Moment of Inertia Calculator

It measures how hard it is to change an object's rotation about a chosen axis. It depends on the mass and on how that mass is distributed relative to the axis, scaling with the square of the distance.

A hollow shell keeps more of its mass far from the center, and inertia scales with the square of distance, so it resists rotation more. For the same mass and radius, I_hollow = (2/3)MR² is larger than I_solid = (2/5)MR².

Enter the full length of the rod and choose whether the axis runs through its center or its end. For a center axis I = (1/12)ML²; for an end axis I = (1/3)ML², four times larger. This is why a door is much harder to swing when pushed near the hinge than near the handle.

It shifts a moment of inertia to a parallel axis offset from the center of mass: I = I_cm + Md², where I_cm is the value about the center-of-mass axis, M is the mass, and d is the distance between the two axes. This calculator returns I about each shape's standard (center-of-mass or stated) axis; to find I about an off-center parallel axis, take that result as I_cm and add Md² yourself. For example, the rod's end-axis value (1/3)ML² is just its center value (1/12)ML² plus M(L/2)².