Question

Fan A ceiling fan with 16 -in. blades rotates at45rpm.
(a) Find the angular speed of the fan in rad/min

(b) Find the linear speed of the tips of the blades in inch/min.

Short answer

(a) $\mathbf{90}\mathit{\pi }\mathbf{\text{radian}}\mathbf{/}\mathbf{\text{min}}$

(b)4521.6 inches/min.

Step-by-step solution

a. Step 1. Given information.

To find:
The angular speed of the fan

Convert the frequency in time and use the formula of angular velocity.

The central for the fan$\theta =2\pi .$.

The given values are:

frequency$\left(f\right)=45\text{rpm}$

a. Step 2. Finding angular speed of fan.

Time$t=\frac{1}{45}$ minutes.
The angular velocity $\omega =\frac{2\pi \cdot \left(45\right)}{1}$

$\omega =90\pi \text{radian}/\text{min}$

Thus, this is the value of angular speed.

b. Step 1. Given information

To find:
The linear speed of the fan (v)

The linear speed of the fan is4521.6inches/min.

To convert the angular speed into linear speed multiply it by the length of the fan's blade.

The given values are:

Length of the blade$=16in$

Angular speed$\omega =90\pi \text{rad}/\text{min}$

b. Step 2. Finding linear speed of the tips of the blades.

Time $t=\frac{1}{45}$minutes
The angular velocity$\omega =\frac{2\pi \cdot \left(45\right)}{1}$

$\omega =90\pi$radian/min

Thus, this is the value of angular speed.