Moment of inertia calculator
Moment of inertia of a rod, disk, ring or sphere about an axis.
Fill in the fields and the result will appear here automatically.
Moment of inertia depends on mass distribution and the chosen axis, not just mass. Bodies here are uniform: a thin rod uses a perpendicular axis through its centre or end; a disk and thin ring use a central axis normal to their plane; a solid sphere and thin spherical shell use a diameter. Size means length L for rods and radius r for the other forms.
How it works
Formula and logic
Rod about centre mL²/12, about end mL²/3, disk mr²/2, ring mr², solid sphere 2 mr²/5, hollow sphere 2 mr²/3.
Example
A disk of 2 kg and 15 cm radius has a moment of inertia of 0.0225 kg·m².
Fields and units
- Body — list option
- Mass — kg
- Radius or length — m
How to use
- — Pick the body: the axis of rotation is set by that choice.
- — For a disk, ring or sphere enter the radius; for a rod enter its length.
- — The radius of gyration tells you how far from the axis all the mass would have to sit to give the same moment.
- — For a compound body work out the parts separately and add their moments about the same axis.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: inertia and the specified axis
- Limitation
- Uniform ideal forms and the listed axes. “Hollow sphere” means a thin shell; a thick shell needs an inner radius. Arbitrary axes, nonuniform density and the parallel-axis theorem are not automatically calculated.
FAQ
Why is a ring twice a disk?
A ring has all of its mass at the radius, while a disk spreads it from centre to rim. The moment grows as the square of distance, so the inner layers of a disk contribute far less.
Why two options for a rod?
Iend=Icentre+m(L/2)²=mL²/3, four times mL²/12. This compares two axes; how easily an applied force turns the body also depends on its lever arm, not only inertia.
What does the radius of gyration show?
The distance from the axis at which the whole mass would have to be concentrated as a point to leave the moment unchanged. It replaces a complicated shape with one number.
How do I handle a compound body?
Add the moments of the parts about the same axis. If the axis does not pass through a part's centre, add its mass times the offset squared — the parallel axis theorem, which is not computed here.