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.
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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