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.