Moment of inertia calculator

What is moment of inertia?

It measures how hard an object is to spin up or slow down about an axis — the rotational counterpart of mass. For point masses I = Σ m r², the sum of each mass times the square of its distance from the axis. Mass far from the axis counts much more than mass near it.

Formulas for common shapes

From OpenStax University Physics (Figure 10.20): hoop about its axis M R²; solid disk or cylinder ½ M R²; annular cylinder ½ M (R₁² + R₂²); thin rod about its center (1/12) M L², about one end ⅓ M L²; solid sphere ⅖ M R²; thin spherical shell ⅔ M R²; hoop about a diameter ½ M R²; rectangular slab about the perpendicular axis through its center (1/12) M (a² + b²).

Example: a child on a merry-go-round

OpenStax treats the merry-go-round as a 500 kg disk of radius 2.0 m: ½ × 500 × 2.0² = 1,000 kg·m². A 25 kg child sitting 1.0 m from the center adds 25 × 1.0² = 25 kg·m², for 1,025 kg·m² in total. Add the parts one at a time here — moments of inertia about the same axis simply add up.

What is the parallel-axis theorem?

If you know I about an axis through the center of mass, the moment about any parallel axis a distance d away is I = I_cm + M d². Example from OpenStax: a 1.0 kg sphere of radius 0.2 m whose center is 0.7 m from the axis has 0.016 + 1.0 × 0.7² = 0.506 kg·m². The offset field appears only for shapes whose axis passes through the center of mass.

What units does it use?

SI units are kg·m². In US units the answer is shown in lb·ft² (1 lb·ft² ≈ 0.04214 kg·m²), with lb·in² in the table. The radius of gyration √(I/M) is the distance at which all the mass could sit to give the same moment of inertia.

How does moment of inertia relate to torque and kinetic energy?

It plays the role of mass in the rotational laws: net torque = I × angular acceleration, and rotational kinetic energy = ½ I ω². With I = 0.225 kg·m² (a 5 kg disk of 0.3 m radius), a torque of 0.5 N·m gives an angular acceleration of 2.22 rad/s², and spinning at 10 rad/s stores 11.25 J.

Do I enter the radius or the diameter?

The radius, R, for every round shape. Entering a diameter makes the result four times too big, because I grows with the square of the distance. For a ring (annular cylinder) enter the inner and outer radii, R₁ and R₂.

Why is the same mass harder to spin on a hoop than on a disk?

Because the hoop has all its mass at the full radius, while a disk's mass is spread from the center out. For 5 kg and a 0.3 m radius the hoop has 0.45 kg·m², the disk 0.225 kg·m² and a solid sphere 0.18 kg·m².

Worked example: a rod about its center and its end

A uniform rod of mass 2 kg and length 1.2 m. About an axis through its center, perpendicular to the rod, I = (1/12) M L² = (1/12) × 2 × 1.2² = 0.24 kg·m². About one end it is (1/3) M L² = (1/3) × 2 × 1.2² = 0.96 kg·m², four times as much, because the axis has moved from the middle of the rod to its end.

The parallel-axis theorem gives the same result from the center value: moving the axis a distance d = L/2 = 0.6 m gives I = I_cm + M d² = 0.24 + 2 × 0.6² = 0.96 kg·m². The radius of gyration for the end axis is √(I/M) = 0.693 m. The two routes agree, which is a useful check whenever a direct formula exists.

Assumptions and limits

The shapes are uniform (constant density) rigid bodies, and the formulas are for the specific axes shown in the OpenStax figure: a rod about an axis perpendicular to its length, a cylinder or disk about its symmetry axis, a sphere about any diameter. A real flywheel with spokes or a hollow casting needs to be split into parts; since moments of inertia about the same axis add, you can compute each part and sum them. The parallel-axis theorem requires that the starting axis go through the center of mass, so the calculator offers the offset field only for those shapes.

Common mistakes

Shifting an axis from one off-center position to another does not work with the theorem; go back to the center of mass first. Another is mixing mass and weight: in US units the answer is in lb·ft² only when the input is a mass in pounds, not a force. A third is combining a moment of inertia taken about different axes: only values about the same axis can be added.

Sources: OpenStax, University Physics Volume 1, sections 10.4 (Figure 10.20, rotational kinetic energy), 10.5 (parallel-axis theorem) and 10.7 (Newton's second law for rotation). Related: torque calculator, angular velocity calculator, angular momentum calculator, kinetic energy calculator.