91Ó°ÊÓ

The initial angular speed is${\mathrm{Ó\neg }}_{0_z}=12rad/s$ .

The final angular speed is${\mathrm{Ó\neg }}_z=16rad/s$ .

The angular displacement is$\mathrm{θ}=7rad$ .

The equation (9.7) is given by,

${\mathit{Ó\neg }}_{\mathbf{z}}\mathbf{=}{\mathit{Ó\neg }}_{{\mathbf{0}}_{\mathbf{z}}}\mathbf{+}{\mathit{Î\pm }}_{\mathbf{z}}\mathit{t}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

Here,${\mathit{Î\pm }}_{\mathbf{z}}$ is the angular acceleration,${\mathit{Ó\neg }}_{{\mathbf{0}}_{\mathbf{z}}}$ is the angular velocity of the body at time 0, t is the time.

The equation (9.11) is given by,

$\mathit{θ}\mathbf{=}{\mathit{θ}}_{\mathbf{0}}\mathbf{+}{\mathit{Ó\neg }}_{{\mathbf{0}}_{\mathbf{z}}}\mathit{t}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{Î\pm }}_{\mathbf{z}}{\mathit{t}}^{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(2\right)$

Here,${\mathit{θ}}_{\mathbf{0}}$ is the Angular position of the body at time 0.

Rewrite the equation (1) as follows.

$$\begin{gathered} {\mathrm{Ó\neg }}_z={\mathrm{Ó\neg }}_{0_z}+{\mathrm{Î\pm }}_{z}t \\ {\mathrm{Î\pm }}_{z}t={\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z} \\ t=\frac{{\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}}{{\mathrm{Î\pm }}_z} \end{gathered}$$

Substitute$t=\frac{{\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}}{{\mathrm{Î\pm }}_z}$ in equation (2).

$$\begin{gathered} \mathrm{θ}={\mathrm{θ}}_0+{\mathrm{Ó\neg }}_{0_z}\left(\frac{{\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}}{{\mathrm{Î\pm }}_z}\right)+\frac{1}{2}{\mathrm{Î\pm }}_z{\left(\frac{{\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}}{{\mathrm{Î\pm }}_z}\right)}^2 \\ =\frac{1}{{\mathrm{Î\pm }}_z}\left[{\mathrm{Ó\neg }}_{0_z}\left({\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}\right)+\frac{1}{2}{\left({\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}\right)}^2\right] \\ =\frac{\left({\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}\right)}{{\mathrm{Î\pm }}_z}\left[{\mathrm{Ó\neg }}_{0_z}+\frac{1}{2}\left({\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}\right)\right] \\ =\frac{\left({\mathrm{Ó\neg }}_z-{\mathrm{Ó\neg }}_{0_z}\right)\left({\mathrm{Ó\neg }}_z+{\mathrm{Ó\neg }}_{0_z}\right)}{2{\mathrm{Î\pm }}_z} \end{gathered}$$

Simplify further.

$$\begin{gathered} \mathrm{θ}=\frac{\left({\mathrm{Ó\neg }}_z^2-{\mathrm{Ó\neg }}_{0_2}^2\right)}{2{\mathrm{Î\pm }}_z} \\ {\mathrm{Ó\neg }}_z^2-{\mathrm{Ó\neg }}_{0_2}^2=2{\mathrm{Î\pm }}_z\mathrm{θ} \\ {\mathrm{Ó\neg }}_z^2={\mathrm{Ó\neg }}_{0_2}^2+2{\mathrm{Î\pm }}_z\mathrm{θ} \end{gathered}$$

Therefore, the required expression is${\mathrm{Ó\neg }}_z^2={\mathrm{Ó\neg }}_{0_2}^2+2{\mathrm{Î\pm }}_z\mathrm{θ}$ .

Substitute ${\mathrm{Ó\neg }}_{0_z}^2=12rad/s,\mathrm{θ}=7rad,and{\mathrm{Ó\neg }}_z=16rad/s$and in equation${\mathrm{Ó\neg }}_z^2={\mathrm{Ó\neg }}_{0_z}^2+2{\mathrm{Î\pm }}_z\mathrm{θ}$ .

$$\begin{gathered} {\left(16rad/s\right)}^2={\left(12rad/s\right)}^2+2{\mathrm{Î\pm }}_z\left(7rad\right) \\ {\mathrm{Î\pm }}_z=\frac{{\left(16rad/s\right)}^2-{\left(12rad/s\right)}^2}{2\left(7rad\right)} \\ =8rad/s^2 \end{gathered}$$

Therefore, the angular acceleration is$8rad/s^2$ .