Chapter 9.2, Problem 51PS

Solution

To determine the radians did people rotate each second.

The people rotated at a rate of $\frac{\pi }{2}\text{ radians per second }\text{.}$

Given:

People were placed in a $58$ feet long centrifuge and the centrifuge spun at a rate of about $15$ revolution per minute.

Calculation:

The people in the centrifuge must have turned an angle of $2\pi \text{ radians}$ when it completed one revolution. As the centrifuge spun at rate of about $15$ revolution per minute, then the angle rate for the centrifuge is:

  $\begin{array}{l}{A}_R=15\times 2\\ A_R=30\pi \text{ radians per minute}\end{array}$

Now, convert the rotation rate into radians per second as:

  $\begin{array}{l}{A}_R=30\pi \text{ radians per minute}\\ A_R=30\pi \times \frac{1}{60}\text{ radians per second}\\ A_R=\frac{\pi }{2}\text{ radians per second}\end{array}$

Hence, the people in the centrifuge rotated at a rate of $\frac{\pi }{2}$ radians per second.

To find: the length of the arc rotated in a second.

  $45.53\text{ feet}$

Given:

People were placed in a $58$ feet long centrifuge and the centrifuge spun at a rate of about $15$ revolution per minute.

Calculation:

The total length of the centrifuge is $58$ feet. As the radius of the centrifuge is half of the total length, the radius of the centrifuge is:

  $r=\frac{58}{2}$

In one second, the centrifuge rotates by:

  $\theta =\frac{\pi }{2}$

Now, the length of the arc rotated in one second is given as:

  $\begin{array}{l}s=r\theta \\ s=\frac{58}{2}\times \frac{\pi }{2}\\ s=\frac{58\pi }{4}\\ s=\frac{182.12}{4}\\ s=45.53\text{ feet}\end{array}$