Moment of Inertia & Rotational Energy Calculator
Find the moment of inertia of a common shape and its rotational kinetic energy from the mass, radius and rotation speed.
How it works
I = factor · m·r² (½ disk, 1 ring, 2⁄5 solid sphere, 2⁄3 hollow sphere) · KE = ½·I·ω² · ω = rpm·2π ÷ 60
Rotational kinetic energy is ½Iω², the spinning analogue of ½mv², where the moment of inertia I plays the role of mass. I depends not just on how much mass there is but on how far it sits from the axis — mass squared by radius — so a hoop with all its mass at the rim (I = mr²) stores far more energy than a solid disk (I = ½mr²) spinning at the same rate. This is the principle behind flywheels, which store energy by concentrating mass at a large radius.
Worked example
A solid disk of mass 2 kg and radius 0.15 m has I = ½ × 2 × 0.15² = ½ × 2 × 0.0225 = 0.0225 kg·m². At 300 rpm, ω = 300 × 2π/60 ≈ 31.42 rad/s, so KE = ½ × 0.0225 × 31.42² ≈ 11.1 J and L = 0.0225 × 31.42 ≈ 0.707 kg·m²/s.
Sources: Moment of inertia — Wikipedia · Rotational energy — Wikipedia
Frequently asked questions
Why does the shape change the energy?
The moment of inertia depends on where the mass is, not just how much. A ring keeps all its mass at radius r (I = mr²) while a solid disk spreads it inward (I = ½mr²), so at the same spin rate the ring stores twice the energy.
What is the difference from the rotational kinetic energy calculator?
This tool first computes the moment of inertia for you from a chosen shape, mass and radius, then finds the energy — so you don’t need to look up or calculate I separately.
Which axis does the formula assume?
Each factor is for rotation about the symmetry axis through the centre of mass — a disk about its central axle, a sphere about a diameter. A different axis (for example the parallel-axis case) uses a different moment of inertia.
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This calculator is for educational purposes. Double-check important results before acting on them.