Chapter 13: Q 51. (page 1039)

Let $C = \{(x, y) | x^{2} + y^{2} \leq 1\}$
If the density at each point in C is proportional to the point’s 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

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

Step by step solution

Step 1. Given information

$C = \{(x, y) | x^{2} + y^{2} \leq 1\}$

Step 2. Explanation

$C = \{(x, y) | x^{2} + y^{2} \leq 1\}$

$\begin{array}{rcl} I_{y} & = & 4 \int_{0}^{π / 2} \int_{0}^{1} k \cdot r \cdot r^{2} \cos^{2} θ r d r d θ \\ = & 4 k \int_{0}^{π / 2} \int_{0}^{1} r^{4} \cos^{2} θ d r d θ \\ I_{y} & = & 4 k \int_{0}^{π / 2} {[\frac{r^{5}}{5}]}_{0}^{1} \cos^{2} θ d θ \\ = & \frac{4}{10} k \int_{0}^{π / 2} (1 + \cos 2 θ) d θ (Use \cos^{2} θ = \frac{1 + \cos 2 θ}{2}) \\ I_{y} & = & \frac{4}{10} k {[θ + \frac{1}{2} \sin 2 θ]}_{0}^{π / 2} \\ = & \frac{1}{5} k π \end{array}$

$\begin{array}{rcl} I_{x} & = & \int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y^{2} ρ (x, y) d y d x (use ρ (x, y) = k \sqrt{x^{2} + y^{2}}) \\ = & \int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k y^{2} \sqrt{x^{2} + y^{2}} d y d x \\ I_{x} & = & 4 \int_{0}^{π / 2} \int_{0}^{1} k r \cdot r^{2} \sin^{2} θ r d r d θ \\ = & 4 k \int_{0}^{π / 2} \int_{0}^{1} r^{4} \sin^{2} θ d r d θ \\ I_{x} & = & 4 k \int_{0}^{π / 2} {[\frac{r^{5}}{5}]}_{0}^{1} \sin^{2} θ d θ \\ = & \frac{4 k}{5} \int_{0}^{π / 2} \sin^{2} θ d θ \\ = & \frac{4 k}{10} \int_{0}^{π / 2} (1 - \cos 2 θ) d θ (Use \sin^{2} θ = \frac{(1 - \cos 2 θ)}{2}) \\ I_{x} & = & \frac{4 k}{10} {[θ - \frac{1}{2} \sin 2 θ]}_{0}^{π / 2} \\ = & \frac{1}{5} k π \end{array}$

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Let C=x,y|x2+y2≤1If the density at each point in C is proportional to the point’s 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

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Most popular questions from this chapter

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Applied Mathematics

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