Answer
(a) The force of static friction pushes up on the people and prevents them from falling out the bottom of the cylinder.
(b) The minimum angular speed is $3.13~rad/s$
Work Step by Step
(a) The force of static friction pushes up on the people and prevents them from falling out the bottom of the cylinder.
(b) Let $m$ be a person's mass. In order for people to stay inside the cylinder, the force of static friction must be equal in magnitude to the person's weight. We can find an expression for the normal force $F_N$:
$F_N~\mu = mg$
$F_N = \frac{mg}{\mu}$
The normal force $F_N$ provides the centripetal force to keep each person moving in a circle. We can find the minimum angular speed:
$F_N = m~\omega^2~r$
$\frac{mg}{\mu} = m~\omega^2~r$
$\frac{g}{\mu} = \omega^2~r$
$\frac{g}{r~\mu} = \omega^2$
$\omega = \sqrt{\frac{g}{r~\mu}}$
$\omega = \sqrt{\frac{9.80~m/s^2}{(2.5~m)(0.40)}}$
$\omega = 3.13~rad/s$
The minimum angular speed is $3.13~rad/s$
