recall and use F = mrω2 and F = mv2 / r

Centripetal Acceleration 🚀

What is Centripetal Acceleration?

When an object moves in a circle, it keeps changing direction. That change in direction means it’s accelerating even if its speed stays the same. The force that keeps it moving in a circle is called the centripetal force.

Key Formulae

• Angular form: $F = m r \omega^2$
• Linear form: $F = \dfrac{m v^2}{r}$

Analogy: Swinging a Ball on a Rope 🎾

Imagine holding a ball on a string and swinging it around. The string pulls the ball toward you – that pull is the centripetal force. The faster you swing (higher $v$ or $\omega$), the stronger the pull you feel.

Centripetal Acceleration Formula

Acceleration toward the centre: $a_c = r \omega^2 = \dfrac{v^2}{r}$

Example Problem

1. A car travels in a circular track of radius 50 m at a constant speed of 20 m/s. Find the centripetal acceleration.
2. Calculate the required centripetal force if the car’s mass is 1500 kg.

Solution:

1. $a_c = \dfrac{v^2}{r} = \dfrac{20^2}{50} = \dfrac{400}{50} = 8 \text{ m/s}^2$
2. $F = m a_c = 1500 \times 8 = 12{,}000 \text{ N}$

Quick Reference Table

Common Mistakes to Avoid

• Mixing up $\omega$ (rad/s) with $v$ (m/s). Remember $v = r \omega$.
• Using the wrong radius: use the radius of the circular path, not the distance from the centre to the object’s centre of mass if it’s a rotating body.
• Forgetting that the force points toward the centre, not away from it.

Quick Quiz

1️⃣ If a 0.5 kg ball is swung in a circle of radius 0.2 m at 10 rad/s, what is the centripetal force?

Answer: $F = 0.5 \times 0.2 \times 10^2 = 0.5 \times 0.2 \times 100 = 10 \text{ N}$.

2️⃣ A cyclist is turning around a roundabout of radius 30 m at 5 m/s. What is the centripetal acceleration?

Answer: $a_c = v^2 / r = 25 / 30 \approx 0.83 \text{ m/s}^2$.