**Given:**

The angular speed of the CD is $omega = 22 mathrm{~rad/s}$.

**Required:**

Finding the angular speed in revolutions per minute ($rpm$).

**Calculation:**

In order to evaluate the angular speed in $rpm$, we should convert from radians per second to revolutions per minute.

$$begin{align*}
omega &= 22 mathrm{~rad/s} times dfrac{ 1 mathrm{~rev} }{ 2 pi mathrm{~rad} } times dfrac{ 60 mathrm{~s} }{ 1 mathrm{~min} } \
&= 210.085 mathrm{~rpm}
end{align*} $$

So, The angular speed in revolutions per minute ($rpm$) is $210.085 mathrm{~rpm}$.

$$omega=210.085 mathrm{~rpm}$$

$textbf{Given: }$

The angular position change of the ceiling fan is $Delta theta = 45^{circ}$. The time interval that the fan takes to rotate is $Delta t = 0.75 mathrm{~s}$.

$textbf{Required: }$

Finding the angular speed of the fan in radians per second.

$textbf{Calculation: }$

As we know the average angular velocity $omega_{avg}$ is given by the angular displacement divided by the time interval which the displacement occurs.

$$
begin{align*}
omega_{avg} &= dfrac{ Delta theta }{ Delta t } \
&= dfrac{ 45^{circ} }{ 0.75 mathrm{~s} } times dfrac{ 2 pi }{ 360^{circ} } \
&= dfrac{ pi }{3} mathrm{~rad/s}
end{align*}
$$

So, the average speed of the fan in radians per second is $dfrac{ pi }{3} mathrm{~rad/s}$.

The average speed of the fan in radians per second is $dfrac{ pi }{3} mathrm{~rad/s}$.

$textbf{Given: }$

The angular speed of the airplane propeller is $omega = 260 mathrm{~rad/s}$. The time interval that the airplane propeller takes to rotate is $Delta t = 5 mathrm{~s}$.

$textbf{Required: }$

Finding the angular position change.

$textbf{Calculation: }$

As we know the average angular velocity $omega_{avg}$ is given by the angular displacement divided by the time interval which the displacement occurs.

$$
begin{align*}
omega_{avg} &= dfrac{ Delta theta }{ Delta t } \
&= dfrac{ 45^{circ} }{ 0.75 mathrm{~s} } times dfrac{ 2 pi }{ 360^{circ} } \
&= dfrac{ pi }{3} mathrm{~rad/s}
end{align*}
$$

Rearrange and solve for the angular position change:

$$
begin{align*}
Delta theta &= omega_{avg} times Delta t \
&= 260 mathrm{~rad/s} times 5 mathrm{~s} \
&= 1300 mathrm{~rad} \
end{align*}
$$

In order to convert the angle in degrees instead of the radiansm we use the following relation:

$$
begin{align*}
Delta theta &= omega_{avg} times Delta t \
&= 260 mathrm{~rad/s} times 5 mathrm{~s} \
&= 1300 mathrm{~rad} \
&= 1300 mathrm{~rad} times dfrac{ 360^{circ} }{ 2 pi mathrm{~rad}} \
&= 74484.513^{circ} times left( 206 times 360^{circ} right) \
&= 324^{circ}
end{align*}
$$

So, the airplane propeller rotates with an angle equal to $1300 mathrm{~rad}$ in rad and $324^{circ}$ in degrees after $206$ completed cycles.

The airplane propeller rotates with an angle equal to $1300 mathrm{~rad}$ in rad and $324^{circ}$ in degrees after $206$ completed cycles.

$textbf{Given: }$

The angular speed ofthe baseball is $omega = 38 mathrm{~rad/s}$. The angular displacment of the baseball is $Delta theta = 15^{circ}$.

$textbf{Required: }$

Finding the time interval that the baseball takes.

$textbf{Calculation: }$

As we know the average angular velocity $omega_{avg}$ is given by the angular displacement divided by the time interval which the displacement occurs.

$$
begin{align*}
omega_{avg} &= dfrac{ Delta theta }{ Delta t }
end{align*}
$$

Rearrange and solve for the time interval that the baseball takes:

$$
begin{align*}
Delta t &= dfrac{ Delta theta }{ omega_{avg} } \
&= dfrac{15^{circ} }{ 38 mathrm{~rad/s} } times dfrac{ 2 pi }{ 360^{circ} } \
&= 6.89 times 10^{-3} mathrm{~s}
end{align*}
$$

So, the time interval that the baseball takes is $6.89 times 10^{-3} mathrm{~s}$.

The time interval that the baseball takes is $6.89 times 10^{-3} mathrm{~s}$.