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Angular Momentum Calculator

This angular momentum calculator allows you to calculate the angular momentum of an object, either by using the moment of inertia and angular velocity, or by using the mass and velocity of the object along with the radius of the curved path.

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Angular momentum28.274334 kg·m²/s
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
Rotational kinetic energy1.7765 kJ½Iω² at 1,200 rpm
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.