# Centripetal acceleration

> Physics · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9702-u16-centripetal-acceleration/

This sub-topic covers the origin, magnitude and direction of centripetal acceleration for uniform circular motion, including core derivations and problem-solving techniques required for CIE A-Level Physics examinations.

**Prerequisites:** [Uniform circular motion basics](https://www.owlsprep.com/study/cie-9702-u16-uniform-circular-motion/); [Angular velocity](https://www.owlsprep.com/study/cie-9702-u16-angular-velocity/)

## Learning objectives

- Derive expressions for centripetal acceleration in terms of speed/radius and angular velocity/radius
- Calculate centripetal acceleration for uniform circular motion
- Identify common misconceptions about the direction and nature of centripetal acceleration

## Direction and nature of centripetal acceleration

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.

**Centripetal acceleration** — 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.

*Notation:* $a_c$

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

> **info**
>
> "Centripetal" comes from Latin for "center-seeking", which is a simple reminder of its fixed direction relative to the circular path.

**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.

## Derivation and magnitude of centripetal acceleration

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:** 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 $\Delta t$, turning through angle $\Delta \theta$.

1. The magnitude of the velocity vector at both A and B is $v$. For small $\Delta \theta$, the magnitude of the change in velocity $\Delta v$ 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 $\Delta \theta$ into the expression for $\Delta v$:
6. $$v \left( \frac{v \Delta t}{r} \right) = |\Delta v| \implies \frac{|\Delta v|}{\Delta t} = \frac{v^2}{r}$$
7. As $\Delta t \to 0$, the magnitude of acceleration $a_c$ equals the limit of $\frac{\Delta v}{\Delta t}$

*Conclusion:* a_c = \frac{v^2}{r}

Using the relationship between tangential speed and angular velocity $v = \omega r$, 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, $\omega = 0.50$ rad s⁻¹. Use the formula $a_c = \omega^2 r$.
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.

## Problem-solving with centripetal acceleration

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.

**Exam command terms**

Common command terms for this topic have specific expectations in CIE exams:

- **Derive** — Show all steps of the derivation from first principles, do not just state the result *(You must include the limit $\Delta t \to 0$ when deriving $a_c = v^2/r$)*

- **Calculate** — Show full working, give final answer with correct units and significant figures matching the question

**Check your understanding**

Check your understanding before moving on:

1. 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

   *Why:* Correct: use $a_c = v^2/r = (4^2)/2 = 8$ m s⁻²

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

   *Why:* 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.

## Common pitfalls

- **Wrong:** Claiming centripetal acceleration is fully constant for uniform circular motion
  - Why it fails: While the magnitude is constant, direction changes continuously, so the acceleration vector is not constant
  - Correct: State that only the magnitude of centripetal acceleration is constant for uniform circular motion
- **Wrong:** Using angular velocity in degrees per second in $a_c = \omega^2 r$
  - Why it fails: All circular motion formulas in CIE Physics require angular velocity in radians per second
  - Correct: Convert any angular speed given in degrees per second to radians per second before substitution
- **Wrong:** Stating centripetal acceleration points away from the center
  - Why it fails: This confuses real centripetal acceleration with fictitious centrifugal acceleration
  - Correct: Always remember centripetal acceleration points towards the center of the circular path
- **Wrong:** Using diameter instead of radius when calculating centripetal acceleration
  - Why it fails: Questions often give diameter to test attention to detail
  - Correct: Always halve the diameter to get radius before substituting into the formula
- **Wrong:** Claiming no acceleration because speed is constant
  - Why it fails: Acceleration depends on change in velocity (a vector), not just change in speed
  - Correct: Recognize that any change in velocity direction means non-zero acceleration

## Cheatsheet

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

## 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.

- [Centripetal force](https://www.owlsprep.com/study/cie-9702-u16-centripetal-force/)
- [Gravitational fields](https://www.owlsprep.com/study/cie-9702-u17-overview/)
- [Gravitational field concepts](https://www.owlsprep.com/study/cie-9702-u17-gravitational-field-concepts/)

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