Well, what a twist and turn this question is! It's like a wild ride in a clothes dryer!
To figure out how many revolutions per second the cylinder should make, we need to consider the forces at play here. When the clothes lose contact with the wall, that means there's a balance between the gravitational force and the normal force acting on the clothes.
Now, we can use a little humor to calculate the acceleration of the clothes when they lose contact. We'll call it Circus Acceleration!
Since we know the radius of the vertical circle (r = 0.40 m) and the angle at which the clothes lose contact (θ = 68.0°), we can use some good old trigonometry to find the vertical component of the gravitational force (mg) acting on the clothes.
Using the right triangle formed by the radius and the angle, we find that the vertical component of the gravitational force is equal to mg * sin(θ). The normal force acting on the clothes is then equal to this vertical component, since they're in balance.
Now, here comes the showstopper! We can set the centripetal force equal to the normal force, since at that moment the clothes lose contact. The centripetal force is given by mv^2/r, where v is the velocity of the clothes and m is the mass.
Now, if we substitute the normal force with mg * sin(θ), we can solve for v.
What's that? You're not following? Don't worry, I'll break it down for you:
mv^2/r = mg * sin(θ)
Now solving for v:
v = sqrt(g * r * sin(θ))
Okay, folks, now we're getting somewhere! We know that the period (T) of the circular motion is equal to 1/f, where f is the frequency of the motion. The frequency is the number of revolutions per second, so we want to solve for f.
T = 1/f
Now, we know that the velocity of the clothes (v) is equal to the circumference of the circle (2πr) divided by the period (T):
v = 2πr/T
But wait, there's more! Remember that equation we found earlier?
v = sqrt(g * r * sin(θ))
So, we can equate these equations:
sqrt(g * r * sin(θ)) = 2πr/T
Now, we can solve for T:
T = 2π * sqrt(r/g * sin(θ))
Finally, we can solve for the frequency (f) by taking the reciprocal of T:
f = 1/T
So, my dear friend, in order for the clothes to lose contact with the wall at θ = 68.0°, the cylinder should make approximately f = 1/(2π * sqrt(r/g * sin(θ))) revolutions per second.
I hope this wild ride in the clothes dryer didn't leave you spinning!