Rotational Kinetic Energy Calculator
This rotational kinetic energy calculator is a tool that lets you determine the energy of a rotating object, such as a spindle, a wheel, or a carousel. Enter the object's moment of inertia and its angular velocity, and the calculator instantly returns the rotational kinetic energy. It supports both the metric (SI) and imperial (US customary) unit systems and displays the result in multiple energy units — joules, kilojoules, kilocalories, watt-hours, BTU, and foot-pounds.
What is the rotational kinetic energy?
Rotational energy is a form of kinetic energy. 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 expressed with a straightforward equation, rotational kinetic energy can too. In this case:
RE = 0.5 × I × ω²
where:
- RE — rotational kinetic energy, expressed in joules (J);
- I — moment of inertia of the object, expressed in kg·m²; and
- ω — angular velocity of the body, expressed in radians per second (rad/s). It can also be given in hertz (rotations per second) or RPM (revolutions per minute) after the appropriate unit conversion.
The calculator accepts angular velocity in rad/s, RPM, rev/s (Hz), or °/s and converts it to rad/s automatically before computing the energy.
How to calculate the rotational kinetic energy?
Let's analyze the following example together: a wheel of radius R = 0.5 m and mass M = 1 kg is rolling along a track, making 30 revolutions per minute. What is its rotational energy?
Step 1 — Angular velocity. We know that it is equal to 30 RPM, which is equivalent to about 3.1416 rad/s:
ω = 30 RPM × (2π / 60) ≈ 3.1416 rad/s
Step 2 — Moment of inertia. As the wheel has the shape of a circle (treated as a hoop), we can express its moment of inertia with the equation I = MR²:
I = M × R² = 1 × 0.5² = 0.25 kg·m²
Step 3 — Rotational kinetic energy. 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:
RE = 0.5 × 0.25 × 3.1416² ≈ 1.2337 J
We have found that the rotational kinetic energy of the rolling wheel equals about 1.2337 J.
Energy Unit Conversions
| Unit | Equivalent in Joules |
|---|---|
| 1 Joule (J) | 1 J |
| 1 Kilojoule (kJ) | 1,000 J |
| 1 Kilocalorie (kcal) | 4,184 J |
| 1 Watt-hour (Wh) | 3,600 J |
| 1 BTU | 1,055.06 J |
| 1 Foot-pound (ft·lbf) | 1.35582 J |
Frequently Asked Questions
What is the difference between kinetic and rotational kinetic energy?
Regular (translational) kinetic energy is the energy of an object moving along a straight line, KE = ½mv². Rotational kinetic energy is the energy of an object spinning about an axis, RE = ½Iω². An object that both moves and spins (like a rolling wheel) possesses both, and its total kinetic energy is the sum of the two.
What is the moment of inertia?
The moment of inertia (I) is the rotational analogue of mass. It measures how the mass of an object is distributed around its axis of rotation and therefore how hard it is to change the object's rotation. For a thin hoop or ring it equals MR², for a solid disk ½MR², and for a solid sphere ⅖MR², where M is mass and R is radius.
How do I convert RPM to rad/s?
Multiply the value in revolutions per minute by 2π and divide by 60: ω(rad/s) = RPM × 2π / 60 ≈ RPM × 0.10472. For example, 30 RPM ≈ 3.1416 rad/s. To convert rev/s (Hz) to rad/s, multiply by 2π.
How does doubling the angular velocity affect the energy?
Because the energy depends on the square of the angular velocity, doubling ω quadruples the rotational kinetic energy. Doubling the moment of inertia, on the other hand, only doubles the energy (a linear relationship).
What units does the calculator use?
In the metric (SI) system the moment of inertia is entered in kg·m² and the energy is reported in joules (J). In the imperial (US customary) system the moment of inertia is entered in lb·ft² and the primary result is shown in foot-pounds force (ft·lbf). Either way, the calculator also lists conversions into kJ, kcal, Wh, BTU, and ft·lbf.