Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 26

Applications of Integration - all with Video Answers

Educators

Section 5

Moments of Inertia

Problem 1

Make the given changes in the indicated examples of this section, and then solve the resulting problems.
In Example $2,$ change $y=4 x^{2}$ to $y=4 x$

Stanley E.

Stanley E.

Numerade Educator

Problem 2

Make the given changes in the indicated examples of this section, and then solve the resulting problems.
In Example 4, change $y^{3}=x$ to $y^{2}=x$

Stanley E.

Numerade Educator

Problem 3

Find the moment of inertia (in $\mathrm{g} \cdot \mathrm{cm}^{2}$ ) and the radius of gyration (in $\mathrm{cm}$ ) with respect to the origin of each of the given arrays of masses located at the given points on the $x$ -axis.
$$4.2 \mathrm{g} \text { at }(1.7,0), 3.2 \mathrm{g} \text { at }(3.5,0)$$

Stanley E.

Numerade Educator

Problem 4

Find the moment of inertia (in $\mathrm{g} \cdot \mathrm{cm}^{2}$ ) and the radius of gyration (in $\mathrm{cm}$ ) with respect to the origin of each of the given arrays of masses located at the given points on the $x$ -axis.
$$3.4 \mathrm{g} \text { at }(-1.5,0), 6.0 \mathrm{g} \text { at }(2.1,0), 2.6 \mathrm{g} \text { at }(3.8,0)$$

Stanley E.

Numerade Educator

Problem 5

Find the moment of inertia (in $\mathrm{g} \cdot \mathrm{cm}^{2}$ ) and the radius of gyration (in $\mathrm{cm}$ ) with respect to the origin of each of the given arrays of masses located at the given points on the $x$ -axis.
$$82.0 \mathrm{g} \text { at }(-3.80,0), 90.0 \mathrm{g} \text { at }(0.00,0), 62.0 \mathrm{g} \text { at }(5.50,0)$$

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Problem 6

Find the moment of inertia (in $\mathrm{g} \cdot \mathrm{cm}^{2}$ ) and the radius of gyration (in $\mathrm{cm}$ ) with respect to the origin of each of the given arrays of masses located at the given points on the $x$ -axis.
$$\begin{aligned}
&564 \mathrm{g} \text { at }(-45.0,0), 326 \mathrm{g} \text { at }(-22.5,0), 720 \mathrm{g} \text { at }(15.4,0), 205 \mathrm{g}\\ &\text { at }(64.0,0) \end{aligned}$$

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Problem 7

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia of a plate covering the region bounded by $x=-1, x=1, y=0,$ and $y=1$ with respect to the $x$ -axis.

Stanley E.

Numerade Educator

Problem 8

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of a plate covering the region bounded by $x=2, x=4, y=0,$ and $y=4,$ with respect to the $y$ -axis.

Stanley E.

Numerade Educator

Problem 9

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia of a plate covering the first-quadrant region bounded by $y^{2}=x, x=9,$ and the $x$ -axis with respect to the $x$ -axis.

Stanley E.

Numerade Educator

Problem 10

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia of a plate covering the region bounded by $y=2 x, x=1, x=2,$ and the $x$ -axis with respect to the $y$ -axis.

Stanley E.

Numerade Educator

Problem 11

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of a plate covering the region bounded by $y=x^{3}, x=3,$ and the $x$ -axis with respect to the $y$ -axis.

Stanley E.

Numerade Educator

Problem 12

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of a plate covering the first-quadrant region bounded by $y^{2}=1-x$ with respect to the $x$ -axis.

Stanley E.

Numerade Educator

Problem 13

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of a plate covering the region bounded by $y=x^{2}, x=3,$ and the $x$ -axis with respect to the $x$ -axis.

Stanley E.

Numerade Educator

Problem 14

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of a plate covering the region bounded by $y^{2}=x^{3}, y=8,$ and the $y$ -axis with respect to the $y$ -axis.

Stanley E.

Numerade Educator

Problem 15

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of the plate of Exercise 14 with respect to the $x$ -axis.

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Numerade Educator

Problem 16

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration of a plate covering the first-quadrant region bounded by $x=1, y=2-x,$ and the $y$ -axis with respect to the $y$ -axis.

Stanley E.

Numerade Educator

Problem 17

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia with respect to its axis of the solid generated by revolving the region bounded by $y^{2}=4 x, y=2$ and the $y$ -axis about the $x$ -axis.

Stanley E.

Numerade Educator

Problem 18

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration with respect to its axis of the solid generated by revolving the first-quadrant region under the curve $y=4-x^{2}$ about the $y$ -axis.

Stanley E.

Numerade Educator

Problem 19

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration with respect to its axis of the solid generated by revolving the region bounded by $y=4 x-x^{2}$ and the $x$ -axis about the $y$ -axis.

Stanley E.

Numerade Educator

Problem 20

Find the indicated moment of inertia or radius of gyration.
Find the radius of gyration with respect to its axis of the solid generated by revolving the region bounded by $y=2 x$ and $y=x^{2}$ about the $y$ -axis.

Stanley E.

Numerade Educator

Problem 21

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia of the triangular sail in Exercise 29 of Section 26.4, with respect to the vertical side in terms of its mass.

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Numerade Educator

Problem 22

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia of a rectangular sheet of metal of sides $a$ and $b$ with respect to side $a$. Express the result in terms of the mass of the metal sheet.

Stanley E.

Numerade Educator

Problem 23

Find the indicated moment of inertia or radius of gyration.
A top in the shape of an inverted right circular cone has a base radius $r$ (at the top), height $h$, and mass $m$. Find the moment of inertia of the cone with respect to its axis (the height) in terms of its mass and radius. See Fig. 26.57

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Problem 24

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia in terms of its mass of a circular hoop of radius $r$ and of negligible thickness with respect to its center.

Stanley E.

Numerade Educator

Problem 25

Find the indicated moment of inertia or radius of gyration.
A rotating drill head is in the shape of a right circular cone. Find the moment of inertia of the drill head with respect to its axis if its radius is $0.600 \mathrm{cm},$ its height is $0.800 \mathrm{cm},$ and its mass is $3.00 \mathrm{g}$ (See Exercise 23.)

Stanley E.

Numerade Educator

Problem 26

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia (in $\mathrm{kg} \cdot \mathrm{m}^{2}$ ) of a rectangular door $2 \mathrm{m}$ high and $1 \mathrm{m}$ wide with respect to its hinges if $k=3 \mathrm{kg} / \mathrm{m}^{2}$ (See Exercise 22.)

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Problem 27

Find the indicated moment of inertia or radius of gyration.
Find the moment of inertia of a flywheel with respect to its axis if its inner radius is $4.0 \mathrm{cm},$ its outer radius is $6.0 \mathrm{cm},$ and its mass is 1.2 kg. See Fig. 26.58.

Stanley E.

Numerade Educator

Problem 28

Find the indicated moment of inertia or radius of gyration.
A cantilever beam is supported only at its left end, as shown in Fig. $26.59 .$ Explain how to find the formula for the moment of inertia of this beam with respect to a vertical axis through its left end if its length is $L$ and its mass is $m$. (Consider the mass to be distributed evenly along the beam. This is not an area or volume type of problem.) Find the formula for the moment of inertia.

Stanley E.

Numerade Educator