# A Problem from "Physics for scientists and engineers"

#### GeorgeGR

Hello everyone!!
I'm glad I found this forum!!

I need help to the following problem!

In an amusement park in Luna Park, passengers stand in a 16-meter-wide horizontal pan resting their backs on its side wall. The pan starts to rotate around a vertical axis. Suddenly, the floor of the pan, on which the passengers are standing, opens. If all goes well, the passengers will be stuck to the side of the pan and will not slide down. The static friction coefficient between garments and steel ranges from 0.60 to 1.0 and the kinetic friction coefficient ranges from 0.40 to 0.70 . At the entrance to the pan there is a sign indicating that "Children under 30 kg are not allowed to enter." What should be the minimum angular (RPM) speed for which passengers are safe?

#### Cervesa

in the horizontal direction, the normal force on each passenger as they rotate is equal to the centripetal force provided by the back side of the pan

$N = mr\omega^2$

given $m \ge 30 \, kg$, $r = 8 \, m$

for each passenger to remain in a state of equilibrium in the vertical direction,

$mg = F_s \le \mu_s \cdot N$

$mg \le \mu_s \cdot m r\omega^2 \implies \omega \ge \sqrt{\dfrac{g}{\mu_s r}}$

note the value of $\omega$ in the above inequality is in radians per second.

#### GeorgeGR

Thank you for help!

I have a question on this:

This answer tells us that there is an independence of mass!! So what is the purpose of the sign "Children under 30 kg are not allowed to enter." attending on mass instead of age ?

Also, if the coordinate system is [r t z], z for vertical, t for tangential, r for radial component, isn't there a friction force on the tangential component ?

#### Cervesa

A tangential acceleration component occurs as the pan transitions from a state of rest until it reaches its designed rotational speed. The force contributing to that tangential component is a combination of friction force from the floor and the side of the pan. Once it reaches the required value of $\omega$, there is only a radial component for acceleration when the floor is opened.

As far as the mass restriction for kids ... ?

#### GeorgeGR

Now I understand!
Thank you very much!