Centripetal force

CIE A-Level Physics· 16.3 Centripetal force· 15 min read

1. Definition and key properties of centripetal force★★☆☆☆⏱ 5 min

📘 Definition

Centripetal force

$F_{c}$

The net resultant force that acts on an object moving in a circular path, always directed perpendicular to the object's instantaneous velocity towards the centre of the circle.

Example:

Tension in a string spinning a attached mass provides centripetal force.

Because centripetal force is always perpendicular to the direction of motion, it does no work on the object. For uniform circular motion, this means the object's speed stays constant, only the direction of velocity changes, which maintains the circular path.

📐 Worked Example

A 2.0 kg mass is spun at constant speed in a horizontal circle, attached to a fixed central point by a string. State the direction of the centripetal force acting on the mass and identify what provides it.

1
By definition, centripetal force always points towards the centre of the circular path.
2
The only horizontal force acting on the mass is tension from the string, which pulls the mass towards the central fixed point.
3
Answer: Centripetal force is directed towards the central fixed point, and is provided entirely by the tension in the string.

2. Equations for centripetal force★★★☆☆⏱ 6 min

🔬 Derivation

Goal:

Derive the magnitude of centripetal force from first principles

Starting from:

Newton's second law $F_{n e t} = ma$ and the centripetal acceleration formula $a_{c} = \frac{v ^{2}}{r} = ω^{2} r$

1
Centripetal force is the net force towards the centre, so substitute $a_{c}$ into Newton's second law:
2
$F_c = m a_c = \frac{m v^2}{r}$
3
To get the form in terms of angular velocity, substitute $v = ω r$ :
4
$F_c = \frac{m (\omega r)^2}{r} = m \omega^2 r$

Result:

We have two equivalent forms of the centripetal force equation, used depending on whether we know linear speed $v$ or angular speed $ω$ .

📐 Worked Example

A 1500 kg car drives around a flat circular bend of radius 40 m at a constant speed of 12 m/s. Calculate the magnitude of the centripetal force required to keep the car on the bend.

1
List known values: $m = 1500 kg$ , $r = 40 m$ , $v = 12 m s^{- 1}$
2
Use the centripetal force equation for linear speed:
3
$F_c = \frac{mv^2}{r} = \frac{1500 \times (12)^2}{40}$
4
Calculate the result: $F_{c} = \frac{1500 \times 144}{40} = 5400 N$
5
This force is provided by friction between the car's tyres and the road surface.

3. Sources of centripetal force★★★★☆⏱ 5 min

A critical point for exams: centripetal force is not a new, separate force. It is the net resultant of existing physical forces that acts towards the centre of the circle. It can be provided by a single force, or the vector sum of multiple forces.

Tension: masses spinning on strings, ropes, or rods
Gravity: planets orbiting stars, satellites orbiting planets
Friction: cars and cyclists moving around flat bends
Normal reaction: cars on banked tracks, roller coaster loop-the-loops
Electrostatic force: electrons orbiting an atomic nucleus (Bohr model)

📐 Worked Example

A roller coaster car travels around a vertical circular loop of radius 10 m. At the top of the loop the car is upside down on the inside of the track. What forces contribute to the centripetal force at this point?

1
Draw a free-body diagram for the car at the top of the loop. Two vertical forces act:
2
1. Weight of the car ( $m g$ ), acting downwards, 2. Normal reaction from the track ( $R$ ), also acting downwards (track is above the upside down car)
3
The centre of the loop is directly below the car at the top of the loop, so both forces point towards the centre.
4
Answer: Centripetal force is the sum of weight and normal reaction: $F_{c} = m g + R$ , so both forces contribute.

4. Common Pitfalls

Wrong move:

Treating centripetal force as an additional, separate force on a free-body diagram.

Why:

Centripetal force is just the net resultant of existing forces, not a new interaction between objects.

Correct move:

Draw all existing forces first, then sum their components directed towards the centre to get the net centripetal force.

Wrong move:

Drawing centripetal force pointing outwards from the centre of the circle.

Why:

By definition, centripetal means 'centre-seeking', so direction is always towards the centre.

Correct move:

Always draw force components contributing to centripetal force pointing towards the centre of the circular path.

Wrong move:

Substituting angular speed in degrees per second into $F_{c} = m ω^{2} r$ .

Why:

The centripetal force equation is derived for angular speed measured in radians per second.

Correct move:

Always convert angular speed from degrees per second to radians per second before substitution.

Wrong move:

Claiming centripetal force does work to keep the object moving in a circle.

Why:

Centripetal force is always perpendicular to instantaneous velocity, so work done = force × displacement in direction of force = zero.

Correct move:

State that centripetal force only changes the direction of velocity, not its magnitude, and does no work on the object.

5. Quick Reference Cheatsheet

Property	Formula	Key Fact
Direction of $F_{c}$		Always towards the centre of the circle
Magnitude (v known)	$F_{c} = \frac{m v ^{2}}{r}$	$v$ = instantaneous linear speed
Magnitude ( $ω$ known)	$F_{c} = m ω^{2} r$	$ω$ must be in radians per second
Nature of force		Not a separate force, it is the net resultant force
Common sources		Tension, gravity, friction, normal reaction

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Identify source of centripetal force
2023 · 2
Calculate centripetal force on car
2024 · 1
Direction of centripetal force question

What's Next

Centripetal force is the foundational concept for all circular motion problem-solving in CIE A-Level Physics. You will apply this core idea to more complex scenarios including vertical circular motion, conical pendulums, banked tracks, and orbital motion of satellites. Mastery of centripetal force is also required for later topics like rotational dynamics, simple harmonic motion, and gravitational fields. Practise identifying the source of centripetal force in different scenarios, as this is a very common exam question.