# Moment of Inertia Calculator

Calculate moment of inertia for solid/hollow cylinders, spheres, and rods. Select a shape, enter mass and dimensions to find I and radius of gyration.

## What this calculates

Moment of inertia (I) measures how much an object resists rotational acceleration, just like mass resists linear acceleration. A 5 kg solid disk with a 0.5 m radius has I = 0.625 kg*m^2. If you rearrange that same mass into a hollow ring, I jumps to over 1.0 kg*m^2 because more mass is far from the axis. This is why figure skaters spin faster when they pull their arms in.

## Inputs

- **Shape** — options: Solid Cylinder / Disk (I = 1/2 MR²), Hollow Cylinder / Ring (I = 1/2 M(R₁²+R₂²)), Solid Sphere (I = 2/5 MR²), Hollow Sphere / Spherical Shell (I = 2/3 MR²), Rod about Center (I = 1/12 ML²), Rod about End (I = 1/3 ML²) — Select the shape to calculate moment of inertia.
- **Mass (M)** (kg) — min 0 — Mass of the object.
- **Radius (R) or Length (L)** (m) — min 0 — Outer radius for cylinders/spheres, total length for rods.
- **Inner Radius (R₁)** (m) — min 0 — Inner radius for hollow cylinder only (ignored for other shapes).

## Outputs

- **Moment of Inertia** (kg·m²) — Moment of inertia about the axis of symmetry.
- **Formula Used** — formatted as text — The specific formula applied for this shape.
- **Radius of Gyration** (m) — k = sqrt(I/M), the effective distance of mass from the axis.

## Details

The moment of inertia formulas for common shapes about their standard axes:

| Shape | Formula | Notes |
|-------|---------|-------|
| Solid Cylinder/Disk | I = (1/2)MR2 | About the central axis |
| Hollow Cylinder | I = (1/2)M(R12 + R22) | R1 = inner, R2 = outer |
| Solid Sphere | I = (2/5)MR2 | About any diameter |
| Hollow Sphere | I = (2/3)MR2 | Thin spherical shell |
| Thin Rod (center) | I = (1/12)ML2 | About the midpoint |
| Thin Rod (end) | I = (1/3)ML2 | About one end |

The radius of gyration (k = sqrt(I/M)) tells you the effective distance at which you could concentrate all the mass and get the same moment of inertia. It is useful for comparing objects of different shapes and sizes.

For compound objects, the parallel axis theorem lets you calculate I about any axis: I = Icm + Md2, where d is the distance from the center of mass to the new axis. This is how engineers calculate the moment of inertia for complex machine parts.

## Frequently Asked Questions

**Q: What is moment of inertia?**

A: Moment of inertia is the rotational equivalent of mass. It quantifies how much torque is needed to change an object's angular velocity. The farther the mass is distributed from the rotation axis, the higher the moment of inertia. It is measured in kg*m^2 in SI units.

**Q: Why does shape matter for moment of inertia?**

A: Moment of inertia depends on how mass is distributed relative to the rotation axis. A solid sphere (I = 2/5 MR^2) has a lower I than a hollow sphere (I = 2/3 MR^2) of the same mass and radius because more mass is concentrated near the center. This is why a solid ball rolls down a hill faster than a hollow ball.

**Q: What is the parallel axis theorem?**

A: The parallel axis theorem states that I about any axis equals I about the center of mass plus M*d^2, where d is the perpendicular distance between the two parallel axes. For example, a rod's I about its center is (1/12)ML^2, but about one end it is (1/12)ML^2 + M(L/2)^2 = (1/3)ML^2.

**Q: How is moment of inertia used in engineering?**

A: Engineers use moment of inertia to design flywheels (energy storage), balance rotating machinery, size motors and brakes, analyze vehicle dynamics, and calculate structural beam deflection (area moment of inertia). In robotics, knowing I for each link is essential for controlling arm movements.

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Source: https://vastcalc.com/calculators/physics/moment-of-inertia
Category: Physics
Last updated: 2026-04-08