# Rotational Kinetic Energy Calculator

Created by Bogna Szyk
Reviewed by Dominik Czernia, PhD and Steven Wooding
Last updated: Jan 12, 2023

This rotational kinetic energy calculator is a tool that lets you determine the energy of a rotating object, such as a spindle or a wheel, or a carousel.

In this article, we will provide you with all information you will need to make your calculations, including the rotational kinetic energy formula.

We will also show you how to calculate rotational kinetic energy, illustrating the guidelines with a step-by-step example.

💡 You might also be interested in our rotational stiffness calculator and the spindle speed calculator.

## What is the rotational kinetic energy?

Rotational energy is a form of kinetic energy (see kinetic energy calculator). It describes the energy of an object in motion. While regular kinetic energy is related to objects moving along a straight line, rotational energy is linked with rotating objects. That is why, instead of a linear speed, we have to use angular velocity in our calculations.

Remember that if an object is both in translation (moving along a straight line) and rotation (turning around its axis), it has both regular and rotational kinetic energy. Therefore, to find out its total energy, you must calculate and add both values.

## Rotational kinetic energy formula

Just as potential energy can be calculated using the potential energy formula, rotational kinetic energy can be expressed with a straightforward equation. In this case:

$\small {\rm RE} = 0.5 I \omega^2$

where:

• $\rm RE$ – Rotational kinetic energy, expressed in joules;
• $I$ – Moment of inertia of the object, expressed in kg·m²; and
• $\omega$ – Angular velocity of the body, expressed in radians per second, hertz (rotations per second), or RPM (revolutions per minute) after the appropriate unit conversion. If you're not sure how to calculate this value for a body in rotary motion, head straight to our centrifugal force calculator to find out.

## How to calculate the rotational kinetic energy?

Let's analyze the following example together: a wheel of radius $R = 0.5\ \rm{m}$ and mass $M = 1\ \rm{kg}$ is rolling along a track, making 30 revolutions per minute. What is its rotational energy?

1. First, we need to determine the wheel's angular velocity. We know that it is equal to 30 RPM, which is equivalent to about 3.1416 rad/s.

2. Then, we need to calculate the wheel's moment of inertia. We already know its radius. As the wheel has the shape of a circle, we can express its moment of inertia with the equation:

\qquad \small \begin{align*} I &= MR^2\\ &= 1 \times 0.5^2 = 0.25\ \rm{kg}\cdot\rm{m}^2 \end{align*}
1. Once we know both the angular velocity and the moment of inertia, all we have to do is plug them into the rotational kinetic energy formula:
\qquad \small \begin{align*} {\rm RE} &= 0.5 \times I \times \omega^2\\ &= 0.5 \times 0.25 \times 3.1416^2\\ &\approx 1.2337\ \rm J \end{align*}
1. We have found that the rotational kinetic energy of the rolling wheel equals about 1.2337 J.

And if you'd like to learn about three-dimensional rotations, make sure to check Omni's quaternion calculator.

💡 To calculate the mass moment of inertia, head over to our mass moment of inertia calculator.

Bogna Szyk
Angular velocity
Moment of inertia
lb·ft²
Rotational kinetic energy
J
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