Question

A bicycle with $24$-inch diameter wheels is traveling at $\text{15 miles/h}$ Find the angular speed of the wheels in $\text{rad/min}$. How many revolutions per minute do the wheels make?

Answer 1

Step- 1: Linear velicity of wheeel:

The diameter of the wheel is $24$ inches, so the radius of the wheel is:

$\text{r}=\frac{24}{2}=12$ inches.

The velocity of the bicycle is $15$ miles per hour.

Now, convert velocity into inches per minutes:

$\begin{array}{rcl}\text{velocity}\left(\text{v}\right)& =& 15\text{mph}\\ & =& \frac{15\times 63360\text{inches}}{1\text{hours}}\left(\because 1\text{miles}=63360\text{inches}\right)\\ & =& \frac{15\times 63360\text{inches}}{60\text{mintues}}\left(\because 1\text{hour}=60\text{mintues}\right)\\ & =& 15840\text{inches per minutes}\end{array}$

Step- 2:Find the angular speed of the wheels and number of revolutions per minute:

Now, the angular speed of the wheel can be calculate as in below:

$\begin{array}{rcl}\text{Angular speed}\left(\text{w}\right)& =& \frac{\text{v}}{\text{r}}\\ & =& \frac{15840}{12}\\ & =& 1320\text{rad/min}\end{array}$

Now, find the number of revolution per minute.

The number of revolutions are calculated in below:

$$\begin{gathered} =1320\times 2\mathrm{\pi } \\ =2640\mathrm{\pi } \end{gathered}$$

Hence, the angular speed is $\mathbf{1320}\mathbf{}\mathbf{\text{rad/min}}$ and the number of revolutions are $\mathbf{2640}\mathbf{\pi }\mathbf{}\mathbf{\text{per minute}}$.