What the moment of inertia describes
The moment of inertia J (also called the moment of mass) describes a body’s resistance to a change in its rotational motion—similar to mass in linear motion. The larger J is, the more torque is required to accelerate or decelerate a body. The unit is kg·m². J depends not only on the mass but especially on how that mass is distributed relative to the axis of rotation: mass located far from the axis increases J disproportionately because the distance to the axis is factored in as a square.
Formulas for standard geometries
For component shapes commonly used in drive technology, J can be calculated directly from mass and dimensions. A selection (all formulas apply to rotation around the axis of symmetry):
| Geometry | Formula J | Typical Application |
|---|---|---|
| Full-bore cylinder | J = ½ · m · r² | Shafts, Rollers, Pulley Wheels |
| Hollow cylinder | J = ½ · m · (ra² + ri²) | Tubes, Hollow Shafts |
| Ball | J = (2/5) · m · r² | Balls, spherical end effectors |
| Point mass | J = m · rdist² | Eccentric weights, cranks |
Additional geometries (disc, rectangular prism, spindle) using the same principles are available in the inertia calculator.
All formulas in the table apply to rotation about the axis of symmetry or the center of mass. If, on the other hand, a component rotates about an axis that is offset parallel to it—such as an eccentric disc that rotates not about its center but about a point on its edge—the so-called Steiner term is added: J = JS + m · d², where JS is the moment of inertia about the center of mass axis and d is the distance between the center of mass axis and the actual axis of rotation. The farther the axis of rotation is from the center of mass, the larger J becomes in addition to the pure shape component.
Mnemonic
Worked example
Given
A steel shaft (solid cylinder) with m = 10 kg and r = 100 mm (0.1 m).
Calculation
J = ½ · m · r² = ½ · 10 · 0.1² = 0.0500 kg·m²
To perform your own calculations using your own values and different geometries: Inertia Calculator (opens in a new tab)
Knowledge check
Answer all three questions, then click "Check". From 2 of 3 correct answers, the unit counts as completed. You can retry at any time.
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