## Chapter 6

## Q. 6.6

An amusement park ride shown in Fig. 6.3 consists of a 20 ft diameter cylindrical room that turns about its axis. People stand again st the rough cylindrical wall. After the room has reached a certain angular speed, the floor drops from under the riders. What must be the angular speed of the room to assure that a person will not slide on the wall? The design coefficient of static friction is μ = 0.4 .

## Step-by-Step

## Verified Solution

To assess the safe angular speed design, we seek a no-slip Coulomb condition sufficient to assure that a rider does not slide on the wall of the rotating room. The free body diagram of a rider represented as a center of mass object *P* is shown in Fig 6.3a. The rider’s weight is \mathbf{W}=-W \mathbf{k}, \text { and } \mathbf{N}=-N \mathbf{e}_r and \mathbf{f}=f_\phi \mathbf{e}_\phi + f_z \mathbf{k} are the normal and the tangential frictional forces exerted by the wall. Thus, the total force **F** on a rider in a cylindrical frame that turns with the room is

\mathbf{F}(P, t)=\mathbf{N} + \mathbf{f}+\mathbf{W}=-N \mathbf{e}_r + f_\phi \mathbf{e}_\phi + \left(f_z-W\right) \mathbf{k} (6.19a)

For the safety of a rider, we require that the rider remain at rest relative to the wall. Then by (6.4) in which \dot{\phi}=\omega, or by (4.48) in which \boldsymbol{\omega}_f=\omega \mathbf{k}, it follows

that m \mathbf{a}_P=-m r \omega^2 \mathbf{e}_r . Equating this to the force in (6.19a), we obtain the scalar equations of motion

\mathbf{F} \equiv F_r \mathbf{e}_r + F_\phi \mathbf{e}_\phi + F_z \mathbf{e}_z=m\left[\left(\ddot{r} – r \dot{\phi}^2\right) \mathbf{e}_r + \frac{1}{r} \frac{d}{d t}\left(r^2 \dot{\phi}\right) \mathbf{e}_\phi + \ddot{z}\mathbf{e}_z\right] . (6.4)

N=m r \omega^2, \quad W=f_z, \quad f_\phi=0 . (6.19b)

In the steady rotation of the room, no circumferential component f_\phi of the frictional force is exerted on the rider by the wall; and the second of these relations shows that the rider will not slide down the wall if the Coulomb condition W=f_z \leq f_c=\mu N holds. Therefore, with the first equation in (6.19b), the design criterion for safety of the riders is given by \mu m r \omega^2 \geq W. That is,

\omega \geq \sqrt{\frac{g}{r \mu}}, (6.19c)

equality holding when slip is imminent; the smallest value \omega^*=\sqrt{g / r \mu} being the critical angular speed of the room . *The result is independent of the weight of the rider; so all persons, fat or thin, will stay on the wall, provided that their coefficient of friction with the wall is not less than the design value chosen for μ.*

For the given conditions *r* = 10 ft and *μ* = 0.4 , the critical angular speed is \omega^*=2.84 \mathrm{rad} / \mathrm{sec}, which is about 27 rpm. Thus, to secure the safety of the riders, the room must spin at a rate greater than 27 rpm.