You are laying on the kitchen floor of your apartment, staring up at the ceiling

Question

You are laying on the kitchen floor of your apartment, staring up at the ceiling, because you can't figure out all of the problems that your physics professor has come up with. Your study-partner suddenly offers you a bet: that you can't figure out the value of your own coefficient of static friction. You reply that someone could drag you across the floor with spring scale. She replies that you're not allowed to ask anyone for assistance. You think for a moment, and then accept the bet. You remembered that ride at the Fair - the one with the round room that spins so fast that you stick to the wall even as the floor drops out from beneath you. You remembered that the surface of the wall of the ride is just like your kitchen floor. And you remembered that after the ride begins its spin-down, the floor does not rise back up. Rather, you slide downward to the lowered floor. You return to the ride at the Fair. When you enter the ride, you measure the radius, R, of the room with your handy tape measure. Near the end of the ride, you use a stopwatch to measure the period of rotation, T, at the instant when you start to slide down the wall of the room. Show that your coefficient of static friction is given by mu_s = gT^2/4 pi^2 R.

Explanation / Answer

T = 2*pi*R/V
V = 2*pi*R/T
V is the speed of revolution

Normal force is centrefugal force
N = m*V^2/R
=(m/R) * (4*pi^2*R^2/T^2)
= (4*pi^2*m*R/T^2)

Now frictional force on wall is balacing your weight
fs = m*g
mius*N = m*g
mius* (4*pi^2*m*R/T^2) = m*g
mius* * (4*pi^2*R/T^2) = g
mius = (g*T^2) / (4*pi^2*R)
proved

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