PRINCIPAL MOMENTS OF INERTIA EXAMPLES

PRINCIPAL MOMENTS OF INERTIA EXAMPLES

Given the principal axes for the symmetrical sphere with mass, m, and radius, r, shown in Figure 1↓, find the principal moments of inertia for this body.
figure images/circle.png
Figure 1 Two-dimensional view of a spherical body

ANSWER: Because the axes are synonymous with the axes of symmetry, the three principal moments of inertia are all the same. Calculating the integrals for moments of inertia, Ixx = Iyy = Izz = 2 ⁄ 5mr2.

To get this answer, the equation for the moment of inertia is
(1) I =  ⌠⌡Vρr⋅rdV
where r is the vector that defines the location of an arbitrary point. In this case, spherical coordinates is the most convenient coordinate system to use for the geometry of this body, so
(2) I =  ⌠⌡Vρr2dV
By evaluating the the integrals from zero to 2 π and where dV = sin θ r2 dr dθ dφ, the answer can be obtained.