91Ó°ÊÓ

i) Angular position${\mathrm{θ}}_1=10.0\text{ }\text{rad}$

ii) Angular position${\mathrm{θ}}_2=70.0\text{ r²¹»å}$

iii) Time is$\text{t}=6.00\text{ s}$

iv) Angular velocity at ${\mathrm{θ}}_2$is$15.0\text{ r²¹»å}/\text{s}$

Use the concept of angular kinematics. Use the angular displacement equation to find the angular velocity at the given theta. Using the angular kinematic equation, which relates initial and final angular velocities and acceleration, find angular acceleration. Draw the graph of angular displacement versus t and angular velocity versus t.

Formulae:

$\mathrm{θ}−{\mathrm{θ}}_0=\frac{1}{2}({\mathrm{Ó\neg }}_0+\mathrm{Ó\neg })\text{t}\mathrm{................}\text{(i)}$

$\mathrm{Ó\neg }={\mathrm{Ó\neg }}_0+\mathrm{Î\pm }\text{t}\mathrm{...........................}\text{(ii)}$

${\mathrm{Ó\neg }}^2={\mathrm{Ó\neg }}_0^2+2\mathrm{Î\pm }(\mathrm{θ}−{\mathrm{θ}}_0)\mathrm{.............}\left(\text{iii}\right)$

Angular velocity at :${\mathrm{θ}}_1$

We can use the equation (i) as follows:

$\begin{array}{c}{\mathrm{θ}}_2−{\mathrm{θ}}_1=\frac{1}{2}({\mathrm{Ó\neg }}_1+{\mathrm{Ó\neg }}_2)\text{t}\\ 70−10=\frac{1}{2}({\mathrm{Ó\neg }}_1+15.0)\left(6.00\right)\\ \frac{60×2}{6.00}−15.0={\mathrm{Ó\neg }}_1\\ {\mathrm{Ó\neg }}_1=5.00\text{ }\text{rad}/\text{s}\end{array}$

Hence, the angular velocity is.$5.00\text{ }\text{rad}/\text{s}$

Angular acceleration:

We use the equation (ii) to find angular acceleration as follows:

$\begin{array}{c}15.0=5.00+\mathrm{Î\pm }\left(6.00\right)\\ \mathrm{Î\pm }=\frac{15.0−5.00}{6.00}\\ =\frac{1.66\text{ r²¹»å}}{{\text{s}}^2}\\ =1.67\text{ r²¹»å}/{\text{s}}^2\end{array}$

Hence, the angular acceleration is$1.67\text{ r²¹»å}/{\text{s}}^2$.

Angular position when disk was at rest:

We have the equation (iii) as follows:

$\begin{array}{c}{\mathrm{Ó\neg }}_1^2={\mathrm{Ó\neg }}_0^2+2\mathrm{Î\pm }({\mathrm{θ}}_1−{\mathrm{θ}}_0)\\ {5.0}^2=0^2+2\left(1.67\right)(10.0−{\mathrm{θ}}_0)\\ {\mathrm{θ}}_0=−\frac{25}{2×1.67}+10.0\\ =2.50\text{ r²¹»å}\\ \end{array}$

Hence, the angular acceleration is.$2.50\text{ r²¹»å}$

Graph of$\mathrm{θ}$ Vs t and$\mathrm{Ó\neg }$ Vs t:

We can draw the given graphs as follows: