Moment of inertia
Measure of rotational inertia, analogous to mass in linear motion.
The moment of inertia is a measure of how difficult it is to change the rotation rate of a rigid body about a given axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis, playing the same role in rotational motion as mass does in linear motion. A body's moment of inertia depends on both its mass and the distribution of that mass relative to the axis, increasing with mass and distance from the axis.
- SI unit
- kilogram metre squared (kg·m2)
- Imperial/US unit
- pound-foot-squared (lb·ft2)
- Symbol
- I or J
- Definition (point mass)
- mass times the square of the perpendicular distance to the axis (mr²)
- Role
- resistance to change in rotational motion
Lore & Background
For a point mass, the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation, and for a rigid composite system it is the sum of the moments of inertia of its component subsystems about the same axis.
Reader's Guide
The moment of inertia is a fundamental concept in rotational dynamics, describing how mass distribution affects a body's resistance to angular acceleration. It appears in momentum, kinetic energy, and Newton's laws of motion for rigid bodies. In planar movement, a single scalar defines the moment of inertia, while spatial movement requires a symmetric three-by-three matrix (inertia tensor) that can be diagonalized into principal axes. Practical applications include the use of flywheels to smooth rotational output in machines, and the determination of how steering forces affect an airplane's roll, pitch, and yaw. The moment of inertia also explains phenomena such as a figure skater spinning faster when pulling in arms, as constant angular momentum leads to increased angular velocity when the moment of inertia decreases.
Did You Know?
- The moment of inertia is defined as the ratio of net angular momentum L to angular velocity ω around a principal axis: I = L/ω.
- For a simple pendulum, the moment of inertia is the product of the mass and the square of its distance to the pivot (I = mr²).
- The radius of gyration k is defined such that I = mk² for an object of mass m about a given axis.
- If a body is allowed to rotate in three dimensions, its moment of inertia is described by a symmetric three-by-three matrix.
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