Moment of Inertia Calculator
Seven standard shapes, with rotational energy.
Work out Moment of Inertia. Seven standard shapes, with rotational energy. Says where the approximation stops holding.
Optional — gives rotational energy and angular momentum.
Moment of inertia
1.25 kg·m²
Solid cylinder or disc, about its axis — I = 0.5mr²
Moment of inertia is rotational mass — how hard something is to spin up. The coefficient encodes where the mass sits relative to the axis, and that placement matters more than the total. A hoop has exactly twice the moment of inertia of a solid disc with the same mass and radius, because all its mass sits at the maximum distance. Race a hoop and a disc down a ramp and the disc wins every time, regardless of which is heavier — the hoop puts more of its energy into spinning and less into moving. The same principle explains a skater pulling their arms in to speed up: angular momentum is conserved, so reducing the moment of inertia must raise the spin rate.
How the Moment of Inertia Calculator works
Pick a shape and enter mass and radius for the moment of inertia, plus rotational energy and angular momentum if you supply a spin rate. The coefficient encodes where the mass sits relative to the axis, and that placement matters more than the total.
Also known as: rotational inertia calculator · rotational kinetic energy calculator · moment of inertia formulas · i = mr2 calculator
Frequently asked questions
What is moment of inertia?
Rotational mass — how hard something is to spin up. It depends on both the mass and how far that mass sits from the axis, since I = kmr² with k set by the shape.
Why does a hoop roll slower than a disc?
Because a hoop has exactly twice the moment of inertia of a disc with the same mass and radius. More of its energy goes into spinning and less into moving, so it loses the race down a ramp regardless of weight.
What are the standard coefficients?
Solid cylinder ½, hoop 1, solid sphere ⅖, hollow sphere ⅔, rod about its centre 1/12, rod about one end ⅓. A point mass at radius r is simply 1.
Why does a skater speed up pulling their arms in?
Angular momentum L = Iω is conserved. Pulling the arms in reduces the moment of inertia, so the angular velocity has to rise to compensate.
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