Q 48. - Calculus

Question

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

If the density at each point in C 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 C about the x- and y-axes.

Step-by-Step Solution

Answer

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

Step 1. Given information

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

Step 2. Explanation

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

$\begin{array}{rcl}{I}_y& =& {\int }_{-1}^1{\int }_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} k{x}^3 dy dx\\ & =& {\int }_{-1}^{1}k{x}^3{\left[y\right]}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} dx\\ & =& 2k{\int }_{-1}^1{x}^3\sqrt{1-x^2 }dx\\ & =& 0\\ & & \end{array}$

$\begin{array}{rcl}{I}_y& =& {\int }_{-1}^1{\int }_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} kx{y}^2 dy dx\\ & =& k{\int }_{-1}^1{\left[\frac{y^3}{3}\right]}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} x dx\\ & =& k{\int }_{-1}^{1}x\left[\frac{2{\left(1-x^2\right)}^{{\displaystyle \frac{3}{2}}}}{3}\right] dx\\ & =& \frac{2}{3}k{\int }_{-1}^1 x{\left(1-x^2\right)}^{\frac{3}{2}}dx\\ & =& 0\\ & & \end{array}$