In an automatic clothes dryer, a hollow cylinder moves the clothes on a vertical circle (radius r = 0.36 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 è 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 è = 73.0°?
To find the number of revolutions per second the cylinder should make in order for the clothes to lose contact with the wall when the angle è is 73.0°, we can use the concept of centripetal force.
The force exerted by the wall on the clothes is the centripetal force required to keep the clothes moving in a circular path. When the clothes lose contact with the wall, the centripetal force becomes zero.
The centripetal force can be calculated using the formula:
F = m * (v^2 / r)
where F is the force, m is the mass of the clothes, v is the velocity, and r is the radius.
In this case, we are given the values for r and è, and we need to find the velocity v.
The vertical component of the velocity can be calculated using the formula:
v = √(g * r * (1 - cos(è)))
where g is the acceleration due to gravity (9.8 m/s^2) and cos(è) is the cosine of the angle.
Substituting the given values into the formula, we get:
v = √(9.8 * 0.36 * (1 - cos(73°)))
Solving this equation, we find that v ≈ 2.89 m/s.
To find the number of revolutions per second, we need to convert the velocity to angular velocity (ω) and then divide by 2π:
ω = v / r
ω = 2.89 / 0.36
Finally, the number of revolutions per second is given by:
n = ω / (2π)
n = (2.89 / 0.36) / (2 * π)
n ≈ 4.03 revolutions per second.
Therefore, the cylinder should make approximately 4.03 revolutions per second for the clothes to lose contact with the wall when the angle è is 73.0°.