Moment of Inertia
Simple Explanation
Moment of inertia is a measure of how much an object resists changes to its rotational motion — it depends not just on an object's mass, but on how that mass is distributed relative to the axis of rotation. Mass farther from the axis contributes much more to the moment of inertia.
Why Do We Need It?
Moment of inertia is the rotational equivalent of mass in Newton's second law — it plays exactly the same role for rotation that ordinary mass plays for straight-line motion.
Formula
Moment of Inertia (Point Mass)
I = mr²
Moment of inertia measures an object's resistance to changes in its rotational motion — for a point mass, it depends on the mass and how far that mass is from the axis of rotation.
- I
- — Moment of inertia, in kg·m²
- m
- — Mass of the point/particle, in kg
- r
- — Distance from the axis of rotation, in metres
When to use it: Use for a point mass or as the building block for finding the total moment of inertia of an extended object (by summing mr² for every mass element).
Worked Example
Finding moment of inertia of a point mass
A 0.5 kg mass is attached to a rotating rod at a distance of 0.8 m from the axis. Find its moment of inertia.
Why Does This Work?
Mass located farther from the axis has to travel a larger circle (and therefore a larger linear distance and speed) for the same angular velocity, requiring more force to accelerate — the r² dependence captures how quickly this resistance to rotational change grows with distance from the axis.
Real-Life Example
Why a figure skater spins faster with arms pulled in
A spinning figure skater speeds up dramatically when they pull their arms close to their body.
Pulling the arms in reduces r for that mass, sharply reducing the total moment of inertia (since it depends on r²) — this connects directly to angular momentum conservation, covered later in this chapter.
Practice
A 2 kg point mass rotates at a radius of 0.3 m. Find its moment of inertia.
MediumCommon mistake
Assuming moment of inertia depends only on mass, like ordinary inertia — it depends on BOTH mass and how far that mass is distributed from the axis (r²), so the same mass can have very different moments of inertia depending on its distribution.
Quick Review
- I = mr² for a point mass — moment of inertia depends on mass AND distance from the axis.
- Moment of inertia is the rotational equivalent of mass.
- Mass farther from the axis contributes disproportionately more (r² dependence).