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 $\mu_s = 0.60$. Therefore;
$F_f = mg$
$F_N~\mu_s = mg$
$m\omega^2~r~\mu_s = mg$
$\omega^2 = \frac{g}{r~\mu_s}$
$\omega = \sqrt{\frac{g}{r~\mu_s}}$
$\omega = \sqrt{\frac{9.80~m/s^2}{(2.5~m)(0.60)}}$
$\omega = 2.56~rad/s$
We then 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.
