10-5 CALCULATING THE ROTATIONAL INERTIA 273Checkpoint 4
The figure shows three small spheres that rotate
about a vertical axis. The perpendicular distance
between the axis and the center of each sphere is
given. Rank the three spheres according to their
rotational inertia about that axis, greatest first.Rotation
axis4 kg3 m2 m1 m9 kg36 kg10-5CALCULATING THE ROTATIONAL INERTIA
After reading this module, you should be able to...
10.20Determine the rotational inertia of a body if it is given in
Table 10-2.
10.21Calculate the rotational inertia of a body by integration
over the mass elements of the body.
10.22Apply the parallel-axis theorem for a rotation axis that is
displaced from a parallel axis through the center of mass of
a body.●Iis the rotational inertia of the body, defined as
for a system of discrete particles and defined as
for a body with continuously distributed mass. The randriin
these expressions represent the perpendicular distance from
the axis of rotation to each mass element in the body, and the
integration is carried out over the entire body so as to include
every mass element.
Ir^2 dm
Imiri^2
●The parallel-axis theorem relates the rotational inertia Iof a
body about any axis to that of the same body about a parallel
axis through the center of mass:IIcomMh^2.Herehis the perpendicular distance between the two axes,
andIcomis the rotational inertia of the body about the axis
through the com. We can describe has being the distance
the actual rotation axis has been shifted from the rotation axis
through the com.Learning Objectives
Key Ideas
Calculating the Rotational Inertia
If a rigid body consists of a few particles, we can calculate its rotational inertia
about a given rotation axis with Eq. 10-33 ; that is, we can find the
productmr^2 for each particle and then sum the products. (Recall that ris the per-
pendicular distance a particle is from the given rotation axis.)
If a rigid body consists of a great many adjacent particles (it is continuous,like
a Frisbee), using Eq. 10-33 would require a computer. Thus, instead, we replace the
sum in Eq. 10-33 with an integral and define the rotational inertia of the body as
(rotational inertia, continuous body). (10-35)Table 10-2 gives the results of such integration for nine common body shapes and
the indicated axes of rotation.
Parallel-Axis Theorem
Suppose we want to find the rotational inertia Iof a body of mass Mabout a
given axis. In principle, we can always find Iwith the integration of Eq. 10-35.
However, there is a neat shortcut if we happen to already know the rotational in-
ertiaIcomof the body about a parallelaxis that extends through the body’s center
of mass. Let hbe the perpendicular distance between the given axis and the axis
Ir^2 dm
(Imiri^2 )