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Chapter 13: Q 30. (page 1039)

Let $T$ be rectangular region with vertices $(0, 0), (1, 1), and (1, - 1)$
If the density at each point in $T$ is proportional to the point鈥檚 distance from the $x$ -axis, find the moments of inertia about the $x$ - and $y$ -axes. Use these answers to find the radii of gyration of $T$ about the $x$ - and $y$ -axes.

Short Answer

Expert verified

Moment of Inertia is $I_{y} = \frac{k}{5}$ and $I_{x} = \frac{k}{10}$

Mass of region is $m = \frac{k}{3}$

Radius of gyration is $R_{x} = \sqrt{\frac{3}{10}}, R_{y} = \sqrt{\frac{3}{5}}$

Step by step solution

01

Given Information

Vertices of triangular region are $(0, 0), (1, 1), and (1, - 1)$

Densityrole="math" localid="1653994140071" $蚁 (x, y) = k y$

02

MOI about y axis

$I_{y} = {鈭�}_{惟} x^{2} 蚁 (x, y) d A$

Putting limits $I_{y} = {鈭�}_{0}^{1} {鈭�}_{- x}^{x} x^{2} k y d y d x [蚁 (x, y) = k y]$

As per figure

$I_{y} = 2 {鈭�}_{0}^{1} {鈭�}_{0}^{x} x^{2} k y d y d x$

$I_{y} = 2 k {鈭�}_{0}^{1} {[\frac{y^{2}}{2}]}_{0}^{x} x^{2} d y$

$I_{y} = k {鈭�}_{0}^{1} x^{4} d y$

$I_{y} = k {[\frac{x^{5}}{5}]}_{0}^{1}$

$I_{y} = \frac{k}{5}$

03

MOI about x axis

It is expressed as:

$I_{x} = {鈭�}_{惟} y^{2} 蚁 (x, y) d A$

$I_{x} = {鈭�}_{0}^{1} {鈭�}_{- x}^{x} y^{2} k y d y d x [蚁 (x, y) = k y]$

As region is symmetric about $x$ axis

$I_{x} = 2 {鈭�}_{0}^{1} {鈭�}_{0}^{x} y^{2} k y d y d x$

$I_{x} = 2 k {鈭�}_{0}^{1} {鈭�}_{0}^{x} y^{3} d y d x$

$I_{x} = \frac{k}{2} {鈭�}_{0}^{1} x^{4} d x$

$I_{x} = \frac{k}{2} {[\frac{x^{5}}{5}]}_{0}^{1}$

$I_{x} = \frac{k}{10}$

04

Mass of triangular region

It is given by $m = 2 {鈭�}_{0}^{1} {鈭�}_{0}^{x} k y d y d x [m = {鈭�}_{惟} 蚁 (x, y) d A]$

$m = 2 k {鈭�}_{0}^{1} {[\frac{y^{2}}{2}]}_{0}^{x} d x$

$m = k {鈭�}_{0}^{1} x^{2} d x$

$m = k {[\frac{x^{3}}{3}]}_{0}^{1}$

$m = \frac{k}{3}$

05

Radius of gyration

It is given by

$R_{x} = \sqrt{\frac{I_{x}}{m}} and R_{y} = \sqrt{\frac{I_{y}}{m}}$

$R_{x} = \sqrt{\frac{k / 10}{k / 3}} and R_{y} = \sqrt{\frac{k / 5}{k / 3}}$

$R_{x} = \sqrt{\frac{3}{10}}, R_{y} = \sqrt{\frac{3}{5}}$

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Most popular questions from this chapter

In Exercises 45鈥�52, rewrite the indicated integral with the specified order of integration.

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$f (x, y) = x y W h e r e R = {(x, y) | - 2 鈮� x 鈮� 3, - 1 鈮� y 鈮� 5}$

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