Q 55. - Calculus

Question

Let $S=\left\{\left(x,y\right)|x^2+y^2\le 1,x\ge 0\right\}$

If the density at each point in S is proportional to the point’s 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 S about the x- and y-axes.

Step-by-Step Solution

Answer

$\begin{array}{rcl}{I}_y& =& \frac{4}{15}k\\ I_x& =& \frac{2}{15}k\\ R_y& =& \frac{\sqrt{10}}{5}\\ R_x& =& \frac{\sqrt{5}}{5}\end{array}$

Step 1. Given information

$S=\left\{\left(x,y\right)|x^2+y^2\le 1,x\ge 0\right\}$

Step 2. Explanation

$S=\left\{\left(x,y\right)|x^2+y^2\le 1,x\ge 0\right\}$

$\begin{array}{rcl}{I}_y& =& {\int }_0^1{\int }_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}{x}^{2}kxdydx\\ & =& {\int }_0^1{\int }_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}k{x}^{3}dydx\\ & =& {\int }_0^{1}k{x}^3[y{]}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}dx\\ & =& 2k{\int }_0^1{x}^3\sqrt{1-x^2}dx\\ & =& 2k\left(\frac{2}{15}\right)\\ & =& \frac{4}{15}k\\ I_x& =& {\int }_0^1{\int }_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}kx{y}^{2}dydx\\ & =& k{\int }_0^1{\left[\frac{y^3}{3}\right]}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}xdx\\ & =& \frac{2}{3}k{\int }_0^{1}x{\left(1-x^2\right)}^{3/2}dx\\ & =& \frac{2}{3}k{\left[-\frac{1}{5}{\left(1-x^2\right)}^{5/2}\right]}_0^1\\ & =& \frac{2}{15}k\\ m& =& 2k{\int }_0^1[y{]}_0^{\sqrt{1-x^2}}xdx\\ & =& 2k{\int }_0^{1}x\sqrt{1-x^2}dx\\ & =& 2k{\left[-\frac{1}{3}{\left(1-x^2\right)}^{3/2}\right]}_0^1\\ & =& \frac{2}{3}k\end{array}$

$$\begin{gathered} R_y=\sqrt{\frac{\frac{4}{15}k}{\frac{2}{3}k}}\text{ and }{R}_x=\sqrt{\frac{\frac{2}{\frac{15}{2}k}}{\frac{2}{3}k}} \\ {R}_y=\sqrt{\frac{2}{5}}\text{ and }{R}_x=\sqrt{\frac{1}{5}} \end{gathered}$$

$\begin{array}{rcl}{I}_y& =& \frac{4}{15}k\\ I_x& =& \frac{2}{15}k\\ R_y& =& \frac{\sqrt{10}}{5}\\ R_x& =& \frac{\sqrt{5}}{5}\end{array}$