Question

Show that the kinetic energy of an object rotating about a fixed axis with angular momentum $\mathit{L}\mathbf{=}\mathit{I}\mathit{w}$ can be written as$\mathit{K}\mathbf{=}\frac{{\mathbf{L}}^{\mathbf{2}}}{\mathbf{2}\mathbf{I}}$ .

Short answer

It is proved that the kinetic energy of an object rotating about a fixed axis with angular momentum $L=I\omega$ can be written as $K=\frac{L^2}{2I}$.

Step-by-step solution

Step 1:

The angular momentum is the amount of rotation of the body, which is the product of its inertia time and angular velocity.

Step 2:

Angular momentum

$L=I\omega$

Here, I is the moment of inertia and $\omega$ is the angular velocity.

As rotational kinetic energy of an object is,

$\begin{array}{c}K=\frac{1}{2}I{\omega }^2\\ =\frac{1}{2}\frac{{\left(I\omega \right)}^2}{I}\\ =\frac{L^2}{2I}\end{array}$

Result

Hence, it is proved that the kinetic energy of an object rotating about a fixed axis with angular momentum $L=I\omega$ can be written as $K=\frac{L^2}{2I}$ .