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Rotational Kinetic Energy Calculator

The rotational kinetic energy calculator finds the energy of an object in rotational motion.

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Rotational kinetic energy1.7765 kJ½Iω² at 1,200 rpm
Moment of inertia0.225 kg·m²0.5 × 5 kg × (0.3 m)²
ShapeSolid disc or cylinderabout its central axis
Coefficient k0.5the fraction of mr² this shape and axis gives
Angular momentum28.274334 kg·m²/s
Angular velocity125.663706 rad/s
Rim speed37.6991 m/sv = ωr at the outer edge
Radius of gyration0.212132 mthe radius at which a point mass would have the same moment of inertia
As a point mass at a distance0.45 kg·m²2× yours, at the same mass and size
As a thin hoop or ring0.45 kg·m²2× yours, at the same mass and size
As a thin hoop, about a diameter0.225 kg·m²1× yours, at the same mass and size
As a solid cylinder, about a diameter0.1125 kg·m²0.5× yours, at the same mass and size
As a solid sphere0.18 kg·m²0.8× yours, at the same mass and size
As a thin spherical shell0.3 kg·m²1.3333× yours, at the same mass and size
As a solid cone0.135 kg·m²0.6× yours, at the same mass and size

The formula

I = kmr², with k depending on the shape and axis

Where the mass sits is what matters

Moment of inertia is rotational mass: it measures how hard a body is to spin up, and it depends not just on how much mass there is but on how far that mass sits from the axis. The r² is why — mass at twice the radius contributes four times as much.

That is why a hoop, with all its mass at the rim, has exactly twice the moment of inertia of a solid disc of the same mass and radius, and why a solid sphere is lower still at two fifths. Race down a slope and they finish in that order reversed: the sphere wins because less of its energy goes into spinning.

One body has a different moment about every axis, so the axis is part of the answer, not a footnote. A rod about its end has four times the moment it has about its centre. The parallel-axis theorem relates them: shifting the axis a distance d from the centre of mass adds md², which means the centre of mass is always the axis of least resistance.