Question

For the solid in Problem 7, Find${\mathit{I}}_{\mathit{Z}}$ I Mif$\mathit{\alpha }\mathbf{=}\mathit{\tau }\mathit{\tau }\mathbf{/}\mathbf{3}$and the density is constant.

Short answer

The required value of $\frac{I_z}{M}is\frac{a^2}{4}$.

Step-by-step solution

Given information

The value of $\alpha$ is $\frac{\tau \tau }{3}$ and the density is constant.

Concept of spherical coordinates

The spherical coordinatesare related to Cartesian coordinates (x,y,z) by:

$$\begin{gathered} \mathit{r}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{\theta }} \\ \mathbf{}\mathbf{}\mathbf{=}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\frac{y}{x}\right)\mathit{\varphi } \\ \mathbf{}\mathbf{}\mathbf{=}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\left(\frac{z}{r}\right) \end{gathered}$$

The cylindrical coordinates are related to Cartesian coordinates (x,y,z)by:

$$\begin{gathered} \mathit{r}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{\theta }} \\ \mathbf{}\mathbf{=}\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\left(\frac{y}{x}\right)\mathit{z} \\ \mathbf{}\mathbf{=}\mathit{z} \end{gathered}$$

The moment of inertia

The moment of inertia is calculated as:

$$\begin{gathered} V=\frac{2\tau \tau }{3}{a}^3\left(1-\cos \alpha \right) \\ =\frac{\tau \tau a^3}{3} \end{gathered}$$

Calculate further as follows:

$$\begin{gathered} I_z=\int \rho \left(x^2+y^2\right)dV \\ =\rho \left(r^2{\sin }^2\theta \right)dV \\ =\rho {\int }_0^{2\tau \tau }d\varphi {\int }_0^{\tau \tau /3}{\sin }^3\theta d\theta {\int }_0^{\tau \tau /3}{r}^{4}dr \\ =-\frac{2\tau \tau \rho a^5}{5}{\int }_0^{\tau \tau }{\sin }^3\theta d\theta \end{gathered}$$

The above integral can be written as follows:

$$\begin{gathered} I_z=-\frac{2\tau \tau \rho a^5}{5}{\int }_0^{\pi /3}\left(1-{\cos }^2\theta \right)d\left(\cos \theta \right) \\ =-\frac{2\tau \tau \rho a^5}{5}{\left(\cos \theta -\frac{{\cos }^3\theta }{3}\right)}_0^{\pi /3} \\ =\frac{\pi }{12}\rho a^5 \end{gathered}$$

Substitute localid="1659146809219" $\frac{\tau \tau }{12}\rho a^5$ for $I_z$ and $\rho \frac{\tau \tau a^3}{3}$ for M in $\frac{I_z}{M}$.

$\frac{I_z}{M}=\frac{a^2}{4}$

Therefore, the required value of $\frac{I_z}{M}$is $\frac{a^2}{4}$.