Chapter 15.6, Problem 43E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Assume that the solid has constant density *k.*

**43**. Find the moments of inertia for a cube with side length *L* if one vertex is located at the origin and three edges lie along the coordinate axes.

To determine

**To find:** The momentum of inertia if the solid has constant density *k*.

Explanation

**Given:**

Side length of a cube is *L* and one of the vertex of a cube located at the origin and the three edges lie along the coordinate axes.

**Calculation:**

The region

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.

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

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

On further simplification find the value of moment of inertia.

Thus, the moment of inertia of the cube 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 the *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

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