Calculators
Rotational Kinetic Energy Calculator
Solve K = ½Iω² for rotational energy, inertia, or angular speed.
Calculate rotational kinetic energy or rearrange K = ½Iω² to solve moment of inertia or angular speed for a rigid body rotating about a fixed axis.
Calculated result
9 J
K = ½Iω²
Energy depends on angular speed squared, so reversing rotation direction does not change K. This model uses the moment of inertia about the actual rotation axis and excludes translational kinetic energy unless you calculate and add it separately.
About This Tool
Rotating objects store kinetic energy because their mass moves around an axis. For a rigid body rotating about a fixed axis, rotational kinetic energy is K = ½Iω², where I is the moment of inertia about that axis and ω is angular velocity in radians per second. This calculator solves for any of those three quantities, handles common rotational units, and keeps all calculations in your browser.
How To Use It
- Choose whether to solve rotational kinetic energy, moment of inertia, or angular speed.
- Enter the two known quantities and select their units. Angular velocity may be signed; energy depends on its square, so direction does not change the energy.
- Choose the desired result unit. The calculator converts internally through kg·m², rad/s, and joules.
- Use the moment of inertia about the actual rotation axis. If the object's axis or mass distribution changes, calculate the appropriate I first.
Examples
2 kg·m² at 3 rad/s
K = ½ × 2 × 3² = 9 J.
1 kg·m² at 60 RPM
60 RPM equals 2π rad/s, so K = ½ × 1 × (2π)² ≈ 19.739 J.
Solve moment of inertia
If K = 9 J and ω = 3 rad/s, I = 2K/ω² = 2 kg·m².
Doubling angular speed
Because K is proportional to ω², doubling angular speed while I stays constant multiplies rotational kinetic energy by four.
Useful Notes
Rotational kinetic energy formula
For fixed-axis rigid-body rotation, K = ½Iω². It mirrors translational kinetic energy ½mv²: moment of inertia plays the rotational role of mass, while angular velocity plays the role of linear speed.
Moment of inertia must match the axis
Moment of inertia depends on both mass distribution and the chosen rotation axis. The same object can have different I values about different axes. Use Hanakash's Moment of Inertia Calculator when you need I for a standard shape and axis.
Radians per second in the formula
The standard SI form uses angular velocity in radians per second. RPM, revolutions per second, and degrees per second are converted to rad/s before the energy equation is evaluated.
Direction does not change rotational energy
Clockwise and counterclockwise angular velocities can be represented with opposite signs, but ω is squared in K = ½Iω². Therefore equal-magnitude angular speeds have equal rotational kinetic energy.
Rotation plus translation
A rolling or otherwise moving object may have both translational and rotational kinetic energy. This calculator reports only the rotational part. For a rigid object whose center of mass translates while it rotates, total kinetic energy can include ½Mv² + ½Iω² when that decomposition applies.
FAQ
What is the SI unit of rotational kinetic energy?
The SI unit is the joule (J), the same unit used for translational kinetic energy and other forms of energy.
What units should I use for I and ω?
The SI formula uses kg·m² and rad/s. This calculator also accepts g·cm² and lb·ft² for inertia plus deg/s, RPM, and rev/s for angular velocity and converts them internally.
Can angular velocity be negative?
Yes. The sign can indicate rotation direction, but rotational kinetic energy remains nonnegative because angular velocity is squared.
Is this the same as the regular kinetic energy calculator?
No. Translational kinetic energy uses ½mv² for linear motion. Rotational kinetic energy uses ½Iω² for rotation. An object can have both at the same time.
Does changing the rotation axis matter?
Yes. Changing the axis usually changes moment of inertia, which changes rotational kinetic energy at the same angular speed. Always use I about the actual axis.
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