The rotating twin of ½mv²
Ordinary kinetic energy uses mass and speed. Rotational kinetic energy uses the same shape of formula with the rotational versions of both: moment of inertia instead of mass, angular velocity instead of speed.
Moment of inertia is where the physics lives. It is not just how much mass an object has, but how far that mass sits from the axis — and distance from the axis counts squared. That is why a hoop is harder to spin than a solid disc of the same weight.
The square is the whole story
Angular velocity is squared, so energy climbs far faster than speed does. Take I = 2 kg·m²:
| Angular velocity | Energy |
|---|---|
| 10 rad/s | 100 J |
| 20 rad/s | 400 J |
| 40 rad/s | 1,600 J |
Four times the speed, sixteen times the energy. This is the arithmetic behind flywheel energy storage, and behind why a failing grinding wheel is so much more dangerous at full speed than at half.
Radians are not decoration
The formula only works with ω in radians per second. RPM plugged straight in gives an answer out by a factor of about 91. The calculator converts for you, but it is the standard way to get this wrong on paper, so it is worth knowing that is what went wrong when a result looks absurd.
Frequently asked questions
What is the rotational kinetic energy formula?
KE = ½ · I · ω², where I is the moment of inertia in kg·m² and ω is the angular velocity in radians per second. With I = 2 kg·m² and ω = 10 rad/s, that is ½ × 2 × 100 = 100 J.
Why is the speed squared?
For the same reason it is in ordinary kinetic energy: energy grows with the square of speed, not in proportion to it. Double the spin rate of the same object and the energy goes up four times, from 100 J to 400 J. It is why a flywheel spun twice as fast is far more than twice as dangerous.
How do I convert RPM to rad/s?
Multiply by 2π and divide by 60, which is roughly ×0.1047. So 1,000 RPM is about 104.7 rad/s. The calculator accepts RPM and rev/s directly so you do not have to convert by hand.
What is moment of inertia?
The rotational equivalent of mass: how hard something is to spin up. It depends not just on how much mass there is but on how far that mass sits from the axis, and distance counts squared. A hoop and a solid disc of the same mass have very different moments of inertia.
Does an object have both kinds of kinetic energy?
Yes. A rolling wheel is moving along and spinning at the same time, so it carries translational energy ½mv² and rotational energy ½Iω² together. That is why a ball rolls down a slope more slowly than it slides — some of the energy goes into spin instead of speed.