Calculate rotational kinetic energy from moment of inertia and angular speed, or use a common shape to find inertia first.
Advanced options
How to use our Rotational Kinetic Energy Calculator
- Choose "Direct moment of inertia" if you already know Moment of inertia (kg*m^2), or choose "Common shape" if you want the calculator to find it from Object shape, Mass (kg), and size.
- Enter Angular speed and then choose the matching Angular speed unit. The unit is binding, so 60 rpm is not the same as 60 rad/s.
- If you picked Direct moment of inertia, enter Moment of inertia (kg*m^2) as a value greater than 0.
- If you picked Common shape, choose Object shape, then enter Mass (kg) and the needed size: Radius (m) for disks, hoops, spheres, and shells, or Length (m) for rods.
- Open Advanced options if you want a different Output energy unit such as kJ, MJ, or Wh, or if you want a specific Number display format.
- Click Calculate.
- Read Rotational kinetic energy as the main answer, then check Angular speed in SI units to confirm your speed was converted the way you expected.
- Sanity-check the result by looking at Moment of inertia used and Formula used. If energy seems way too large or small, the usual cause is a wrong angular-speed unit or using radius when the shape formula needs length.
Definitions
Moment of inertia (kg*m^2): A measure of how strongly an object resists changes in spinning. Bigger values usually mean mass is farther from the axis.
Angular speed: How fast something rotates. The calculator can read it in rad/s, rev/s, rpm, or deg/s.
Angular speed in SI units (rad/s): Your input converted to radians per second, which is the unit needed in the main energy formula.
Rotational kinetic energy: Energy stored in spinning motion. More inertia or more angular speed gives more rotational energy.
Object shape: The rigid-body model used to compute moment of inertia in shape mode, such as a solid disk, hoop, sphere, shell, or rod.
Formula used: A text summary showing whether the calculator used your direct inertia value or first computed inertia from the selected shape.
Common mistakes and quick fixes
Mistake: Typing Angular speed as 60 and leaving Angular speed unit on rad/s when you really mean rpm.
Fix: Change Angular speed unit to rpm and recalculate, then confirm the converted value in Angular speed in SI units.
Mistake: Entering Moment of inertia (kg*m^2) while "Common shape" is selected and expecting that value to be used.
Fix: Either switch What do you want to enter? to "Direct moment of inertia" or stay in shape mode and use Object shape, Mass (kg), and the needed size field.
Mistake: Using Length (m) for a disk or sphere, or using Radius (m) for a rod formula.
Fix: Match the size field to Object shape: use Radius (m) for disk, hoop, sphere, and shell, and use Length (m) for rod options.
Mistake: Entering 0 or a negative value for Mass (kg), Radius (m), Length (m), or Moment of inertia (kg*m^2).
Fix: Use a value greater than 0 for the active input fields. The calculator blocks nonphysical values for these fields.
Mistake: Thinking a negative Angular speed should give a negative Rotational kinetic energy.
Fix: Rotational kinetic energy depends on speed squared, so the result stays nonnegative. Use Angular speed in SI units to check the converted magnitude.
Mistake: Looking only at Rotational kinetic energy and not checking whether the right equation path was used.
Fix: Review Formula used and Moment of inertia used so you can catch a wrong mode, wrong shape, or wrong axis choice before trusting the final energy.
Limitations & Key Assumptions / Boundary Conditions
- This calculator is for rigid bodies rotating about a fixed axis and uses standard textbook moment-of-inertia formulas for the listed shapes.
- In Common shape mode, the result is only as good as the shape match. A real object with holes, uneven density, or a different axis can have a different moment of inertia.
- For rod options, Length (m) means the full rod length, and the axis is perpendicular to the rod either through the center or through one end, depending on Object shape.
- For disk, hoop, sphere, and shell options, Radius (m) is used about the named center axis or diameter. Choosing the wrong axis can change the answer a lot.
- Angular speed may be entered as a negative value, but rotational kinetic energy stays nonnegative because the speed is squared.
- Output energy unit and Number display format change how the result is shown, not the underlying physics calculation.
- Very large computed energies can be mathematically valid while still being unrealistic for real materials, which may bend, break, or fail before reaching that speed.
Methodology
Core method
The calculator first converts Angular speed to radians per second, then it finds the moment of inertia to use, and finally it computes rotational kinetic energy.
KE_rot = (1/2) * I * omega^2
Here, I is moment of inertia in kg*m^2 and omega is angular speed in rad/s [1].
Angular speed conversion
If your Angular speed unit is already rad/s, no conversion is needed.
omega = 2 * pi * f
f is revolutions per second.
omega = 2 * pi * rpm / 60
This is used when Angular speed unit is rpm.
omega = deg_s * pi / 180
This is used when Angular speed unit is deg/s.
How moment of inertia is chosen
If "Direct moment of inertia" is selected, the calculator uses your Moment of inertia (kg*m^2) value directly.
If "Common shape" is selected, it computes moment of inertia from Mass (kg) and the needed size for the chosen Object shape [1].
I = (1/2) * m * r^2
Solid disk or solid cylinder about center axis.
I = m * r^2
Thin hoop or ring about center axis.
I = (2/5) * m * r^2
Solid sphere about center axis.
I = (2/3) * m * r^2
Thin spherical shell about center axis.
I = (1/12) * m * L^2
Slender rod about center.
I = (1/3) * m * L^2
Slender rod about end.
Output unit conversion
The physics calculation is done in joules first. Then the chosen Output energy unit is applied.
KE_kJ = KE_J / 1000
KE_MJ = KE_J / 1000000
KE_Wh = KE_J / 3600
Mini example
Suppose you choose Common shape, then select a solid disk with Mass (kg) = 3, Radius (m) = 0.4, and Angular speed = 10 rad/s.
I = (1/2) * 3 * 0.4^2 = 0.24 kg*m^2
KE_rot = (1/2) * 0.24 * 10^2 = 12 J
So the disk has 12 joules of rotational kinetic energy.
Validation and interpretation
The calculator requires positive values for Moment of inertia (kg*m^2), Mass (kg), Radius (m), and Length (m) when those fields are active. Blank or invalid Angular speed is treated as an error, not as zero. Hidden fields are ignored so the current mode alone controls the result.
Assumptions behind the result
The result assumes the selected shape formula matches the real object and axis. If the mass distribution or axis is different, the true moment of inertia and energy can differ from this result.