Answer:
w=3.05 rad/s or 29.88rpm
Explanation:
k = coefficient of friction = 0.3900
R = radius of the cylinder = 2.7m
V = linear speed of rotation of the cylinder
w = angular speed = V/R or to rewrite V = w*R
N = normal force to cylinder
N=[tex]=\frac{m(V)^{2}}{R}=m*(w)^2*R[/tex]
[tex]Friction force\\Ff = k*N\\Ff= k*m*w^2*R[/tex]
[tex]Gravitational force \\Fg = m*g[/tex]
These must be balanced (the net force on the people will be 0) so set them equal to each other.
[tex]Fg = Ff[/tex]
[tex]m*g = k*m*w^2*R[/tex]
[tex]g=k*w^{2}*R[/tex]
[tex]w^2 =\frac{g}{k*R}[/tex]
[tex]w=\sqrt{\frac{g}{k*R}} \\w =\sqrt{\frac{9.8\frac{m}{s^{2}}}{0.3900*2.7m}}\\ w=\sqrt{9.306}=3.05 \frac{rad}{s}[/tex]
There are 2*pi radians in 1 revolution so:
[tex]RPM=\frac{w}{2\pi }*60\\RPM=\frac{3.05\frac{rad}{s}}{2\pi}*60\\RPM= 0.498*60\\RPM=29.88[/tex]
So you need about 30 RPM to keep people from falling out the bottom
