Q 51. Let C=x,y|x2+y2鈮�1If the densit... [FREE SOLUTION]

Chapter 13: Q 51. (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 moments of inertia about the x-axis, the y-axis, and the origin. Use these answers to find the radii of gyration of C about the x-axis,
the y-axis, and the origin.

Short Answer

Expert verified

$\begin{array}{rcl} I_{x} & = & \frac{1}{5} 蟺办 \\ I_{y} & = & \frac{1}{5} 蟺办 \\ I_{0} & = & \frac{2}{5} k 蟺 \\ R_{x} & = & \frac{\sqrt{30}}{10} \\ R_{y} & = & \frac{\sqrt{30}}{10} \\ R_{0} & = & \frac{\sqrt{15}}{5} \end{array}$

Step by step solution

Step 1. Given information

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

Step 2. Explanation

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

$\begin{array}{rcl} I_{y} & = & 4 {鈭�}_{0}^{蟺 / 2} {鈭�}_{0}^{1} k 路 r 路 r^{2} \cos^{2} 胃 r d r d 胃 \\ = & 4 k {鈭�}_{0}^{蟺 / 2} {鈭�}_{0}^{1} r^{4} \cos^{2} 胃 d r d 胃 \\ I_{y} & = & 4 k {鈭�}_{0}^{蟺 / 2} {[\frac{r^{5}}{5}]}_{0}^{1} \cos^{2} 胃 d 胃 \\ = & \frac{4}{10} k {鈭�}_{0}^{蟺 / 2} (1 + \cos 2 胃) d 胃 (Use \cos^{2} 胃 = \frac{1 + \cos 2 胃}{2}) \\ I_{y} & = & \frac{4}{10} k {[胃 + \frac{1}{2} \sin 2 胃]}_{0}^{蟺 / 2} \\ = & \frac{1}{5} k 蟺 \end{array}$

$\begin{array}{rcl} I_{x} & = & {鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y^{2} 蚁 (x, y) d y d x (use 蚁 (x, y) = k \sqrt{x^{2} + y^{2}}) \\ = & {鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k y^{2} \sqrt{x^{2} + y^{2}} d y d x \\ I_{x} & = & 4 {鈭�}_{0}^{蟺 / 2} {鈭�}_{0}^{1} k r 路 r^{2} \sin^{2} 胃 r d r d 胃 \\ = & 4 k {鈭�}_{0}^{蟺 / 2} {鈭�}_{0}^{1} r^{4} \sin^{2} 胃 d r d 胃 \\ I_{x} & = & 4 k {鈭�}_{0}^{蟺 / 2} {[\frac{r^{5}}{5}]}_{0}^{1} \sin^{2} 胃 d 胃 \\ = & \frac{4 k}{5} {鈭�}_{0}^{蟺 / 2} \sin^{2} 胃 d 胃 \\ = & \frac{4 k}{10} {鈭�}_{0}^{蟺 / 2} (1 - \cos 2 胃) d 胃 (Use \sin^{2} 胃 = \frac{(1 - \cos 2 胃)}{2}) \\ I_{x} & = & \frac{4 k}{10} {[胃 - \frac{1}{2} \sin 2 胃]}_{0}^{蟺 / 2} \\ = & \frac{1}{5} k 蟺 \end{array}$

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

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

Discrete Mathematics

Mechanics Maths

Calculus

Logic and Functions

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.