Q 63. - Calculus

Question

Find the specified quantities for the solids described:

The moment of inertia about the z-axis of the region from Exercise $53$, assuming that the density at every point is inversely proportional to the point’s distance from the $z$-axis.

Step-by-Step Solution

Answer

The moment of Inertia is $I_z=\frac{16k\mathrm{\pi }}{3}$ units

Step 1: Given Information

The density at every point is inversely proportional to the point’s distance from the

$z$-axis. The equations are $x^2+y^2-z^2=1$ and $x^2+y^2=5$.

Step 1: Evaluation of limits

The relation between cylindrical and rectangular coordinates are

$r=\sqrt{x^2+y^2}, \tan \theta =\frac{y}{x}, z=z$

and

$x=r\cos \theta , y=r\sin \theta , z=z$

The equation of sphere $x^2+y^2-z^2=1$ in terms of cylindrical coordinates gives

$r^2-z^2=1$

The equation of cylinder $x^2+y^2=5$ in terms of cylindrical coordinates yields

$r^2=5$

Rectangular limits are

$x^2+y^2-z^2=1 \Rightarrow z=\sqrt{r^2-1}$ ($z=0$ is equation of $xy$ plane)

$x^2+y^2=5 \Rightarrow r=\sqrt{5}$

and $z=0 \Rightarrow r=1$

Cylindrical limits are $0\le z\le \sqrt{r^2-1}, 1\le r\le \sqrt{5}, 0\le \theta \le 2\pi$

Step 3: Calculation of Inertia

Moment of inertia is given by $I_z={\iiint }_E\left(x^2+y^2\right)\rho (x,y,z)dxdydz$

Since the density at every point is inversely proportional to the point’s distance from the $z$ axis.

$\rho (x,y,z)=\frac{k}{\left(x^2+y^2\right)}$

Putting limits

$I_z={\int }_{\theta =0}^{2\pi }{\int }_{r=1}^{\sqrt{5}}{\int }_{z=0}^{\sqrt{r^2-1}}krdzdrd\theta$

$={\left.{\int }_{\theta =0}^{2\pi }{\int }_{r=1}^{\sqrt{5}}kr\left(z\right)\right|}_{z=0}^{2=\sqrt{r^2-1}}drd\theta$

$={\int }_{0=0}^{2\pi }{\int }_{r=1}^{\sqrt{5}}kr\left(\sqrt{r^2-1}\right)drd\theta$

$={\int }_{\theta =0}^{2\pi }\left(\frac{k}{2}\right){\int }_{r=1}^{\sqrt{5}}\left(2r\right)\sqrt{r^2-1}drd\theta$

$={\int }_{\theta =0}^{2\pi }\left(\frac{k}{2}\right){\left(\frac{{\left(r^2-1\right)}^{3/2}}{3/2}\right)}^{\sqrt{5}}d\theta$

$=\left(\frac{8k}{3}\right){\int }_{\theta =0}^{2\pi }d\theta$

Hence, $I_z=\frac{16k\mathrm{\pi }}{3}$ units