Mass Moment of Inertia Calculator

What is the mass moment of inertia?

Mass measures how strongly a body resists linear acceleration. The mass moment of inertia is the same idea for rotation: it sets the torque needed to speed up or slow down a body about an axis. In T = I × α, I is the moment of inertia and α the angular acceleration. Its unit is kg·m².

It depends not only on mass but on how far that mass sits from the axis, and the distance enters squared. That is why a ring carries twice the inertia of a solid disc of the same mass.

Common formulas

• Solid cylinder or disc, own axis: I = ½ × m × r²
• Hollow cylinder: I = ½ × m × (r_outer² + r_inner²)
• Thin ring: I = m × r²
• Solid sphere: I = ⅖ × m × r²
• Slender rod about its middle: I = m × L² / 12
• Rectangular block: I = m × (a² + b²) / 12

Parallel axis theorem

When the axis does not pass through the centre of mass, add m × e² to the value about the parallel axis that does, where e is the distance between the two axes. A 1.2 m, 2 kg rod turning about its middle has I = 0.24 kg·m². Turning about its end it has 0.24 + 2 × 0.6² = 0.96 kg·m².

Example

A solid disc of 10 kg and 200 mm diameter spins on its own axis. Its moment of inertia is ½ × 10 × 0.1² = 0.05 kg·m². Taking it from rest to 1500 rpm in 1 second needs 0.05 × 157 = 7.85 N·m of torque.

Limits

The formulas are for homogeneous bodies of simple shape. For complex parts, split the body into simple pieces, move each to the common axis and add them up, or use the mass properties of a CAD program. Do not confuse this quantity with the area moment of inertia used for beams.