Moment of inertia measures how hard it is to start, stop, or change the rotation of an object. It plays the same role in rotational motion that mass plays in straight-line motion, but it also depends on where the mass is located relative to the rotation axis. A compact object is easier to spin than the same mass spread farther outward.
This idea explains why different shapes rotate differently even when they have the same mass.
Key Facts
- Moment of inertia for point masses: I = Σmr^2
- Rotational form of Newton's second law: τ = Iα
- Rotational kinetic energy: Krot = 1/2 Iω^2
- Hoop or thin ring about center: I = MR^2
- Solid disk or solid cylinder about center: I = 1/2 MR^2
- Parallel axis theorem: I = Icm + Md^2
Vocabulary
- Moment of inertia
- Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a chosen axis.
- Rotation axis
- The rotation axis is the line around which an object spins or could spin.
- Torque
- Torque is the turning effect of a force, equal to the force times the perpendicular lever arm.
- Angular acceleration
- Angular acceleration is the rate at which angular velocity changes with time.
- Mass distribution
- Mass distribution describes how an object's mass is arranged relative to a chosen rotation axis.
Common Mistakes to Avoid
- Treating moment of inertia as only mass is wrong because the distance of each bit of mass from the axis matters through r^2.
- Using the wrong axis for a formula is wrong because the same object can have different moments of inertia about different axes.
- Saying a hoop and disk with equal mass and radius are equally hard to spin is wrong because the hoop has all its mass at radius R, while the disk has much of its mass closer to the center.
- Forgetting units is wrong because moment of inertia is measured in kg m^2, not kg or N.
Practice Questions
- 1 A thin hoop has mass 2.0 kg and radius 0.50 m. Find its moment of inertia about its central axis.
- 2 A solid disk has mass 2.0 kg and radius 0.50 m. Find its moment of inertia about its central axis, then compare it with the hoop from question 1.
- 3 Two objects have the same mass and radius: a thin hoop and a solid disk. If the same torque is applied to both, which has the larger angular acceleration and why?