understand that centripetal acceleration causes circular motion with a constant angular speed

Centripetal Acceleration

What is Centripetal Acceleration?

Imagine a pizza 🍕 spinning on a pizza wheel. Every slice of the pizza is moving in a circle, but it keeps turning around the centre of the wheel. The force that keeps each slice moving in that circle is called centripetal force, and the rate at which the direction of the slice’s velocity changes is called centripetal acceleration. It’s the “pull” that keeps objects moving in a circle rather than flying straight out.

Key Formulae

The magnitude of centripetal acceleration is given by two equivalent formulas:

• $a_c = \dfrac{v^2}{r}$ – where $v$ is linear speed and $r$ is the radius of the circle.
• $a_c = \omega^2 r$ – where $\omega$ is angular speed (in rad s⁻¹).

The units are metres per second squared (m s⁻²). Notice that if the angular speed is constant, the centripetal acceleration is also constant – that’s why a car driving at a constant speed around a roundabout stays on the track.

Quick Reference Table

Real‑World Example: Roller Coaster Loop

1. A roller coaster car moves at 20 m s⁻¹ around a vertical loop of radius 10 m.
2. Calculate the centripetal acceleration at the top of the loop:
3. Using $a_c = \dfrac{v^2}{r}$, we get $a_c = \dfrac{(20)^2}{10} = 40$ m s⁻².
4. That’s about 4 g’s of force – enough to keep the car pressed against the track.

Exam Tip Box

Remember:

• Always check if the problem gives you linear speed $v$ or angular speed $\omega$.
• If the speed is constant, the centripetal acceleration is constant too.
• Use the formula that matches the variables you have – it saves time.
• Units matter: convert everything to SI units before plugging into the formula.

Quick Quiz

A bicycle wheel of radius 0.3 m is turning at 10 rad s⁻¹. What is the centripetal acceleration of a point on the rim?

Answer: $a_c = \omega^2 r = (10)^2 \times 0.3 = 30$ m s⁻².