Chapter 12.2, Problem 47PPS

a.

To determine

Calculate the angle $\theta$ in radians through which the carousel rotates in one second.

Answer to Problem 47PPS

The angle in radian is $\frac{\pi }{6}$ .

Explanation of Solution

Given:

It is given in the question that a carousel makes $5$ revolution per minute, The circle formed by riders sitting in the outside row has a radius $17.2$ feet and the circle formed by riders in the inside row has a radius $13.1$ feet.

Concept Used:

In this, use the concept of doing unitary method and converting the units from minute to second and also know that one revolution is equals to $2\pi$ .

Calculation: Since,

one revolution = $2\pi$

  $5$ revolution = $5\times 2\pi =10\pi$

Now, in radian with second,

  $\frac{10\pi }{\min }\times \frac{\min }{60\sec }=\frac{\pi }{6}$

Conclusion:

The angle is $\frac{\pi }{6}$ .

b.

To determine

Find the difference in arc length for both the riders.

Answer to Problem 47PPS

The difference in arc length is $2.15ft$ .

Explanation of Solution

Given:

It is given in the question that a carousel makes $5$ revolution per minute, The circle formed by riders sitting in the outside row has a radius $17.2$ feet and the circle formed by riders in the inside row has a radius $13.1$ feet.

Concept Used:

In this, use the concept of using the formula of arc length i.e. $s=r\cdot \theta$

Calculation:

For 1^st rider:

  $\begin{array}{l}{s}_1=r_1\cdot \theta \\ s_1=17.2\cdot \frac{\pi }{6}\\ s_1=\frac{17.2\times 3.14}{6}ft\\ s_1=9.00ft\end{array}$

For 2^nd rider:

  $\begin{array}{l}{s}_2=r_2\cdot \theta \\ s_2=13.1\cdot \frac{\pi }{6}\\ s_2=\frac{13.1\times 3.14}{6}ft\\ s_2=6.85ft\end{array}$

Net difference between the two = $s_1-s_2=(9-6.85)ft=2.15ft$

Conclusion:

The difference is $2.15ft$ .