Q 50. Let C=x,y|x2+y2鈮�1If the densit... [FREE SOLUTION]

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Chapter 13: Q 50. (page 1039)

Let $C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$
If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the center of mass of C.

Short Answer

Expert verified

Answer is (0,0)

Step by step solution

Step 1. Given information

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

Step 2. Explanation

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

$\begin{array}{rcl} \overset{炉}{x} & = & \frac{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} x k \sqrt{x^{2} + y^{2}} d y d x}{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k \sqrt{x^{2} + y^{2}} d y d x} \\ \overset{炉}{x} & = & \frac{2 {鈭�}_{- 1}^{1} {[\frac{1}{2} y \sqrt{x^{2} + y^{2}} + \frac{1}{2} x^{2} \ln (y + \sqrt{x^{2} + y^{2}})]}_{0}^{\sqrt{1 - x^{2}}} x d x}{2 {鈭�}_{- 1}^{1} {[\frac{1}{2} y \sqrt{x^{2} + y^{2}} + \frac{1}{2} x^{2} \ln (y + \sqrt{x^{2} + y^{2}})]}_{0}^{\sqrt{1 - x^{2}}} d x} \\ \overset{炉}{x} & = & \frac{2 {鈭�}_{- 1}^{1} [\frac{1}{2} \sqrt{1 - x^{2}} + \frac{1}{2} x^{2} \ln (\frac{1 + \sqrt{1 - x^{2}}}{x})] x d x}{2 {鈭�}_{- 1}^{1} [\frac{1}{2} \sqrt{1 - x^{2}} + \frac{1}{2} x^{2} \ln (\frac{1 + \sqrt{1 - x^{2}}}{x})] d x} \\ = & 0 \\ \overset{炉}{y} & = & \frac{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y k \sqrt{x^{2} + y^{2}} d y d x}{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k \sqrt{x^{2} + y^{2}} d y d x} \\ = & \frac{0}{\frac{3}{2} 蟺} \\ = & 0 \end{array}$

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91影视

Let $C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$
If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the center of mass of C.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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91影视

Let C=x,y|x2+y2鈮�1If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the center of mass of C.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Decision Maths

Discrete Mathematics

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Let $C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$
If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the center of mass of C.