What is moment of inertia?

Moment of inertia is the rotational equivalent of mass. Just as mass resists changes in linear velocity, moment of inertia resists changes in rotational speed. It depends on how the mass is distributed relative to the axis: mass further from the axis contributes much more because the distance enters the formula as a square.

What is the moment of inertia of a solid cylinder?

For a solid cylinder rotating about its central axis, I = 0.5 times m times r squared, where m is the mass and r is the radius. A 10 kg cylinder of radius 0.3 m has I = 0.5 times 10 times 0.09 = 0.45 kg m squared.

Why does a hollow cylinder have a higher moment of inertia than a solid one?

Because the hollow cylinder concentrates all its mass at the rim (distance r from the axis), giving I = m r squared. The solid cylinder has mass spread between zero and r, averaging to a lower effective distance, so its I = 0.5 m r squared. Mass at a greater distance from the axis always contributes more to inertia.

Moment of Inertia Calculator

The moment of inertia is to rotation what mass is to straight-line motion: it tells you how strongly an object resists having its spin changed. Apply the same torque to a small dense flywheel and a large spread-out disc and the flywheel will spin up much faster, because its moment of inertia is smaller. The key difference from ordinary mass is that moment of inertia depends not just on how much mass there is but on where that mass sits relative to the rotation axis. Mass at a larger radius contributes much more than the same mass close to the axis, because the distance enters the formula as a square. This is why figure skaters speed up when they pull their arms in (reducing the effective radius) and why hollow tubes are used in engineering when stiffness is more important than mass. Every common rigid shape has a standard formula. A solid cylinder or disk rotating about its central axis: I = 0.5mr2. A solid sphere: I = (2/5)mr2. A thin rod about its centre: I = (1/12)mL2. A thin rod about one end: I = (1/3)mL2. A thin hoop or hollow cylinder: I = mr2. This calculator lets you choose any of these shapes, enter the mass and the relevant dimension, and it returns the moment of inertia in kg m2 and shows the formula being used. The default example is a solid cylinder with mass m = 10 kg and radius r = 0.3 m, giving I = 0.5 times 10 times 0.09 = 0.45 kg m2. Use this tool for physics homework, for flywheel and gear design, for understanding rotational dynamics, or for any problem involving angular acceleration.

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Shape

Mass (m)

Radius (r)

0.450 kg·m²

moment of inertia

Formula usedI = 0.5 m r²

Coefficient0.5

m x dim²0.900 kg·m²

Moment of inertia in kg·m². Use kg for mass and m for dimensions. Higher I means more resistance to changes in rotation. Rounded for display.

How it works

Each shape has a standard formula derived by integrating r² dm over the body. The results: solid cylinder or disk: I = 0.5 m r²; solid sphere: I = 0.4 m r²; thin rod (centre): I = (1/12) m L²; thin rod (end): I = (1/3) m L²; thin hoop or hollow cylinder: I = m r²; hollow sphere: I = (2/3) m r². The hoop has the highest coefficient (1) because all its mass sits at the maximum radius.

Worked example

A solid cylinder with mass m = 10 kg and radius r = 0.3 m. I = 0.5 times 10 times 0.3 squared = 0.5 times 10 times 0.09 = 0.450 kg m². A solid sphere of the same mass and radius would have I = 0.4 times 10 times 0.09 = 0.36 kg m², lower because its mass is more concentrated toward the centre. These match the default values pre-filled above.

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Centripetal Force Calculator: inward force for circular motion.
Kinetic Energy Calculator: translational kinetic energy from mass and velocity.

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