Centripetal acceleration

Physics· 16.2· 25 min read

1. Direction and nature of centripetal acceleration★★☆☆☆⏱ 8 min

Even when an object moves at constant speed along a circular path, its velocity is constantly changing direction. Since acceleration is defined as the rate of change of velocity, the object is accelerating even if its speed is constant.

📘 Definition

Centripetal acceleration

$a_{c}$

The acceleration responsible for keeping an object in circular motion, always directed towards the center of the circular path, perpendicular to the tangential velocity of the object.

Example:

A car turning a corner at constant speed has centripetal acceleration towards the center of the turn.

📐 Worked Example

A cyclist moves at constant speed around a circular roundabout. What is the direction of their centripetal acceleration? Explain why acceleration exists even at constant speed.

1
Acceleration is defined as any change in velocity, which is a vector quantity. Constant speed only means the magnitude of velocity is constant, but direction changes continuously around the circular path.
2
Centripetal acceleration always points towards the center of the circular path, so it points directly to the center of the roundabout, perpendicular to the cyclist's instantaneous tangential velocity.

Exam tip:

Always mention direction when asked to describe centripetal acceleration in exam questions.

2. Derivation and magnitude of centripetal acceleration★★★☆☆⏱ 10 min

We can derive the magnitude of centripetal acceleration using vector geometry and the definition of acceleration as rate of change of velocity. For uniform circular motion, the magnitude depends on both the tangential speed of the object and the radius of the path.

🔬 Derivation

Goal:

Derive the magnitude of centripetal acceleration for uniform circular motion

Starting from:

Consider an object moving at constant speed $v$ around a circle of radius $r$ , moving from point A to point B in time $Δ t$ , turning through angle $Δ θ$ .

1
The magnitude of the velocity vector at both A and B is $v$ . For small $Δ θ$ , the magnitude of the change in velocity $Δ v$ is approximately:
2
$v \Delta \theta \approx |\Delta v|$
3
The arc length along the object's circular path is:
4
$v \Delta t = r \Delta \theta \implies \Delta \theta = \frac{v \Delta t}{r}$
5
Substitute $Δ θ$ into the expression for $Δ v$ :
6
$v \left( \frac{v \Delta t}{r} \right) = |\Delta v| \implies \frac{|\Delta v|}{\Delta t} = \frac{v^2}{r}$
7
As $Δ t \to 0$ , the magnitude of acceleration $a_{c}$ equals the limit of $\frac{Δ v}{Δ t}$

Result:

a_c = \frac{v^2}{r}

Using the relationship between tangential speed and angular velocity $v = ω r$ , we can substitute into the formula to get a second expression:

a_c = \omega^2 r

📐 Worked Example

A merry-go-round has radius 4.0 m and rotates with constant angular velocity of 0.50 rad s⁻¹. Calculate the magnitude of the centripetal acceleration of a person standing on the edge.

1
List known values: $r = 4.0$ m, $ω = 0.50$ rad s⁻¹. Use the formula $a_{c} = ω^{2} r$ .
2
$a_c = (0.50)^2 \times 4.0 = 0.25 \times 4.0$
3
Final answer, with correct units: $a_{c} = 1.0$ m s⁻², directed towards the center of the merry-go-round.

3. Problem-solving with centripetal acceleration★★★☆☆⏱ 7 min

Centripetal acceleration problems in CIE exams often ask you to calculate acceleration for real-world scenarios, or compare accelerations of different objects in circular motion.

✓ Quick check

Check your understanding before moving on:

An object moves in a circle of radius 2 m at constant speed 4 m s⁻¹. What is the magnitude of centripetal acceleration?
- 2 m s⁻²
- 8 m s⁻²
- 16 m s⁻²
- 32 m s⁻²
Reveal answer
8 m s⁻² —
Correct: use $a_{c} = v^{2} / r = (4^{2}) /2 = 8$ m s⁻²
The direction of centripetal acceleration is:
- Tangent to the circle
- Away from the center of the circle
- Towards the center of the circle
- Parallel to the velocity vector
Reveal answer
Towards the center of the circle —
Correct: centripetal means center-seeking, so it always points towards the center

📐 Worked Example

A car travels at 15 m s⁻¹ around a horizontal bend of radius 60 m. What is the centripetal acceleration of the car?

1
Identify known values: $v = 15$ m s⁻¹, $r = 60$ m. Use the formula $a_{c} = v^{2} / r$ .
2
$a_c = \frac{(15)^2}{60} = \frac{225}{60}$
3
Calculate the result with correct units and significant figures:
4
$a_c = 3.8 \text{ m s}^{-2}$
5
The direction of the acceleration is towards the center of the bend.

4. Common Pitfalls

Wrong move:

Claiming centripetal acceleration is fully constant for uniform circular motion

Why:

While the magnitude is constant, direction changes continuously, so the acceleration vector is not constant

Correct move:

State that only the magnitude of centripetal acceleration is constant for uniform circular motion

Wrong move:

Using angular velocity in degrees per second in $a_{c} = ω^{2} r$

Why:

All circular motion formulas in CIE Physics require angular velocity in radians per second

Correct move:

Convert any angular speed given in degrees per second to radians per second before substitution

Wrong move:

Stating centripetal acceleration points away from the center

Why:

This confuses real centripetal acceleration with fictitious centrifugal acceleration

Correct move:

Always remember centripetal acceleration points towards the center of the circular path

Wrong move:

Using diameter instead of radius when calculating centripetal acceleration

Why:

Questions often give diameter to test attention to detail

Correct move:

Always halve the diameter to get radius before substituting into the formula

Wrong move:

Claiming no acceleration because speed is constant

Why:

Acceleration depends on change in velocity (a vector), not just change in speed

Correct move:

Recognize that any change in velocity direction means non-zero acceleration

5. Quick Reference Cheatsheet

Property	Formula	Key Notes
Magnitude (speed given)	$a_{c} = \frac{v ^{2}}{r}$	$v$ = tangential speed, $r$ = radius
Magnitude (angular velocity given)	$a_{c} = ω^{2} r$	$ω$ must be in rad s⁻¹
Direction	Towards center of circle	Always perpendicular to tangential velocity
Uniform circular motion	$∣ a_{c} ∣ = constant$	Acceleration vector is not constant

6. Frequently Asked

Is centripetal acceleration constant for uniform circular motion?

The magnitude of centripetal acceleration is constant for uniform circular motion, but the direction is always changing (it always points towards the center), so the acceleration vector is not constant.

What is the difference between centripetal and centrifugal acceleration?

Centripetal acceleration is a real acceleration measured in an inertial frame, directed towards the center of rotation. Centrifugal acceleration is a fictitious pseudo-acceleration that only appears when working in a rotating non-inertial frame.

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 · 22
Calculate centripetal acceleration of car
2023 · 12
Derive $a_{c} = v^{2} / r$ from first principles
2024 · 21
Compare linear and centripetal acceleration

Going deeper

SyllabusCIE 9702 A-Level Physics SyllabusCheck topic 16 for full learning outcomes

What's Next

Centripetal acceleration is the foundational concept for understanding centripetal force, the net force required to produce this acceleration for objects moving in circular motion. Mastery of the formulas, derivation and direction of centripetal acceleration is essential for solving all circular motion problems in CIE exams, including problems involving banked curves, vertical circles, and orbital motion of planets and artificial satellites. You will directly apply these concepts to solve force-based circular motion problems in the next sub-topic, and they also appear in topics like simple harmonic motion later in the syllabus.