Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 8 - Dynamics II: Motion in a Plane - Exercises and Problems: 49

Answer

The minimum angular speed for which the ride is safe is 24 rpm.

Work Step by Step

The normal force $F_N$ from the wall provides the centripetal force to keep people moving in a circle. $F_N = m\omega^2~r$ The force of static friction keeps the people from sliding down the wall while they are spinning in a circle. To find the minimum angular speed, we can assume that the force of static friction is at its maximum possible value while using the smallest value of $\mu_s = 0.60$ $F_f = mg$ $F_N~\mu_s = mg$ $m\omega^2~r~\mu_s = mg$ $\omega^2 = \frac{g}{r~\mu_s}$ $\omega = \sqrt{\frac{g}{r~\mu_s}}$ $\omega = \sqrt{\frac{9.80~m/s^2}{(2.5~m)(0.60)}}$ $\omega = 2.56~rad/s$ We can convert $\omega$ to units of rpm. $\omega = (2.56~rad/s)(\frac{1~rev}{2\pi~rad})(\frac{60~s}{1~min})$ $\omega = 24~rpm$ The minimum angular speed for which the ride is safe is 24 rpm.
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