Chapter 15.6, Problem 44E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Assume that the solid has constant density *k.*

**44.** Find the moments of inertia for a rectangular brick with dimensions *a*, *b,* and *c* and mass *M* if the center of the brick is situated at the origin and the edges arc parallel to the coordinate axes.

To determine

**To find:** The moment of inertia of the rectangular brick if the solid has constant density *k*.

Explanation

**Given:**

The rectangular brick has dimensioned as *a, b* and *c*. which is length, breadth and height respectively.

The mass is *M* and the center is situated at origin and also parallel to the coordinate axes.

**Calculation:**

The edges of the brick parallel to the coordinate axes and also center is at the origin. Therefore *x* varies from
*y* varies from
*z* varies from

Initially, calculate the moment of inertia over the *x*-axis. Therefore, the integration will be over

The moment of inertia over the *x*-axis is,

Integrate the above integral with respect to *x* and apply the limit of it.

On further simplification the above integral becomes,

Integrate the above integral with respect to *y* and apply the limit of it.

On further simplification the above integral becomes,

Integrate the above integral with respect to *z* and apply the limit of it.

On further simplification the above integral becomes,

Thus, the moment of inertia of the rectangular brick is

Thus, the moment of inertia of the cylindrical brick about the *x*-axis is

Calculate the moment of inertia over the *y*-axis. Therefore, the integration will be over

The moment of inertia over they *y*-axis is,

Integrate the above integral with respect to *x* and apply the limit of it.

Integrate the above integral with respect to *y* and apply the limit of it.

On further simplification the above integral becomes,

Integrate the above integral with respect to *z-* and apply the limit of it

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