Q 48. Let聽C=x,y|x2+y2鈮�1If the densi... [FREE SOLUTION]

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Chapter 13: Q 48. (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 y-axis, find the moments of inertia about the x- and y-axes. Use these answers to find the radii of gyration of C about the x- and y-axes.

Short Answer

Expert verified

Moment of inertia about y-axis and x-axis are equal to 0,0 respectively.

Step by step solution

Step 1. Given information

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

Step 2. Explanation

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

$\begin{array}{rcl} I_{y} & = & {鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x^{3} d y d x \\ = & {鈭�}_{- 1}^{1} k x^{3} {[y]}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} d x \\ = & 2 k {鈭�}_{- 1}^{1} x^{3} \sqrt{1 - x^{2}} d x \\ = & 0 \end{array}$

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

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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 y-axis, find the moments of inertia about the x- and y-axes. Use these answers to find the radii of gyration of C about the x- and y-axes.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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91影视

LetC=x,y|x2+y2鈮�1If the density at each point in C is proportional to the point鈥檚 distance from the y-axis, find the moments of inertia about the x- and y-axes. Use these answers to find the radii of gyration of C about the x- and y-axes.

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

Decision Maths

Discrete Mathematics

Probability and Statistics

Pure Maths

Geometry

Theoretical and Mathematical Physics

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 y-axis, find the moments of inertia about the x- and y-axes. Use these answers to find the radii of gyration of C about the x- and y-axes.