Phys. Chapter 08 - Beau Chene High School Home Page

The calculation of the moment of inertia is a straightforward but oftentedious process. Fortunately, some simple formulas are available for commonshapes. Table 8-1 gives the moments of inertia for some common shapes.When the need arises, you can use this table to determine the moment of inertiaof a body having one of the listed shapes.The units for moment of inertia are kg •m 2 . To get an idea of the size of thisunit, note that bowling balls typically have moments of inertia about an axisthrough their centers ranging from about 0.7 kg •m 2 to 1.8 kg •m 2 , dependingon the mass and size of the ball.Notice that the moment of inertia for the solid sphere is indeed smallerthan the moment of inertia for the thin hoop, as expected. In fact, themoment of inertia for the thin hoop about the symmetry axis through thecenter of mass is the largest moment of inertia that is possible for any shape.Also notice that a point mass in a circular path, such as a ball on a string,has the same moment of inertia as the thin hoop if the distance of the pointmass from its axis of rotation is equal to the hoop’s radius. This shows thatonly the distance of a mass from the axis of rotation is important in determiningthe moment of inertia for a shape. At a given radius from an axis, it doesnot matter how the mass is distributed around the axis.Finally, recall the example of the rotating baseball bat that began this section.A bat can be modeled as a rotating thin rod. Table 8-1 shows that the moment ofTable 8-1The moment of inertia for a few shapesShape Moment of inertia Shape Moment of inertiaRthin hoop aboutsymmetry axisMR 2lthin rod about1perpendicular axis ⎯ 1 2 ⎯ Ml 2through centerRthin hoop aboutdiameter1⎯ ⎯ MR 22lthin rod aboutperpendicular axis ⎯ 31 ⎯ Ml 2through endRpoint mass about axis MR 2R disk or cylinder about 1⎯ ⎯symmetry axisMR 22RRsolid sphere ⎯ 2 5 ⎯ MR 2about diameterthin spherical shellabout diameter2⎯3 ⎯ MR 2Copyright © by Holt, Rinehart and Winston. All rights reserved.Rotational Equilibrium and Dynamics285