Q. 60 - Calculus

Question

In Exercises 57–60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.

Assuming that the density at each point in R is proportional to the distance of the point from the xy-plane, find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Step-by-Step Solution

Answer

The moment of inertia about the x-axis is $I_x=104k$ and the radius of gyration about the x-axis is $R_x\approx 2.0816.$

Step 1. Given Information.

The given rectangular solid is defined by $\mathfrak{R}=\left\{\right(x,y,z)\mid 0\le x\le 4, 0\le y\le 3, 0\le z\le 2\}.$

Step 2. Find the moment of inertia about the x- axis.

It is given that the density at each point in R is proportional to the distance of the point from the xy-plane, so $\rho (x,y,z)=kz.$

Now, the moment of the inertia about the x-axis is,  

$$\begin{gathered} I_x={\iiint }_T(y^2+z^2)\rho (x,y,z)dV \\ {I}_x={\int }_{x=0}^4{\int }_{y=0}^3{\int }_{z=0}^2(y^2+z^2)\left(kz\right)dxdydz \\ {I}_x=={k{\int }_{x=0}^4{\int }_{y=0}^3\overline{)y^2\left(\frac{z^2}{2}\right)}}_{z=0}^{2}dxdy \\ {I}_x==k{\int }_{x=0}^4{\int }_{y=0}^{3}2y^2+\left(\frac{8}{3}\right)dxdy \end{gathered}$$

Integrate with respect to 'y'

$$\begin{gathered} I_x=k{{\int }_{x=0}^4\overline{)\left(\frac{{\displaystyle 2y^3}}{{\displaystyle 3}}\right)}}_{y=0}^{3}dx+{{\int }_{x=0}^4\left(\frac{{\displaystyle 8y}}{{\displaystyle 3}}\right)}_{y=0}^{3}dx \\ {I}_x=k{\int }_{x=0}^4\left(18+8\right)dx \\ {I}_x=104k \end{gathered}$$

Step 3. Find the radius of the gyration.

To find the radius of gyration about the x-axis, we have to find the mass of the rectangular solid:

$$\begin{gathered} M={\iiint }_R\rho (x,y,z)dxdydz \\ M={\int }_0^4{\int }_0^3{\int }_0^2\left(kz\right)dxdydz \\ M==k\left(4\right)\left(3\right){\overline{)\left(\frac{z^2}{2}\right)}}_{z=0}^2 \\ M=24k \end{gathered}$$

So, the radius of gyration is:

$$\begin{gathered} R_x=\sqrt{\frac{I_x}{m}} \\ {R}_x=\sqrt{\frac{104k}{24k}} \\ {R}_x\approx 2.0816 \end{gathered}$$