Q. 58 In Exercises 57鈥�60, let R be t... [FREE SOLUTION]

Chapter 13: Q. 58 (page 1056)

In Exercises 57鈥�60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.
Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Short Answer

Expert verified

The moment of inertia about the x-axis is $I_{x} = 104$ and the radius of gyration about the x-axis is $R_{x} 鈮� 2.0816 .$

Step by step solution

Step 1. Given Information.

The given rectangular solid is defined by $R = {(x, y, z) 鈭� 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2} .$

Step 2. Find the moment of inertia about the x-axis.

It is given that the density of R is uniform throughout, so $蚁 (x, y, z) = 1 .$

Now, the moment of the inertia about the x-axis is,

$I_{x} = {鈭�}_{惟} (y^{2} + z^{2}) 蚁 (x, y, z) d v I_{x} = {鈭�}_{0}^{4} {鈭�}_{0}^{3} {鈭�}_{0}^{2} (y^{2} + z^{2}) d z d y d x I_{x} = {鈭�}_{0}^{4} {鈭�}_{0}^{3} [{鈭�}_{0}^{2} (y^{2} + z^{2}) d z] d y d x I_{x} = {鈭�}_{0}^{4} {鈭�}_{0}^{3} [y^{2} {鈭�}_{0}^{2} d z + {鈭�}_{0}^{2} z^{2} d z] d y d x I_{x} = {鈭�}_{0}^{4} {鈭�}_{0}^{3} [2 y^{2} + \frac{8}{3}] d y d x I_{x} = {鈭�}_{0}^{4} [2 {鈭�}_{0}^{3} y^{2} d y + \frac{8}{3} {鈭�}_{0}^{3} d y] d x I_{x} = {鈭�}_{0}^{4} 26 d x I_{x} = 104$

Step 3. Find the radius of the gyration.

To find the radius of gyration about the x-axis, we have to find the mass of the rectangular solid:

$M = {鈭�}_{R} 蚁 (x, y, z) d x d y d z M = {鈭�}_{0}^{4} {鈭�}_{0}^{3} {鈭�}_{0}^{2} d x d y d z M = = (4 鈭� 0) (3 - 0) (2 - 0) M = 24$

So, the radius of gyration is:

$R_{x} = \sqrt{\frac{I_{x}}{m}} R_{x} = \sqrt{\frac{104}{24}} R_{x} 鈮� 2.0816$

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91影视

In Exercises 57鈥�60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.
Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the moment of inertia about the x-axis.

Step 3. Find the radius of the gyration.

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91影视

In Exercises 57鈥�60, let R be the rectangular solid defined byR = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the moment of inertia about the x-axis.

Step 3. Find the radius of the gyration.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Probability and Statistics

Geometry

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Exercises 57鈥�60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.
Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.