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Chapter 13: Q 41. (page 1039)

Let $R$ be rectangular region with vertices $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in R is proportional to the point鈥檚 distance from the $y$ -axis, find the moments of inertia about the $x$ - and $y$ -axes. Use these answers to find the radii of gyration of $R$ about the $x$ - and $y$ -axes.

Short Answer

Expert verified

The moment of inertia is $I_{y} = \frac{1}{4} k h b^{4}$ and $I_{x} = \frac{1}{6} k b^{2} h^{3}$ .

The mass is $m = \frac{1}{2} k h b^{2}$

The radius of gyration is $R_{y} = \frac{b}{\sqrt{2}} and R_{x} = \frac{h}{\sqrt{3}}$

Step by step solution

01

Given Information

It is given that vertices of rectangular region is $(0, 0), (b, 0), (0, h) and (b, h)$

$蚁 (x, y) = k x$

02

Calculation of Iy

The formula is $I_{y} = {鈭�}_{惟} x^{2} 蚁 (x, y) d A$

$I_{y} = {鈭�}_{0}^{b} {鈭�}_{0}^{h} x^{2} 蚁 (x, y) d y d x$

$I_{y} = {鈭�}_{0}^{b} {鈭�}_{0}^{h} x^{2} k x d y d x [蚁 (x, y) = k x]$

$I_{y} = k {鈭�}_{0}^{b} {鈭�}_{0}^{h} x^{3} d y d x$

Solving inner integral

$I_{y} = k {鈭�}_{0}^{b} [y]_{0}^{h} x^{3} d x = k {鈭�}_{0}^{b} [h] x^{3} d x = k h {鈭�}_{0}^{b} x^{3} d x$

$I_{y} = k h {[\frac{x^{4}}{4}]}_{0}^{b} = k h [\frac{b^{4}}{4}]$

Hence, $I_{y} = \frac{1}{4} k h b^{4}$

03

Calculating Ix

The formula is $I_{x} = {鈭�}_{惟} y^{2} 蚁 (x, y) d A$

Imposing limits

$I_{x} = {鈭�}_{0}^{b} {鈭�}_{0}^{h} y^{2} 蚁 (x, y) d y d x = {鈭�}_{0}^{b} {鈭�}_{0}^{h} y^{2} k x d y d x = k {鈭�}_{0}^{b} x {[\frac{y^{3}}{3}]}_{0}^{h} d x$

$I_{x} = k {鈭�}_{0}^{b} x [\frac{h^{3}}{3}] d x = \frac{1}{3} k h^{3} {鈭�}_{0}^{b} x d x$

$I_{x} = \frac{1}{6} k b^{2} h^{3}$

04

Mass of Lamina

Mass of Lamina is given by $m = {鈭�}_{惟} 蚁 (x, y) d A$

As $蚁 (x, y) = k x$

$鈬� m = {鈭�}_{0}^{b} {鈭�}_{0}^{h} k x d y d x$

$m = {鈭�}_{0}^{b} k x [y]_{0}^{h} d x$

$m = {鈭�}_{0}^{b} k x h d x$

$m = k h {鈭�}_{0}^{b} x d x$

Solving further

$m = k h {[\frac{x^{2}}{2}]}_{0}^{b}$

Mass is $m = \frac{1}{2} k h b^{2}$

05

Radius of Gyration

Radius of gyration is $R_{y} = \sqrt{\frac{I_{y}}{m}} and R_{x} = \sqrt{\frac{I_{x}}{m}}$

$R_{y} = \sqrt{\frac{\frac{1}{4} k h b^{4}}{\frac{1}{2} k h b^{2}}} and R_{x} = \sqrt{\frac{\frac{1}{6} k b^{2} h^{3}}{\frac{1}{2} k h b^{2}}}$

$鈬� R_{y} = \frac{b}{\sqrt{2}} and R_{x} = \frac{h}{\sqrt{3}}$

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Most popular questions from this chapter

Let $惟$ be a lamina in the xy-plane. Suppose $惟$ is composed of n non-overlapping lamin忙 role="math" localid="1650321722341" $惟_{1}, 惟_{2}, 鈥�, 惟_{n} .$ Show that if the masses of these lamin忙 are $m_{1}, m_{2}, 鈥�, m_{n}$ and the centers of masses are $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}), ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2}), 鈥�, ({\overset{炉}{x}}_{n}, {\overset{炉}{y}}_{n}),$ then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{x}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} and \overset{炉}{y} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{y}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} .$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$f (x, y) = \sin x \cos y W h e r e R = {(x, y) | 0 鈮� x 鈮� 蟺, 0 鈮� y 鈮� 蟺}$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$鈭� {鈭�}_{R} \sin (x + 2 y) d A W h e r e R = {(x, y) | 0 鈮� x 鈮� 蟺, 0 鈮� y 鈮� \frac{蟺}{2}}$

Let $f (x, y, z)$ be a continuous function of three variables, let localid="1650352548375" $惟_{y z} = {(y, z) | a 鈮� y 鈮� b a n d h_{1} (y) 鈮� z 鈮� h_{2} (y)}$ be a set of points in the $y z$ -plane, and let localid="1650354983053" $惟 = {(x, y, z) | (y, z) 鈭� 惟_{y z} a n d g_{1} (y, z) 鈮� x 鈮� g_{2} (y, z)}$ be a set of points in $3$ -space. Find an iterated triple integral equal to the triple integral localid="1650353288865" $\underset{惟}{鈭�} f (x, y, z) d V$ . How would your answer change iflocalid="1650352747263" $惟_{y z} = {(y, z) | a 鈮� z 鈮� b a n d h_{1} (z) 鈮� x 鈮� h_{2} (z)}$ ?

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.

$鈭� {鈭�}_{R} x \sin x \cos y d A, w h e r e R = {(x, y) | 鈭� 3 鈮� x 鈮� 2 a n d 鈭� 2 鈮� y 鈮� 2}$

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