5.4 Rotational Inertia

Describe the rotational inertia of a rigid system rotating about an axis that does not pass through the system’s center of mass. TOPIC 5.4 Rotational Inertia

• A rigid system’s rotational inertia in a given plane is at a minimum when the rotational axis passes through the system’s center of mass.
• The parallel axis theorem uses the following equation to relate the rotational inertia of a rigid system about any axis that is parallel to an axis through its center of mass: BOUNDARY STATEMENT AP Physics C: Mechanics only expects students to use calculus in the derivations of the rotational inertia of thin rods of uniform or nonuniform density about an arbitrary axis perpendicular to the rod, as well as derivations of the rotational inertia of a thin cylindrical shell, disk, or rigid bodies that can be considered to be made up of coaxial rings or shells about an axis that passes through their centers (e.g., annular rings). Students should have a qualitative understanding of the factors that affect rotational inertia; for example, how rotational inertia is greater when mass is farther from the axis of rotation, which is why a hoop has more rotational inertia than a solid puck of the same mass and radius.

Describe the rotational inertia of a rigid system relative to a given axis of rotation.

• Rotational inertia measures a rigid system’s resistance to changes in rotation and is related to the mass of the system and the distribution of that mass relative to the axis of rotation.
• The rotational inertia of an object rotating a perpendicular distance r from an axis is described by the equation Im r2= .
• The total rotational inertia of a collection of objects about an axis is the sum of the rotational inertias of each object about that axis.
• For a solid that can be considered as a collection of differential masses, dm, the solid’s rotational inertia can be calculated using the equation where r is the perpendicular distance from dm to the axis of rotation.