Chapter 6.1, Problem 111AYU

To determine

To find: The angular speed of the wheel in revolutions per minute.

Answer to Problem 111AYU

The angular speed of the wheel in revolutions per minute is $31.47\text{rev}/\min$ .

Explanation of Solution

Given information:

The speed of the cable is $9.55\text{miles}/\text{h}$ and the diameter of the wheel is $8.5\text{ft}$ .

Concept used:

An object moves around a circular path of radius $r$ at a constant speed .If $s$ is the distance travelled in time $t$ around this circles, then the linear speed $v$ of the object is defined as,

  $v=\frac{s}{t}$

The angular speed $\omega$ of this object is the angle $\theta$ (measures in radians ) swept out divided by the elapsed time $t$ that is

  $\omega =\frac{\theta }{t}$

The relation between linear speed and angular speed;

  $v=r\omega$

Here, $\omega$ is measured in radius per unit time.

The angular speed of the wheel is calculated as,

  $\begin{array}{l}v=r\omega \\ \omega =\frac{v}{r}\\ \omega =\frac{9.55\text{miles}/\text{h}\left|\frac{1\text{miles}/\text{min}}{60\text{miles}/\text{h}}\right|}{\frac{8.5\text{ft}}{2}\left|\frac{1\text{mile}}{5280\text{ft}}\right|}\\ \omega =197.74\text{rad}/\min \left|\frac{1\text{rev}/\min }{2\pi \text{rad}/\min }\right|\end{array}$

Simplify further,

  $\omega =31.47\text{rev}/\min$

Thus, the angular speed of the wheel in revolutions per minute is $31.47\text{rev}/\min$ .