Chapter 6.1, Problem 107AYU

To determine

To find:the angular speed of the $8\text{inches}$ pulley to rotate at revolution per minute.

Answer to Problem 107AYU

The revolution per minuteof $8\text{inches}$ pulley is $\omega =\frac{3}{4}\text{revolution per minute}$

Explanation of Solution

Given information:

The giventhe radiusoftwo pulleys are $2\text{inches}$ and $8\text{inches}$ . The $2\text{inches}$ pulley is caused torotateat $3$ revolution per minute.

Concept used:

The object travels around the circles;

That an object moves around a circles 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 its defined as.

  $v=\frac{s}{t}$

The relation between linear speed and angular speed;

  $v=r\omega$

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

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 angular speed of $2\text{inches}$ pulley is equal to rotate at $3$ revolution per minute

  ${\omega }_1=3\text{revolution}\text{per}\text{minute}$

The linear speed of $2\text{inches}$ pulley is calculated as,

  $\begin{array}{c}{v}_1=r_1{\omega }_1\\ v_1=2\times 3\\ v_1=6\end{array}$

The linear speed of $8\text{inches}$ pulley is calculated as,

  $\begin{array}{l}{v}_2=r_2\omega \\ v_2=8\omega \end{array}$

The linear speed of two pulleys connect by a belt are equal.

  $v_1=v_2\mathrm{.....}(1.)$

Substitute the values to calculate the angular speed of two pulleys in this equation (1.)

  $\begin{array}{l}{v}_1=v_2\\ 6=8\omega \\ \omega =\frac{3}{4}\end{array}$

Therefore, the revolution per minute of $8\text{inches}$ pulley is $\omega =\frac{3}{4}\text{revolution per minute}$ .