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# Moments of Inertia for Various Objects Table 2 shows the moments of inertia for various objects, given their shape, axis of rotation, and location. | Object | Location of Axis | Diagram | Moment of Inertia | |----------------...

# Moments of Inertia for Various Objects Table 2 shows the moments of inertia for various objects, given their shape, axis of rotation, and location. | Object | Location of Axis | Diagram | Moment of Inertia | |----------------------------------------|-----------------------|----------------------------|---------------------------------| | Thin hoop of radius *r* | through central diameter |  | *mr*² | | Solid uniform cylinder of radius *r* | through center |  | (1/2) *mr*² | | Uniform sphere of radius *r* | through center |  | (2/5) *mr*² | | Long, uniform rod of length *l* | through center |  | (1/12) *ml*² | | Long, uniform rod of length *l* | through end |  | (1/3) *ml*² | | Thin, rectangular plate of length *l* | through center |  | (1/12) *m(l*² + *w*²) | ## Moment of Inertia and Mass The moments of inertia for extended objects depend on how far the masses are from the axis of rotation. For example, a bicycle wheel has most of its mass in the rim and tire. Its moment of inertia about its axle is almost exactly equal to *mr*², where *r* is the radius of the wheel. For most objects, however, the mass is distributed closer to the axis so the moment of inertia is less than *mr*². For example, a solid cylinder of radius *r* has a moment of inertia of *I = (1/2)* *mr*², while a solid sphere has *I = (2/5)* *mr*². ## Moment of Inertia and Rotational Axis The moment of inertia also depends on the location and direction of the rotational axis. - To observe this effect, hold a book in the upright position by placing your hands at the bottom of the book. Feel the torque needed to rock the book toward you and then away from you. - Now put your hands in the middle of the book and feel the torque needed to rock the book. - Note that much less torque is needed when your hands are placed in the middle of the book because the average distance of the book's mass from the rotational axis is much less.