How Do You Calculate Rotational Kinetic Energy?

Rotational kinetic energy is calculated using $E_{\text{k,rot}}=\frac{1}{2}I\omega^2$, where $I$ is the moment of inertia about the chosen axis and $\omega$ is angular speed. This is an HL-only concept in A.4 Rigid body mechanics.

The Reasoning

The moment of inertia measures an object's resistance to angular acceleration. It depends on both mass and how that mass is distributed relative to the selected axis. Therefore, the same object can have different moments of inertia about different axes.

For a system of point masses:

$$I=\sum mr^2$$

where $m$ is each mass and $r$ is its perpendicular distance from the axis. Add every $mr^2$ contribution to obtain the system's total moment of inertia.

Then substitute $I$ and the angular speed into $E_{\text{k,rot}}=\frac{1}{2}I\omega^2$. Angular speed must be in radians per second. The result is in joules because has units of and radians are dimensionless.

If frequency $f$ or period $T$ is given instead, first use $\omega=2\pi f=\frac{2\pi}{T}$. This conversion is often required before applying the energy equation.

Worked Example

Two point masses, each of mass $0.50\,\text{kg}$, rotate at a radius of $0.30\,\text{m}$ with angular speed $4.0\,\text{rad s}^{-1}$.

First calculate $I$:

$$I=2mr^2=2(0.50)(0.30)^2=0.090\,\text{kg m}^2$$

Then calculate the energy:

$$E_{\text{k,rot}}=\frac{1}{2}(0.090)(4.0)^2=0.72\,\text{J}$$

This is the rotational kinetic energy of the whole system. A common misconception is that it depends only on total mass. Moving the same mass farther from the axis increases $I$, so it increases the energy at the same angular speed.

Exam Technique

For a Calculate question, show the equation, substitution, and final value. Identify the correct rotation axis, convert angular speed to $\text{rad s}^{-1}$, and distinguish rotational energy from translational kinetic energy, $\frac{1}{2}mv^2$.