How to Use Moment of Inertia Calculator
The Moment of Inertia Calculator computes the rotational inertia for eight standard rigid-body shapes. Select the object type, enter the mass and geometry, and the result with the formula is shown instantly.
- Select the object type — choose from Point Mass, Rod (Center), Rod (End), Solid Disk, Ring, Solid Sphere, Hollow Sphere, or Rectangular Plate.
- Enter the mass — type the total mass M in kilograms (or any consistent mass unit).
- Enter the geometry — depending on the object, enter the radius R, length L, distance to axis r, or plate dimensions w and h.
- Read the result — the Moment of Inertia Calculator shows the moment of inertia I with the formula, the substituted expression, and the unit.
For the rectangular plate, the calculator uses the axis through the center perpendicular to the plate.
Formula & Theory - Moment of Inertia Calculator
The Moment of Inertia Calculator uses the following standard formulas derived from I = ∫r² dm:
Point mass: I = m·r²
Rod (center axis): I = (1/12)·M·L²
Rod (end axis): I = (1/3)·M·L²
Solid disk: I = (1/2)·M·R²
Ring: I = M·R²
Solid sphere: I = (2/5)·M·R²
Hollow sphere: I = (2/3)·M·R²
Rectangular plate: I = (1/12)·M·(w² + h²)| Symbol | Meaning |
|---|---|
| I | Moment of inertia (kg·m²) |
| M | Total mass of the object |
| R | Radius of disk, ring, or sphere |
| L | Length of the rod |
| r | Distance from point mass to rotation axis |
| w, h | Width and height of rectangular plate |
These closed-form solutions assume uniform mass distribution. The Moment of Inertia Calculator applies them directly for fast educational and engineering estimation.
Assumptions and Limits
All shapes assume uniform density. The rod formulas apply to a thin, uniform rod. The disk and ring formulas apply to rotation about the central axis perpendicular to the face. The sphere formulas assume rotation about a diameter. For composite objects or non-uniform density, the parallel axis theorem and integration are needed.
Use Cases for Moment of Inertia Calculator
The Moment of Inertia Calculator is useful for students and engineers working with rotational dynamics. Common uses include:
- Physics courses — Compute I for standard shapes as part of torque, angular acceleration, and kinetic energy problems.
- Mechanical engineering — Estimate the rotational inertia of flywheels, pulleys, and rotating machine components.
- Robotics — Calculate link inertias for dynamic simulation of robot arms and actuators.
- Exam preparation — Quickly verify which formula applies to a given shape and confirm numerical answers before an exam.
The Moment of Inertia Calculator displays both the symbolic formula and the numerically substituted expression, helping you understand the relationship between the object’s geometry and its rotational resistance.