Moment of Inertia
Moment of inertia is the rotational analogue of mass — it measures an object's resistance to changes in its rotational motion about a given axis. It depends on both the total mass of the object and how that mass is distributed relative to the rotation axis; mass farther from the axis contributes more. Moment of inertia is fundamental in the design of flywheels, spinning tops, gyroscopes, and all rotating mechanical systems.
Key Formula
I = Σ(mᵢ × rᵢ²) = integral of r² dm
LaTeX: I = \sum m_i r_i^2 = \int r^2 \, dm
| Symbol | Meaning | Unit |
|---|---|---|
| I | Moment of inertia | Kilogram-metre squared (kg·m²) |
| mᵢ | Mass of each point mass element | Kilogram (kg) |
| rᵢ | Perpendicular distance of each element from the rotation axis | Metre (m) |
Worked Example
Problem
Calculate the moment of inertia of a solid uniform disc of mass 3 kg and radius 0.5 m rotating about its central axis.
Solution
Step 1: Recall the formula for a solid disc. I = ½ × m × R² Step 2: Substitute values. I = ½ × 3 × (0.5)² I = ½ × 3 × 0.25 I = 0.375 kg·m²
Answer
I = 0.375 kg·m²
Moment of Inertia Formulas for Common Shapes
| Shape | Axis of Rotation | Formula | Notes |
|---|---|---|---|
| Solid sphere | Through centre | I = 2/5 × m × R² | Ball bearing, planet |
| Hollow sphere (thin shell) | Through centre | I = 2/3 × m × R² | Hollow shell |
| Solid cylinder / disc | Central (longitudinal) | I = 1/2 × m × R² | Flywheel, coin |
| Hollow cylinder (thin shell) | Central | I = m × R² | Pipe, hoop |
| Thin rod | Through centre, perpendicular | I = 1/12 × m × L² | Beam, ruler |
| Thin rod | Through one end | I = 1/3 × m × L² | Pendulum rod |
Interactive Tools
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Related Terms
Torque
Torque is the rotational equivalent of force — it is the tendency of a force to cause rotation about a pivot or axis. Mathematically, it is the cross product of the position vector (from the axis to the point of force application) and the force vector. Torque is essential in engineering design of engines, gears, wrenches, and any rotating machinery.
Angular Acceleration
Angular acceleration is the rate of change of angular velocity with respect to time. Like its linear counterpart, it is a vector quantity and represents how quickly a rotating object is speeding up or slowing down its rotation. Angular acceleration is produced by a net torque and is related to it by the rotational analogue of Newton's second law.
Angular Velocity
Angular velocity is the rate of change of angular displacement of a rotating object with respect to time. It is a vector quantity whose direction is given by the right-hand rule along the axis of rotation. Angular velocity is the rotational analogue of linear velocity and is central to the analysis of rotating machinery, celestial bodies, and rigid body dynamics.
The term "moment" in mechanics derives from Latin "momentum" (movement), used since Archimedes to describe leverage effects. "Inertia" comes from Latin "iners" meaning "idle" or "inactive". The concept was developed by Leonhard Euler and Christiaan Huygens in the 17th–18th centuries.