Chapter 21, Problem 1P

Show that the sum of the moments of inertia of a body, I_xx + I_yy + I_zz, is independent of the orientation of the x, y, z axes and thus depends only on the location of the origin.

To determine

Show that the sum of moments of inertia of a body, $I_{xx}+I_{yy}+I_{zz}$ is independent of the orientations of $x,y,z$ axes and depends only on location of its origin.

Answer to Problem 1P

The given equation is proved.

Explanation of Solution

Given:

The moment of inertia of a body with respect to $xx$ axes is $I_{xx}$.

The moment of inertia of a body with respect to $yy$ axes is $I_{yy}$.

The moment of inertia of a body with respect to $zz$ axes is $I_{zz}$.

Conclusion:

Express the summation of moments of inertia of body with respect to $x,y,z$ axes.

$\begin{array}{c}{I}_{xx}+I_{yy}+I_{zz}={\displaystyle \underset{m}{\int }\left(y^2+z^2\right)dm+}{\displaystyle \underset{m}{\int }\left(x^2+z^2\right)dm+}{\displaystyle \underset{m}{\int }\left(x^2+y^2\right)dm}\\ =2{\displaystyle \underset{m}{\int }\left(x^2+y^2+z^2\right)dm}\\ ={\displaystyle \underset{m}{\int }\left(r^2\right)dm}\end{array}$

Here, the distance from the origin to $dm$ is $r$ and differential mass is $dm$.

Since the distance $\left|r\right|$ is constant and does not depend on the orientation of the $x,y,z$ axes. Thus, $I_{xx}+I_{yy}+I_{zz}$ is also independent of the orientation of the respective axes and depends only on location of its origin.

Hence, the given expression is satisfied and proved.