Q 43. Let R聽be rectangle with coordin... [FREE SOLUTION]

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Chapter 13: Q 43. (page 1039)

Let $R$ be rectangle with coordinates $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the square of 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 $\overset{炉}{x} = \frac{3}{4} b, \overset{炉}{y} = \frac{h}{2}$ .

Step by step solution

Given Information

The vertices of rectangular region is $(0, 0), (b, 0), (0, h) and (b, h)$ .

$蚁 (x, y) = k x^{2}$

Calculation of x-

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

Also $蚁 (x, y) = k x^{2}$

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

$\overset{炉}{x} = \frac{{鈭�}_{0}^{b} {鈭�}_{0}^{h} k x^{3} d y d x}{{鈭�}_{0}^{b} {鈭�}_{0}^{h} k x^{2} d y d x}$

$\overset{炉}{x} = \frac{{鈭�}_{0}^{b} k x^{3} [y]_{0}^{h} d x}{{鈭�}_{0}^{b} k x^{2} [y]_{0}^{h} d x} = \frac{{鈭�}_{0}^{b} k h x^{3} d x}{{鈭�}_{0}^{b} k h x^{2} d x} = \frac{k h {鈭�}_{0}^{b} x^{3} d x}{k h {鈭�}_{0}^{b} x^{2} d x}$

$\overset{炉}{x} = \frac{{[\frac{x^{4}}{4}]}_{0}^{b}}{{[\frac{x^{3}}{3}]}_{0}^{b}} = \frac{[\frac{b^{4}}{4}]}{[\frac{b^{3}}{3}]}$

$\overset{炉}{x} = \frac{3}{4} b$

Calculation of y-

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

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

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

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

$\overset{炉}{y} = \frac{h}{2}$

Hence, $\overset{炉}{x} = \frac{3}{4} b, \overset{炉}{y} = \frac{h}{2}$

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91影视

Let $R$ be rectangle with coordinates $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the square of the point鈥檚 distance from the $y$ -axis, find the center of mass of $R$ .

Short Answer

Step by step solution

Given Information

Calculation of x-

Calculation of y-

One App. One Place for Learning.

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91影视

Let Rbe rectangle with coordinates (0,0),(b,0),(0,h),and(b,h)If the density at each point in Ris proportional to the square of the point鈥檚 distance from the y-axis, find the center of mass of R.

Short Answer

Step by step solution

Given Information

Calculation of x-

Calculation of y-

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

Applied Mathematics

Statistics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Let $R$ be rectangle with coordinates $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the square of the point鈥檚 distance from the $y$ -axis, find the center of mass of $R$ .