>[!summary] Rotational motion is a type of motion that has both rotational force and linear kinematic force. > We can break the motion into linear and tangential motion. > **Key equations:** > Change in motion: $\Delta s = r\Delta \theta$ $v = \omega R$ >$a_t = r\alpha$ > General equations: $\Delta \theta = \omega \Delta t$ $\Delta \omega = \alpha \Delta t$ # Deriving Rotational Kinematics Rotational motion is any type of motion that follows rotates around some axis. ![[krot_1.png]] [^1] >[!note] Explanation Example of rotational motion In general rotational motion will rotate around some axis, and the amount of rotation can be determined by this general formula derived from above. $\Delta s = r\Delta \theta$ >[!warning] Finding Rotational Motion In rotation motion we break up the angular and tangential motion separately to make calculations easier. We base off derivations below from the example above In rotational motion we break up displacement into two parts: $\begin{array}{c} Angular: \Delta \theta = \theta_f - \theta_i \\ Tangential: \Delta s =s_f -s_i = r\Delta \theta \\ \end{array}$ Like in displacement we break up angular and tangential velocity where the velocity is defined the same way in [[Kinematics]] $\begin{array}{c} Angular:\omega = \frac{d \theta}{dt} \\ Tangential: v_t = \frac{ds}{dt} = \frac{rd\theta}{dt} = r\omega \end{array}$ Like in [[Kinematics]] the acceleration is found the same way $\begin{array}{c} Angular: \alpha = \frac{d \omega}{dt} \\ Tangental : a_t = \frac{dv}{dt} = r\alpha \end{array}$ >[!info] General Equation If we assume the above equations are true we can derive a general equation if time is the same in both situations. Note this is the same for [[Kinematics]] which are always true. >$\begin{array}{c} >\int \frac{d\theta}{dt} = \int \omega dt\\ >\Delta \theta = \omega \Delta t \\ \\ \int \frac{d \omega}{dt} = \int \alpha dt \\ \Delta \omega = \alpha \Delta t \end{array}$ [^1]: Taken from R. Epp Lecture notes.