> [!summary] Centripetal acceleration is the acceleration to keep an object rotating around a circle > **Key Equations:** > Centripetal acceleration: $\vec{a}_{c} = \frac{v^2}{r}= \omega^2 r$ >[!info]+ Read Time **⏱ 2 mins** # Definition Centripetal acceleration is the acceleration towards the centre of a circle that keeps an object rotating around a circle at constant [[Angular Speed|angular speed]]. ## Derivation > [!warning] Assumptions To derive centripetal acceleration, assume the following: > - [[Acceleration|Acceleration]] is the [[Rate of Change|change]] in [[Velocity|velocity]] over the change in time > - [[Angular Velocity|Angular velocity]] is described as $\omega = \frac{v}{r} \Rightarrow v=\omega r$ > - The [[Arc Length of Circles|arc length]] of a circle is $s = r\theta$ > - Use the image below as a visual aid for the derivation ![[ca.png|300]] > - $\triangle ABC, \triangle PQR$ are [[Similar Triangles|similar triangles]] > - Angle in $\triangle PQR$ is the same angle in $\triangle ABC$ > **Note this derivation does not use the calculus approach** $ \begin{array}{c} \text{The arc length $\vec{v_{1}}$ and $\vec{v_{2}}$ is denoted as $v\Delta t$ or $\Delta s = v\Delta t$}\\ \\ \text{If $\Delta t$ between the two points are very small then}\\ \text{for $\triangle ABC$ $\Delta r= \Delta s$ because the time between two points are so small}\\ \\ \text{Because $\triangle ABC$ and $\triangle PQR$ are similar the ratio between the two are:} \\ \frac{\Delta v}{v}= \frac{v\Delta t}{r} \\ \frac{\Delta v}{\Delta t} = \frac{v^2}{r} \\ \vec{a} = \frac{v^2}{r} \\ \\ \text{Described in terms of angular velocity:} \\ \vec{a}_{c} = \omega^2 r \end{array} $ [^1]: Centripetal Acceleration. (n.d.). In _Uniform Circular Motion and Gravitation_. Lumen Learning. Retrieved July 19, 2025, from [https://courses.lumenlearning.com/suny-physics/chapter/6-2-centripetal-acceleration/](https://courses.lumenlearning.com/suny-physics/chapter/6-2-centripetal-acceleration/)