> [!summary] Rotational kinetic energy is the energy of rotating bodies > **Key Equations:** > Rotational kinetic energy $K = \frac{1}{2}\omega^2 I_{0}$ > Rotational work-kinetic energy theorem: $W =\Delta K$ >[!info]+ Read Time **⏱ 3 mins** # Definition Rotational kinetic energy is the [[Kinetic Potential Energy & Work-Energy Theorem|kinetic energy]] of rotating objects. It's the [[Rotational Work|rotational work done]] on an object to keep it rotating or to cause rotation. ## Derivation > [!warning] Assumptions To derive rotational kinetic energy, take the example to find the rotational kinetic energy of the rigid body below. > This can be found by taking the sum of very small mass elements and their respective displacement element from an axis. ![[rot_1.png|300]] > As well, assume the following: > - The definition of [[Angular Velocity|angular velocity]] is $\omega = \frac{v}{r} \Rightarrow v=rw$ > - [[Kinetic Potential Energy & Work-Energy Theorem|Kinetic energy]] is defined as $K = \frac{1}{2}mv^2$ $\begin{array}{c} \\ K_ = \sum _{i}\frac{1}{2}m_{i}v_{i}^2 \\ v_{i} = r_{i }\omega \\ \begin{align*} K &= \sum _{i}\frac{1}{2}m_{i}(r_{i}\omega)^2 \\ &=\frac{1}{2} \omega ^2 \underbrace{ \sum_{i} m_{i} r_{i} }_{ I_{0} } \\ &= \frac{1}{2}\omega^2 I_{0} \end{align*} \end{array}$ > [!note] In this derivation we also define the [[Moment of Inertia|moment of inertia]] for rotating bodies. $\sum_{i} m_{i} r_{i} =I_{0}$ # Rotational Work-Energy Theorem > [!warning] Assumptions To derive an equation to relate [[Rotational Work|rotational work]] and kinetic energy. Create an equation to describe work of a rotating body from point a to b. Assume the following: > - [[Rotational Work|Rotational work]] is defined as $W= \int_{a}^b \tau \cdot d\theta$ > - Assume [[Torque|torque]] and [[Angular Displacement|angular displacement]] are always in the same direction > - [[Torque|Torque]] is defined as $\tau = I\alpha$ > - [[Angular Acceleration|Angular acceleration]] is defined as $\alpha= \frac{d\omega}{dt}$ > - [[Angular Velocity|Angular velocity]] is defined as $\omega = \frac{d\theta}{dt}\Rightarrow d\theta=\omega dt$ > - The [[Moment of Inertia|moment of inertia]] is the same through time $t_{1}$ and $t_{2}$ > - [[Rotational Kinetic Energy & Work-Energy Theorem#Derivation|Rotational kinetic energy]] is defined as $\frac{1}{2}\omega I^2$ $ \begin{align*} W &= \int_{t_{1}}^{t_{2}} \tau \cdot d\theta \\ &= \int_{t_{1}}^{t_{2}} \tau d\theta \\ &= \int_{t_{1}}^{t_{2}}I\alpha d\theta\\ &= I \int_{t_{1}}^{t_{2}} \frac{d\omega}{\cancel{ dt }} (\omega \cancel{ dt }) \\ &= I \int_{t_{1}}^{t_{2}} \omega d\omega \\ &= I \int_{t_{1}}^{t_{2}} \omega d \omega \\ &= \frac{1}{2}I[\omega ^2(t_{2})-\omega ^2(t_{1})] \\ &= \frac{1}{2} I (\omega_{f} - \omega_{i}) \\ &= \Delta K \end{align*} $