Math Calculator
Moment of Inertia Calculator
Last Updated On
Calculate the moment of inertia (rotational inertia) for common shapes. Enter mass and geometry, choose an axis, and optionally apply the Parallel Axis Theorem for shifted axes. Results include step-by-step calculations and exact formulas.
Moment of Inertia Calculator
Choose a shape and enter mass and dimensions to compute rotational inertia.
Shape and Axis
Results
Common Shapes and Their Formulas
Moment of inertia measures how hard an object is to spin about an axis. It depends on the mass and on how far that mass sits from the axis, which is why the same rod is three times harder to spin about its end than about its centre.
Understanding Moment of Inertia
Moment of inertia measures how an object's mass is distributed relative to an axis of rotation. It determines how difficult it is to change the object's angular velocity and depends on both mass and geometry.
The SI unit is kilogram meter squared (kg·m²). Use the forms below to compute values for point masses, rods, plates, disks, cylinders, and spheres.
Standard Formulas by Shape
| Shape | Axis | Formula |
|---|---|---|
| Point Mass | At distance r | I = m × r² |
| Thin Rod | Perpendicular through center | I = (1/12) × m × L² |
| Thin Rod | Perpendicular through one end | I = (1/3) × m × L² |
| Rectangular Plate | Perpendicular to plate through center | I = (1/12) × m × (a² + b²) |
| Disk / Solid Cylinder | About central axis | I = (1/2) × m × R² |
| Hollow Cylinder / Ring | About central axis | I = m × R² |
| Solid Sphere | About any diameter | I = (2/5) × m × R² |
| Hollow Sphere (thin shell) | About any diameter | I = (2/3) × m × R² |
Parallel Axis Theorem
The Parallel Axis Theorem allows you to calculate the moment of inertia about any axis parallel to an axis through the center of mass:
Where Icm is the moment of inertia about the center of mass, m is the mass, and d is the perpendicular distance between the two parallel axes.
Frequently Asked Questions
- What is moment of inertia?
- Moment of inertia is a measure of an object's resistance to angular acceleration about a given axis. It is the rotational equivalent of mass in linear motion. The larger the moment of inertia, the more torque (rotational force) is needed to achieve the same angular acceleration.
- What are the SI units for moment of inertia?
- The SI unit for moment of inertia is kilogram meter squared (kg·m²). This unit represents the product of mass and the square of the distance from the axis of rotation.
- What is the Parallel Axis Theorem?
- The Parallel Axis Theorem states that the moment of inertia about any axis parallel to an axis through the center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the axes. Formula: I = I_cm + m × d²
- How does moment of inertia relate to rotational kinetic energy?
- Rotational kinetic energy is directly related to moment of inertia through the formula KE_rot = (1/2) × I × ω², where ω is the angular velocity. A larger moment of inertia means more kinetic energy is stored in rotation at the same angular velocity.
- Why is moment of inertia important in engineering?
- Moment of inertia is critical in engineering for designing rotating machinery, flywheels, turbines, and other systems. It determines how much energy is needed to start, stop, or change the rotation speed of an object, directly impacting efficiency and safety.
- How does the distance from the axis affect moment of inertia?
- Moment of inertia increases with the square of the distance from the axis of rotation. This means that mass distributed farther from the axis contributes much more significantly to the moment of inertia than mass close to the axis.
Related Concepts
Angular Velocity (ω)
The rate of change of angular position, measured in radians per second (rad/s).
Torque (τ)
The rotational force, equivalent to force in linear motion. Formula: τ = I × α
Rotational Kinetic Energy
Energy of a rotating object: KE = (1/2) × I × ω²
Angular Acceleration (α)
The rate of change of angular velocity, in rad/s². Formula: α = τ / I