens to the centripetal force?
In the whirling bung experiment, why is it better to time many revolutions rather than one?

What you'll learn

Why an object moving at constant speed in a circle is still accelerating.
How angular speed links to linear speed using $\omega r$ .
How to calculate centripetal acceleration and centripetal force.
How the whirling bung experiment investigates circular motion.

The key prerequisite: velocity is a vector

A vector is a quantity with both magnitude and direction. Velocity is a vector: its magnitude is speed, but it also has a direction.

So an object can have a constant speed while its velocity changes, if its direction changes. That is exactly what happens in circular motion.

Definition

Resultant force

Why a perpendicular force makes circular motion

For an object to move in a circle at constant speed, its velocity must keep changing direction. The required acceleration points towards the centre of the circle.

Circular motion showing tangent velocity and inward centripetal acceleration and force

Key Idea

The direction is the whole point

In uniform circular motion, the velocity is tangent to the circle, while the acceleration and resultant force point towards the centre.

Example

Deciding the direction of the force

A stone on a string is moving in a horizontal circle. At one instant, it is at the right-hand side of the circle and moving upwards. What is the direction of the resultant force?

The resultant force for circular motion must point towards the centre of the circle.
From the right-hand side of the circle, the centre is to the left of the stone.
Therefore, the resultant force is to the left. The velocity is upwards, so the force is perpendicular to the velocity at that instant.

Constant speed and angular speed

In circular motion, the object travels around a circumference. If the radius is $r$ , the distance for one complete revolution is $2πr2\pi r$ .

The period, $T$ , is the time for one complete revolution, measured in seconds.

Definition

Angular speed

Angular speed, $ω\omega$ , is the angle swept out per unit time. It is measured in radians per second, written as $s−1\text{rad s}^{-1}$ . One complete circle is $2π2\pi$ radians.

For one complete revolution:

ω=2πT\omega = \frac{2\pi}{T}

The linear speed around the circle is:

\frac{2\pi r}{T}

Combining these gives the key OCR equation:

\omega r

Example

Finding linear speed from angular speed

A point on a rotating platform is 0.40 m from the centre. The platform has angular speed $s−16.0\ \text{rad s}^{-1}$ . Calculate the linear speed of the point.

Use the relationship linking linear speed and angular speed:
$\omega r$
Substitute the values:
$6.0\ \text{rad s}^{-1} \times 0.40\ \text{m}$
Since radians are dimensionless, the unit becomes metres per second:
$2.4\ \text{m s}^{-1}$

Centripetal acceleration

Centripetal means “centre-seeking”. In circular motion, the acceleration is directed towards the centre of the circle, even when the speed is constant.

Definition

Centripetal acceleration

Centripetal acceleration is the acceleration of an object moving in a circular path, directed towards the centre of the circle.

The OCR equations are:

\frac{v^2}{r}

and

\omega^2 r

You can see the link between them by substituting $\omega r$ into $\frac{v^2}{r}$ :

\frac{(\omega r)^2}{r} = \omega^2 r

Example

Calculating centripetal acceleration

A model aircraft flies in a horizontal circle of radius 25 m at a constant speed of $s−118\ \text{m s}^{-1}$ . Calculate its centripetal acceleration.

The speed and radius are given, so choose:
$\frac{v^2}{r}$
Substitute the values:
$\frac{(18\ \text{m s}^{-1})^2}{25\ \text{m}}$
Calculate and include the direction:
$13\ \text{m s}^{-2}$
The acceleration is towards the centre of the circle.

Common Mistake

Thinking constant speed means no acceleration

Acceleration means change in velocity, not just change in speed. In circular motion at constant speed, the velocity is changing because its direction is changing.