Q 40. Let R聽be rectangular region wit... [FREE SOLUTION]

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Chapter 13: Q 40. (page 1059)

Let $R$ be rectangular region with vertices role="math" localid="1654011069759" $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the point鈥檚 distance from the $y$ -axis, find the center of mass of $R$ .

Short Answer

Expert verified

The center of mass is $\overset{炉}{x} = \frac{2}{3} b, \overset{炉}{y} = \frac{h}{2}$

Step by step solution

Given Information

It is given that vertices of rectangular region are $\overset{炉}{x} = \frac{2}{3} b, \overset{炉}{y} = \frac{h}{2}$

Also $Density 蚁 (x, y) = k x$

Calculating x-

The formula is $\overset{炉}{x} = \frac{{鈭�}_{惟} x 蚁 (x, y) d A}{{鈭�}_{惟} 蚁 (x, y) d A}$

As $蚁 (x, y) = k x$

$\overset{炉}{x} = \frac{{鈭�}_{0}^{b} {鈭�}_{0}^{h} x k x d y d x}{{鈭�}_{0}^{b} {鈭�}_{0}^{h} k x d y d x} = \frac{{鈭�}_{0}^{b} {鈭�}_{0}^{h} k x^{2} d y d x}{{鈭�}_{0}^{b} {鈭�}_{0}^{h} k x d y d x}$

Solving further

$\overset{炉}{x} = \frac{{鈭�}_{0}^{b} k x^{2} [y]_{0}^{h} d x}{{鈭�}_{0}^{b} k x [y]_{0}^{h} d x} = \frac{{鈭�}_{0}^{b} k h x^{2} d x}{{鈭�}_{0}^{b} k h x d x} = \frac{k h {鈭�}_{0}^{b} x^{2} d x}{k h {鈭�}_{0}^{b} x d x}$

$鈬� \overset{炉}{x} = \frac{{[\frac{x^{3}}{3}]}_{0}^{b}}{{[\frac{x^{2}}{2}]}_{0}^{b}} = = \frac{[\frac{b^{3}}{3}]}{[\frac{b^{2}}{2}]}$

Hence, $\overset{炉}{x} = \frac{2}{3} b$

Calculating y-

Similarly the formula is $\overset{炉}{y} = \frac{{鈭�}_{惟} y 蚁 (x, y) d A}{{鈭�}_{惟} 蚁 (x, y) d A}$

$\overset{炉}{y} = \frac{{鈭�}_{0}^{b} {鈭�}_{0}^{h} y k x d y d x}{{鈭�}_{0}^{b} {鈭�}_{0}^{h} k x d y d x} = \frac{{鈭�}_{0}^{b} k x {[\frac{y^{2}}{2}]}_{0}^{h} d x}{{鈭�}_{0}^{b} k x [y]_{0}^{h} d x} = \frac{{鈭�}_{0}^{b} k x [\frac{h^{2}}{2}] d x}{{鈭�}_{0}^{b} k h x d x}$

$\overset{炉}{y} = \frac{k h^{2} {鈭�}_{0}^{b} x d x}{2 k h {鈭�}_{0}^{b} x d x} = = \frac{h {[\frac{x^{2}}{2}]}_{0}^{b}}{2 {[\frac{x^{2}}{2}]}_{0}^{b}} = \frac{h [\frac{b^{2}}{2}]}{2 [\frac{b^{2}}{2}]}$

Hence, $y = \frac{h}{2}$

Center of Mass is $\overset{炉}{x} = \frac{2}{3} b, \overset{炉}{y} = \frac{h}{2}$

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91影视

Let $R$ be rectangular region with vertices role="math" localid="1654011069759" $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the point鈥檚 distance from the $y$ -axis, find the center of mass of $R$ .

Short Answer

Step by step solution

Given Information

Calculating x-

Calculating y-

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91影视

Let Rbe rectangular region with vertices role="math" localid="1654011069759" (0,0),(b,0),(0,h),and(b,h)If the density at each point in Ris proportional to the point鈥檚 distance from the y-axis, find the center of mass of R.

Short Answer

Step by step solution

Given Information

Calculating x-

Calculating y-

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Logic and Functions

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

Let $R$ be rectangular region with vertices role="math" localid="1654011069759" $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the point鈥檚 distance from the $y$ -axis, find the center of mass of $R$ .