>[!summary] Rotational motion is a type of motion where an object experiences angular displacement, velocity and/or acceleration > We can break the motion into linear and tangential motion. > **Key equations:** > General equations: $\Delta \theta = \omega \Delta t$ $\Delta \omega = \alpha \Delta t$ >[!info]+ Read Time **⏱ 2 mins** # Definition Rotational kinematics describes the motion of angular [[Angular Displacement|displacement]], [[Angular Velocity|velocity]] and [[Angular Acceleration|acceleration]]. This is used to describe an unknown function (like angular displacement), knowing one more of its [[Derivative|derivatives]] (angular velocity/acceleration). Rotational kinematics assumes the velocity or acceleration is always constant from point a to b or can be accurately described as the average of that function. ## Deriving Rotational Kinematics > [!warning] Assumptions Assume the [[Angular Velocity|angular velocity ]] and [[Angular Acceleration|acceleration]] are always constant from time $t_{1}$ to $t_{2}$. As well assume the following: > - [[Angular Velocity|Angular velocity]] is defined as $\omega = \frac{d\theta}{dt}$ > - [[Angular Acceleration|Angular acceleration]] is defined as $\alpha=\frac{d\omega}{dt}$ > - [[Angular Displacement|Angular displacement]] ($\theta$) is the area under [[Angular Velocity|angular velocity]] > - [[Angular Velocity|Angular velocity]] is the area under [[Angular Acceleration|angular acceleration]] If the goal was to find the change in displacement know angular velocity over $t_{1},t_{2}$ $ \begin{array}{c} dt(\frac{d\theta}{dt} = \omega) \\ d\theta = \omega dt \\ \int_{\theta (t_{1})}^{\theta (t_{2})}{d\theta} = \int_{t_{1}}^{t_{2}}\omega dt \\ \Delta \theta = \omega \int_{t_{1}}^{t_{2}} dt \\ \Delta \theta = \omega \Delta t \end{array} $ If the goal was to find the change in angular velocity knowing angular acceleration over $t_{1},t_{2}$ $ \begin{array}{c} dt \left( \frac{d\omega}{dt} = \alpha \right) \\ \int_{\omega(t_{1})}^{\omega (t_{2})} d\omega =\int_{t_{1}}^{t_{2}} \alpha dt \\ \Delta \omega = \alpha \Delta t \end{array} $ --- > 📂 Want to see more structured notes like these? > Help grow the project by [starring Math & Matter on GitHub](https://github.com/rajeevphysics/Obsidian-MathMatter). --- [^1]: Taken from R. Epp Lecture notes.