In an automatic clothes drier, a hollow cylinder moves the clothes on a vertical circle (radius r = 0.39 m), as the drawing shows. The appliance is designed so that the clothes tumble gently as they dry. This means that when a piece of clothing reaches an angle of above the horizontal, it loses contact with the wall of the cylinder and falls onto the clothes below. How many revolutions per second should the cylinder make in order that the clothes lose contact with the wall when = 72.0°?

as the drawing shows....I have no idea how the 72deg is measured. Make your drawing, and at the release point, mg downward has to be greater than the vertical component of mv^2/r

To determine the number of revolutions per second the cylinder should make, we can use the concept of centripetal force and gravitational force acting on the clothes.

To start, let's identify the forces acting on the clothes at the point of losing contact with the wall of the cylinder. At this point, the only force acting on the clothes is the gravitational force, which is directed downwards.

Using the free body diagram, we can write the equation for the gravitational force acting on the clothes:

mg = (m * v^2) / r

Where:
m = mass of the clothes
g = acceleration due to gravity
v = velocity of the clothes
r = radius of the circular motion

We need to find the velocity (v) in terms of revolutions per second (ω):

v = ω * 2π * r

Where:
ω = angular velocity in radians per second

Substituting this value of velocity into our equation gives:

mg = (m * (ω * 2π * r)^2) / r

Simplifying the equation further:

g = (4π^2 * r) * ω^2

Now, we can solve for ω:

ω^2 = (g) / (4π^2 * r)

ω = √[(g) / (4π^2 * r)]

Substituting the given values:
g = 9.8 m/s^2 (acceleration due to gravity)
r = 0.39 m (radius)

ω = √[(9.8 m/s^2) / (4π^2 * 0.39 m)]

ω ≈ √[(9.8 m/s^2) / (4 * 3.14^2 * 0.39 m)]

ω ≈ √[0.247 m^-1] ≈ 0.497 m/s

To convert the angular velocity to revolutions per second, divide by 2π:

Revolution per second = ω / (2π)

Revolution per second ≈ 0.497 m/s / (2π) ≈ 0.079 rev/s

Therefore, the cylinder should make approximately 0.079 revolutions per second in order for the clothes to lose contact with the wall when the angle is 72.0°.