What it is and when to use it
Mass moment of inertia measures how strongly an object resists changes to its rotation about a given axis. It plays the same role for rotation that mass plays for straight-line motion: the larger it is, the more torque is needed to produce a given angular acceleration. It depends on both the object's mass and how far that mass is spread from the axis.
Use this calculator when you need I for common shapes in mechanics homework, flywheel and pulley sizing, robotics or rotational energy problems. It covers standard uniform, rigid shapes about their symmetry axes. For composite parts, add the moments of the pieces, and for an axis that is not through the centre, use the parallel-axis theorem.
Formulas and variables
Pick a shape and the calculator applies the standard result, where m is mass in kilograms, r is radius and L is length, both in metres:
- Solid cylinder or disc about its axis: I = ½ m r²
- Thin ring or hoop: I = m r²
- Solid sphere about a diameter: I = 2/5 m r²
- Thin hollow sphere: I = 2/3 m r²
- Thin rod about its centre: I = 1/12 m L²; about one end: I = 1/3 m L²
- Rectangular plate about the perpendicular axis through its centre: I = 1/12 m (a² + b²)
It also returns the radius of gyration k = √(I / m), which is the distance from the axis at which all the mass could be concentrated to give the same I. The lb·ft² output multiplies kg·m² by 23.73.
Worked example: a 5 kg solid sphere of 0.2 m radius
I = 2/5 × 5 × 0.2² = 0.4 × 5 × 0.04 = 0.0800 kg·m².
Radius of gyration: k = √(0.08 / 5) = √0.016 = 0.1265 m. In imperial units, 0.08 × 23.73 = 1.898 lb·ft².
For comparison, a 5 kg thin ring of the same 0.2 m radius gives 5 × 0.04 = 0.2 kg·m², which is 2.5 times larger, because all of its mass sits at the full radius.
Common mistakes and how to interpret the result
- Using diameter instead of radius. Since r is squared, entering the diameter makes the result four times too large.
- Choosing the wrong axis. A rod's I about its end is four times its I about its centre, so identify the axis of rotation before choosing a formula.
- Mixing units. The formulas expect kilograms and metres. Convert grams, millimetres or pounds first, or the answer will be off by large factors.
- Confusing mass moment of inertia with the area moment of inertia used for beam bending. They share a name but have different units and different uses.