Engineering Mechanics: Dynamics (14th Edition)
Engineering Mechanics: Dynamics (14th Edition)
14th Edition
ISBN: 9780133915389
Author: Russell C. Hibbeler
Publisher: PEARSON
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Chapter 21.1, Problem 1P

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

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To determine

Show that the sum of moments of inertia of a body, Ixx+Iyy+Izz 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 Ixx .

The moment of inertia of a body with respect to yy axes is Iyy .

The moment of inertia of a body with respect to zz axes is Izz .

Conclusion:

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

Ixx+Iyy+Izz=m(y2+z2)dm+m(x2+z2)dm+m(x2+y2)dm=2m(x2+y2+z2)dm=m(r2)dm

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

Since the distance |r| is constant and does not depend on the orientation of the x,y,z axes. Thus, Ixx+Iyy+Izz 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.

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Chapter 21 Solutions

Engineering Mechanics: Dynamics (14th Edition)

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