Chapter 4.1, Problem 55E

Solution

a.

To find: The angular speed of the larger wheel in radians per second.

The angular speed of the larger wheel in radians per second is $4\pi$ .

Given information:

Angular speed of the wheel = $120\text{rpm}$

Radii,

  $\begin{array}{c}r=4\text{cm}\\ R=7\text{cm}\end{array}$

Calculation:

To get the revolution per second, divide the given RPM by $60$

Revolution per second $=120÷60$

Revolution per second = $2$

To get the radians per second, multiply the above equation by $2\pi$ ,

  $2\cdot 2\pi =4\pi$

Therefore, the angular speed of the larger wheel in radians per second is $4\pi$ .

b.

To find: The linear speed of the belt in centimeters per second.

The linear speed of the belt in centimeters per second is $87.96$ .

Given information:

Angular speed of the wheel = $120\text{rpm}$

Radii,

  $\begin{array}{c}r=4\text{cm}\\ R=7\text{cm}\end{array}$

Calculation:

The angular speed in revolutions per minute is converted into revolutions per second.

  $\frac{120\text{rev}}{\text{min}}\times \frac{1\text{min}}{60\text{s}}=2\frac{\text{rev}}{\text{s}}$

Now to find the circumference of the larger wheel,

  $\begin{array}{c}C=2\pi r\\ =2\pi \cdot 7\\ =14\pi \end{array}$

Thus, one revolution of the wheel is equal to $14\pi$ centimeters.

Substituting the circumference value in above equation to get,

  $\begin{array}{c}v=\frac{2.14\pi \text{cm}}{1s}\\ =\frac{28\pi \text{cm}}{1s}\\ \approx 87.96\frac{\text{cm}}{s}\end{array}$

Therefore, the linear speed of the belt in centimeters per second is $87.96$ .

c.

To find: The angular speed of the smaller wheel in radians per second.

The angular speed of the smaller wheel in radians per second is $7\pi$ .

Given information:

Angular speed of the wheel = $120\text{rpm}$

Radii,

  $\begin{array}{c}r=4\text{cm}\\ R=7\text{cm}\end{array}$

Calculation:

The circumference of the smaller wheel is given by,

  $\begin{array}{c}C=4\cdot 2\pi \\ =8\pi \end{array}$

Revolution per second,

  $28\pi ÷8\pi =3.5$

Revolution per second = $3.5$

To get the radians per second, multiply the above equation by $2\pi$ ,

  $3.5\cdot 2\pi =7\pi$

Therefore, the angular speed of the larger wheel in radians per second is $7\pi$ .