Chapter 8.6, Problem 46E

To determine

To select: the angular velocity of smaller pulley from the given options.

Answer to Problem 46E

D.

Explanation of Solution

Given information:

The diagram of the pulley is shown below.

  

Let the subscript $l$ and $s$ represent the larger and smaller pulley.

Diameter of larger pulley $\left(2r_l\right)=9\text{ inches}$

Diameter of smaller pulley $\left(2r_s\right)=6\text{ inches}$

Angular velocity of larger pulley $\left({\omega }_l\right)=120\text{ rpm}$

The options are as follows:

1. 80 rpm
2. 100rpm
3. 160rpm
4. 180rpm
5. 240rpm

Calculation:

The belt moves with a constant speed, so the linear velocity of both the pulleys will be same. The linear velocity of larger pulley is evaluated as follows:

  $\begin{array}{c}{v}_l=v_s\\ {\omega }_l{r}_l={\omega }_s{r}_s\\ \left(120\text{ rpm}\right)\left(\frac{9}{2}\text{ inches}\right)={\omega }_s\left(\frac{6}{2}\text{ inches}\right)\left[\because r=\frac{\text{Diameter}}{2}\right]\\ \left(60\text{ rpm}\right)\left(9\text{ inches}\right)={\omega }_s\left(3\text{ inches}\right)\\ {\omega }_s=\frac{\left(60\text{ rpm}\right)\left(9\text{ inches}\right)}{3\text{ inches}}\\ =\left(60\text{ rpm}\right)\left(3\right)\\ =180\text{ rpm}\end{array}$

The angular velocity of smaller pulley is 180 rpm, so option D. is correct.