Moment of Inertia
Rotational inertia of common shapes
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What the defaults give
A solid disk of 10 kg and 0.5 m radius has a moment of inertia of 1.2500 kg·m², from I = ½mr².
The mass distribution matters more than the mass. The same 10 kg shaped as a ring at the same radius gives 2.5 kg·m² — double — because all of it sits at the outer edge.
The formulas by shape
- Solid disk or cylinder (about its axis): I = ½ m r².
- Thin ring or hoop: I = m r².
- Solid sphere: I = ⅖ m r².
- Rod (about its centre): I = 1/12 m L².
Worked example: disk
m = 10 kg, r = 0.5 m
I = ½ × 10 × 0.5²
0.5² = 0.25, so I = 0.5 × 10 × 0.25 = 1.25 kg·m²
For the same inputs as a ring: I = 10 × 0.25 = 2.5. As a solid sphere: ⅖ × 10 × 0.25 = 1.0. The disk sits between them, which is what makes it the textbook default.
What the number is for
Moment of inertia is rotational mass: torque needed to spin something up is τ = Iα, and rotational kinetic energy is ½Iω². Double the inertia and the same torque takes twice as long to reach a given spin rate.
Where people go wrong
- Wrong axis. A rod spun about its end has I = ⅓mL², not 1/12; the value depends entirely on the axis, which the shape name alone does not fix.
- Radius versus diameter. Entering 500 mm as a 500 m radius produces an answer eight orders of magnitude out.
- Adding masses incorrectly. Inertias about the same axis add, so a motor plus its load is I₁ + I₂ — not the inertia of the combined mass at an averaged radius.
- Forgetting units. kg·m² only works if mass is in kilograms and length in metres.
Beyond the simple shapes
Real components are assemblies. Use the parallel axis theorem, I = Icm + md², to shift a body’s inertia to a different axis — a gearbox flange is rarely a thin disk about its own centre.
Frequently Asked Questions
Why is a disk 1.25 and a ring 2.5 for the same mass and radius?
Because a ring has all its mass at radius r, giving I = mr². A disk spreads mass inward toward the axis, where it contributes less, so half the ring’s value: ½mr².
What are the units?
kg·m² in SI, from mass in kilograms times length squared in metres. Using grams or centimetres without converting gives a value that is wrong by orders of magnitude.
Does a rod use its length or its radius?
Length — for a rod about its centre I = 1/12 mL². The third input is the length, not a radius, and about the end the coefficient changes to 1/3.
How does moment of inertia affect acceleration?
Directly through τ = Iα: for a given torque, doubling I halves the angular acceleration. In a motor-driven system the load inertia is usually expressed as a ratio to the motor’s own inertia.
What if my object is hollow?
Use the ring formula for a thin-walled cylinder. A thick-walled tube needs ½m(R² + r²), which sits between the disk and the ring values.
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