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’s distance from the $y$ -axis, find the center of mass of $R$ .

Short Answer

Expert verified

The center of mass is $\bar{x} = \frac{3}{4} b, \bar{y} = \frac{h}{2}$ .

Step by step solution

Given Information

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

$ρ (x, y) = k x^{2}$

Calculation of x-

The formula is $\bar{x} = \frac{\iint_{Ω} x ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A} and \bar{y} = \frac{\iint_{Ω} y ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A}$

Also $ρ (x, y) = k x^{2}$

$\bar{x} = \frac{\int_{0}^{b} \int_{0}^{h} x k x^{2} d y d x}{\int_{0}^{b} \int_{0}^{h} k x^{2} d y d x}$

$\bar{x} = \frac{\int_{0}^{b} \int_{0}^{h} k x^{3} d y d x}{\int_{0}^{b} \int_{0}^{h} k x^{2} d y d x}$

$\bar{x} = \frac{\int_{0}^{b} k x^{3} [y]_{0}^{h} d x}{\int_{0}^{b} k x^{2} [y]_{0}^{h} d x} = \frac{\int_{0}^{b} k h x^{3} d x}{\int_{0}^{b} k h x^{2} d x} = \frac{k h \int_{0}^{b} x^{3} d x}{k h \int_{0}^{b} x^{2} d x}$

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

$\bar{x} = \frac{3}{4} b$

Calculation of y-

The formula is $\bar{y} = \frac{\iint_{Ω} y ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A}$

$\bar{y} = \frac{\int_{0}^{b} \int_{0}^{h} y k x^{2} d y d x}{\int_{0}^{b} \int_{0}^{h} k x^{2} d y d x}$

$\bar{y} = \frac{\int_{0}^{b} k x^{2} {[\frac{y^{2}}{2}]}_{0}^{h} d x}{\int_{0}^{b} k x^{2} [y]_{0}^{h} d x} = \frac{\int_{0}^{b} k x^{2} [\frac{h^{2}}{2}] d x}{\int_{0}^{b} k h x^{2} d x} = \frac{k h^{2} \int_{0}^{b} x^{2} d x}{2 k h \int_{0}^{b} x^{2} d x}$

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

$\bar{y} = \frac{h}{2}$

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

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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’s 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-

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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’s 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

Geometry

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

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’s distance from the $y$ -axis, find the center of mass of $R$ .