Rotational Dynamics
5.0 Rotational kinetic energy
5.0 Rotational kinetic energy
Rotational kinetic energy is defined as the kinetic energy of a rigid body rotating about a fixed axis.
Consider a rigid body is rotating about a fixed axis with angular velocity $\omega $
Assume a $i^{th}$ particle of mass $m_i$ moving in a circle of radius $r_i$ with a velocity ${\overrightarrow v _i}$ as shown in the figure.
Kinetic energy associated with this particle is, $${K_i} = \frac{1}{2}{m_i}v_i^2$$
So, kinetic energy of the whole rigid body is given by, $$\begin{equation} \begin{aligned} \sum {{K_i}} = \sum {\frac{1}{2}{m_i}v_i^2} \\ \sum {{K_i}} = \sum {\frac{1}{2}{m_i}\left( {{\omega ^2}r_i^2} \right)} \quad \quad \left( {As,\;{v_i} = \omega {r_i}} \right) \\ \sum {{K_i}} = \frac{1}{2}{\omega ^2}\sum {{m_i}r_i^2} \\ {K_R} = \frac{1}{2}I{\omega ^2}\quad \left( {As,\;I = \sum {{m_i}r_i^2} } \right) \\\end{aligned} \end{equation} $$
$I$ is the moment of inertia of the rigid body about an axis of rotation.
Unit of kinetic energy in $MKS$ system is Joules $(J)$ and $ergs$ in $CGS$ system.
As we know, $${K_R} \propto {\omega ^2}$$
Therefore, the graph between rotational kinetic energy $\left( {{K_R}} \right)$ and angular velocity $\left( \omega \right)$ is a parabola as shown in the figure.
