recall and use a = rω2 and a = v2 / r

What is Centripetal Acceleration? 🚗

Centripetal acceleration is the acceleration that keeps an object moving in a circular path. It always points toward the centre of the circle, just like how a car turns around a roundabout and feels pushed toward the middle.

It is given by two equivalent formulas:

• Using angular velocity: $a = r \omega^2$
• Using linear speed: $a = \dfrac{v^2}{r}$

Here, $r$ is the radius of the circle, $\omega$ is the angular velocity in radians per second, and $v$ is the linear speed in metres per second.

Exam Tip 🎯

When you see a problem with a rotating object, first decide whether you have angular or linear data. Use the appropriate formula:

1. If you have $\omega$, use $a = r \omega^2$.
2. If you have $v$, use $a = \dfrac{v^2}{r}$.

Check units: $r$ in metres, $\omega$ in rad s⁻¹, $v$ in m s⁻¹. The result will be in m s⁻².

Analogy: The Merry‑Go‑Round 🎠

Imagine a child on a merry‑go‑round. As the ride spins, the child feels a force pushing them toward the centre. That feeling is due to centripetal acceleration. The faster the ride spins (higher $\omega$), the stronger the push.

Example Problem 🧮

A car is moving in a circle of radius $r = 20\,\text{m}$ at a constant speed of $v = 10\,\text{m s}^{-1}$. Calculate the centripetal acceleration.

Solution:

Use $a = \dfrac{v^2}{r}$:

$a = \dfrac{(10\,\text{m s}^{-1})^2}{20\,\text{m}} = \dfrac{100}{20} = 5\,\text{m s}^{-2}$

So the car experiences a centripetal acceleration of $5\,\text{m s}^{-2}$ toward the centre.

Key Points to Remember

• Centripetal acceleration always points toward the centre of the circle.
• Both formulas give the same result if you convert $\omega$ to $v$ using $v = r \omega$.
• Units: m s⁻² for acceleration.

Common Mistakes ❌

• Using $r$ in centimetres instead of metres.
• Confusing $\omega$ (rad s⁻¹) with $v$ (m s⁻¹).
• Forgetting that centripetal acceleration is always directed inward.

Quick Review

Remember: $a = r \omega^2 = \dfrac{v^2}{r}$. Pick the formula that matches the data you have.

Glossary

• Radius ($r$): Distance from the centre to the object.
• Angular velocity ($\omega$): Rate of rotation in radians per second.
• Linear speed ($v$): Tangential speed along the circular path.
• Centripetal acceleration ($a$): Acceleration directed toward the centre.

Further Practice

1. A bicycle wheel of radius $0.35\,\text{m}$ rotates at $12\,\text{rev min}^{-1}$. Find the centripetal acceleration of a point on the rim.
2. A satellite orbits Earth at a speed of $7.8\,\text{km s}^{-1}$ in a circular orbit of radius $6.6 \times 10^6\,\text{m}$. Calculate its centripetal acceleration.