The rotational kinetic energy calculator above returns the energy stored in a spinning body from its moment of inertia and its angular speed. You can let it build the moment of inertia from one of eight standard shape-and-axis combinations, or type in a figure you already have. It also reports angular momentum, converts your angular speed into radians per second, and — if the body is also travelling rather than spinning on the spot — adds the translational term to give a rolling total.
Arb Digital builds free calculators that make the structure of a formula visible rather than hiding it behind a single output. Rotational energy is a good example of why that matters: two numbers go in, but almost every mistake people make with it happens in the moment of inertia, not in the arithmetic. Splitting the moment of inertia out into its own result, and letting you see it change as you switch the axis, is what turns this from a black box into something you can check.
What This Rotational Kinetic Energy Calculator Does
It evaluates ½Iω², where I is the moment of inertia about the axis of rotation in kilogram metres squared and ω is the angular speed in radians per second. The hero shows that energy, scaled automatically into millijoules, joules, kilojoules or megajoules so the number stays readable across the enormous range of scales this formula is used at, from a spinning coin to a turbine rotor.
The subtitle repeats that energy in watt-hours and foot-pounds. The grid gives four supporting quantities: the moment of inertia the shape selector produced, the angular speed in radians per second, the angular momentum L = Iω, and the total including any translational motion.
Angular momentum earns its place because it is conserved when no external torque acts, while energy is not — a skater pulling their arms in conserves L and gains energy from the work their muscles do.
How to Use It
- Choose the body and the axis together. The list names both, because a moment of inertia without a stated axis is meaningless. If your part is not on the list, use the last option and enter a figure directly.
- Enter mass and the defining dimension. For discs, hoops and spheres that dimension is the radius; for the two rod entries it is the full length. The unit selectors handle inches and feet without conversion.
- Set the angular speed in whatever unit you have. Motor data sheets give rpm, physics problems give rad/s, and the tool converts either way so you never have to remember the 2π ÷ 60 factor.
- Add a centre-of-mass speed only if the body is moving. Leave it at zero for a fixed flywheel. For something rolling, use the rolling button, which sets the speed to ωR for you.
- Read the split bars. They show how the total energy divides between spinning and travelling, which is the number that decides how much of a rolling object's energy a brake has to absorb.
The Formula: How Rotational Kinetic Energy Is Calculated
The rotational form is KE = ½Iω², the exact structural twin of the translational ½mv², with moment of inertia standing in for mass and angular speed standing in for velocity. That correspondence is not a coincidence — it falls straight out of summing ½mivi² over every particle in the body and substituting vi = ωri, which is the derivation given in section 10.4 of OpenStax University Physics Volume 1, Moment of Inertia and Rotational Kinetic Energy. The ω comes out of the sum because every particle in a rigid body shares the same angular speed, leaving Σmiri² behind as the moment of inertia.
The shape presets are the standard results of that sum for uniform bodies: ½MR² for a solid disc or cylinder about its central axis, MR² for a thin hoop or a point mass at radius R, ⅖MR² for a solid sphere about a diameter, ⅔MR² for a thin spherical shell, ML² ÷ 12 for a thin rod about its centre and ML² ÷ 3 for the same rod about one end. The hollow cylinder option is treated as a thick-walled tube using the outer radius with an inner radius of zero unless you override the inertia directly.
Work the defaults. A solid disc of 10 kg and 0.5 m radius has I = ½ × 10 × 0.5² = 1.25 kg·m². At 3,000 rpm, ω = 3,000 × 2π ÷ 60 = 314.159 rad/s. So KE = 0.5 × 1.25 × 314.159² = 0.625 × 98,696.04 = 61,685 J, which the tool shows as 61.685 kJ, or 17.135 Wh. Angular momentum is 1.25 × 314.159 = 392.70 kg·m²/s. At a 1,000 W draw that stored energy lasts about 61.7 seconds.
Why the Axis Matters More Than the Mass
Mass enters the formula linearly, but distance from the axis enters as a square. That single fact is why moving material outward is a far more powerful lever than adding material. Take the two rod entries in the shape list: the same rod, the same mass, the same length, but four times the moment of inertia when spun about its end rather than its centre. Nothing physical changed except where the axis sits.
The same asymmetry explains why a hoop has exactly twice the moment of inertia of a solid disc of identical mass and radius. In the disc, most of the material sits well inside the rim and contributes little; in the hoop it is all at the maximum radius. If you are designing a flywheel you want the hoop distribution, and if you are designing something that must accelerate quickly you want the opposite. Section 10.5 of OpenStax University Physics Volume 1, Calculating Moments of Inertia, works these integrals through and sets out the parallel-axis theorem that lets you shift a known result to an offset axis.
The parallel-axis theorem is the tool to reach for when your axis is not one of the nine on the list. It says I = Icm + Md², where d is the distance from the centre of mass to the new axis. Compute the centre-of-mass value here, add Md² by hand, then re-enter the result using the direct-entry option. Our moment of inertia calculator and mass moment of inertia calculator handle a wider library of shapes if you would rather not do that step manually.
Rolling Objects Carry Two Kinds of Energy at Once
A wheel rolling along a road is doing two things simultaneously: its centre of mass is travelling, and the body is spinning about that centre. Its total kinetic energy is the sum of ½mv² and ½Iω², and neither term is negligible. For rolling without slipping the two are linked by v = ωR, which is what the rolling button on this page enforces.
Substitute that link and something clean falls out. For a solid disc, the rotational term is exactly half the translational term, so a third of the total energy is in the spin. For a hoop the two terms are equal, so half the energy is in the spin. That fraction depends only on the shape, not on the mass, the radius or the speed — which is why a hoop and a disc released together on a ramp always finish in the same order regardless of size. Section 11.1 of OpenStax University Physics Volume 1, Rolling Motion, sets out the rolling constraint and the race result it implies.
That is the whole argument for lightweight wheels: their rotational energy must be supplied on every acceleration and dumped by the brakes on every stop, on top of moving the vehicle's mass. The split bars put a number on the penalty.
Flywheels: Why Speed Beats Mass, and What That Costs
If you want to store energy in a rotating mass, the square on ω tells you where to spend your effort. Doubling the mass doubles the stored energy. Doubling the speed quadruples it. Doubling the radius of a disc also quadruples it, because the radius is squared inside the moment of inertia. Speed and radius are therefore worth four times as much per unit of change as mass is.
The limit is not the energy formula but the material. A spinning rim is held together against centrifugal loading by its own hoop stress, and that stress rises with the square of the rim's surface speed. Push the speed up and you eventually reach the point where the rotor tears itself apart, which is why real flywheel design is a stress problem with an energy calculation attached rather than the other way round. The hoop stress calculator handles that side of it, and the centripetal force calculator gives the loading on any single mass element.
The spin-down figure in the note assumes constant power drawn until the store is empty, ignoring bearing drag, windage and the fact that no real converter extracts the last of the energy as speed falls, so treat it as an upper bound. For scale, the default flywheel holds 61.7 kJ — about what a 100 W lamp uses in ten minutes.
Reading Angular Speed Correctly: RPM Is Not Rad/s
The most common numerical error with this formula is feeding it revolutions per minute where it expects radians per second. The conversion factor is 2π ÷ 60, roughly 0.10472, so an rpm figure used raw overstates ω by about 9.55 times and therefore overstates the energy by about 91 times. That is not a small error you might miss; it is a two-order-of-magnitude one.
Radians are the natural unit here because they are arc length divided by radius, which is what makes v = ωr true with no conversion constant. Degrees and revolutions both carry an arbitrary factor that must be cleared first. Our angular velocity calculator does the same conversion on its own.
One more reading trap: the moment of inertia in this formula is the mass moment of inertia, in kilogram metres squared, which is a property of a body's mass distribution. It shares a name with the area moment of inertia used in beam bending, measured in metres to the fourth, which is a property of a cross-section's geometry and has nothing to do with energy. If your figure came off a structural drawing, check the units before you use it. The polar moment of inertia calculator covers the torsional version of that geometric quantity.
How This Differs From the Site's Other Mechanics Tools
The boundary in one sentence: this page computes the energy of rotation, ½Iω², while the kinetic energy calculator covers only translation, ½mv², and will not tell you anything about a body spinning on the spot.
The moment of inertia calculator stops at I itself and does not take an angular speed. The angular momentum calculator returns Iω rather than ½Iω², which matters because momentum is conserved in collisions where energy is not. The angular acceleration calculator and torque calculator handle the rate of change rather than the accumulated energy, and the motor torque calculator works back from shaft power. For speeds where relativity matters, the relativistic kinetic energy calculator replaces the classical expression entirely.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using rpm directly as ω — the energy comes out about 91 times too large. Convert with 2π ÷ 60 first, or let the unit selector do it.
- Quoting a moment of inertia without an axis — the same rod differs by a factor of four between its centre and its end, so the axis is part of the value.
- Confusing mass and area moment of inertia — one is in kg·m² and belongs in this formula, the other is in m⁴ and belongs in beam bending.
- Forgetting the translational term for a rolling body — a rolling disc has half again as much energy as its spin alone, and a rolling hoop has twice as much.
- Assuming energy is conserved when angular momentum is — pulling mass inward conserves Iω but raises ½Iω², with the difference supplied as work.
Related Free Tools From Arb Digital
Start with the moment of inertia calculator or the mass moment of inertia calculator if you need I for a shape not listed here, then bring the result back. The kinetic energy calculator handles straight-line motion, the angular velocity calculator and angular acceleration calculator cover the kinematics, and the angular momentum calculator covers what is conserved in a collision. For the machinery side, see the torque calculator, the motor torque calculator and the gear ratio calculator; for energy bookkeeping, the work calculator, the potential energy calculator and the energy converter. The full free online tools hub lists everything.
Frequently Asked Questions
It is one half times the moment of inertia times the angular speed squared, with the moment of inertia in kilogram metres squared and the angular speed in radians per second. The result is in joules.
Yes. The formula only works in radians per second, and the factor is two pi divided by sixty. Using rpm untouched overstates the energy by roughly ninety-one times, so this page converts for you.
Because all of a hoop's mass sits at the maximum radius while a disc's mass is spread inward, and distance from the axis is squared in the moment of inertia. Same mass, same radius, double the inertia.
For a solid disc it is one third of the total, and for a thin hoop it is one half. The fraction depends only on the shape, not on the mass, the radius or the speed.
Angular momentum is the moment of inertia times the angular speed, and it is conserved when no external torque acts. Rotational energy carries a squared angular speed and a factor of one half, and it is not conserved when mass moves inward or outward.
Only if it is a mass moment of inertia in kilogram metres squared. The area moment of inertia used in beam bending is measured in metres to the fourth power and describes a cross-section's geometry, not its mass.
Material strength rather than the energy formula. Hoop stress in the rim rises with the square of surface speed, so the rotor reaches its burst limit long before any energy ceiling, which is why flywheel design is a stress problem first.
This tool is provided for educational and study use. It assumes a rigid body of uniform density rotating about a fixed axis at non-relativistic speed, with the idealised shape formulas listed above and no allowance for bearing friction, windage or material limits. Rotating machinery stores large amounts of energy and can fail violently; any design intended to be built should be checked by a qualified engineer against the applicable standards.
