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°?

Question

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°?

rev/s

Answer 1

To determine the number of revolutions per second that the cylinder should make in order for the clothes to lose contact with the wall at a specific angle, we can consider the centripetal force acting on the clothes.

When the clothes reach an angle of above the horizontal, the gravitational force acting on the clothes becomes larger than the centripetal force required to keep them moving in a vertical circle. This causes the clothes to fall onto the clothes below.

To calculate the centripetal force, we can use the equation:

F = mv^2 / r

Where F is the centripetal force, m is the mass of the clothes, v is the linear velocity of the clothes, and r is the radius of the circular path.

Since the clothes are moving in a vertical circle, the linear velocity can be related to the angular velocity using the equation:

v = rw

Where w is the angular velocity in radians per second.

To determine the number of revolutions per second, we need to convert the given angle 72.0° to radians by using the conversion factor:

1 revolution = 2π radians

Given:
r = 0.39 m
= 72.0° = (72.0° * 2π) / 360° = 1.26 radians

We can now solve for w:

F = mv^2 / r
mg = mw^2r
g = w^2r
w = sqrt(g / r)
w = sqrt(9.8 m/s^2 / 0.39 m)
w ≈ 5.05 rad/s

Now, we can convert the angular velocity to revolutions per second:

1 revolution = 2π radians
1 second = 2π / w revolutions
w = 5.05 rad/s ≈ 2π / w revolutions / s

Therefore, the cylinder should make approximately 2π / (5.05 rad/s) revolutions per second for the clothes to lose contact with the wall when the angle is 72.0°.

Calculating this value:

2π / (5.05 rad/s) ≈ 1.00 rev/s

So, the cylinder should make approximately 1.00 revolution per second in order for the clothes to lose contact with the wall when = 72.0°.