In an automatic clothes dryer, a hollow cylinder moves the clothes on a vertical circle (radius r = 0.40 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 θ 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 θ = 68.0°?

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!

To find the number of revolutions per second that the cylinder should make, we can use the concept of centripetal acceleration.

The centripetal acceleration of an object moving in a circular path is given by the equation:

ac = (v^2) / r

where ac is the centripetal acceleration, v is the linear velocity of the object, and r is the radius of the circular path.

In this case, the clothes lose contact with the wall of the cylinder when they reach an angle of θ above the horizontal. At this point, the normal force acting on the clothes becomes zero, and the weight force (mg) provides the centripetal force to keep the clothes moving in a circle.

The weight force (mg) is given by:

mg = m * g

where m is the mass of the clothes and g is the acceleration due to gravity.

The centripetal force is provided by the weight force (mg), so:

mg = (m * v^2) / r

Simplifying the equation:

v^2 = g * r

Now, we need to find the linear velocity v at the point when the clothes lose contact with the wall. To do this, we can use the relationship between angular velocity (ω) and linear velocity (v):

v = r * ω

Substituting this into the previous equation:

(r * ω)^2 = g * r

Simplifying the equation:

ω^2 = g / r

Now, we can find the angular velocity (ω) in terms of the number of revolutions per second (n):

ω = 2π * n

Substituting this into the previous equation:

(2π * n)^2 = g / r

Simplifying the equation:

4π^2 * n^2 = g / r

Now, we can solve for the number of revolutions per second (n):

n = √(g / (4π^2 * r))

Given that the radius r = 0.40 m and the angle θ = 68.0°, we can use the acceleration due to gravity (g = 9.8 m/s^2) to calculate the number of revolutions per second (n):

n = √(9.8 / (4π^2 * 0.40))

n ≈ 0.532 revolutions per second

Therefore, the cylinder should make approximately 0.532 revolutions per second in order for the clothes to lose contact with the wall when θ = 68.0°.

To determine the number of revolutions per second the cylinder should make in order for the clothes to lose contact with the wall at a specific angle θ, we can use the concept of centripetal acceleration.

First, let's find the velocity at which the clothes lose contact with the cylinder wall when θ = 68.0°. At this point, the weight of the clothes provides the necessary centripetal force to keep them in circular motion. The centripetal force (Fc) is given by the equation:

Fc = m * g * sin(θ)

where m is the mass of the clothes and g is the acceleration due to gravity. Since the clothes lose contact with the wall, the normal force (N) acting on them is zero, and thus the net force on the clothes is only the gravitational force. Therefore, Fc is equal to the gravitational force:

Fc = m * g

Setting these two equations equal to each other and solving for the mass of the clothes, we get:

m * g * sin(θ) = m * g
sin(θ) = 1

Since sin(θ) = 1 corresponds to a maximum angle of 90°, we can conclude that the clothes lose contact with the wall when θ = 90°.

Now, let's find the velocity at which the clothes lose contact with the wall. At any point on the vertical circle, the net centripetal force is provided by the tension in the clothes. The centripetal force can be calculated using the equation:

Fc = m * a
m * g - T = m * a

where T is the tension in the clothes and a is the centripetal acceleration.

The centripetal acceleration (ac) can be calculated using the equation:

ac = v^2 / r

where v is the velocity of the clothes and r is the radius of the vertical circle.

Substituting this into the previous equation, we get:

m * g - T = m * v^2 / r

Rearranging the equation to solve for v, we have:

v^2 = r * (g - T / m)
v = √(r * (g - T / m))

Now, at the point where the clothes lose contact with the wall, the tension in the clothes is zero. Thus, the equation becomes:

v = √(r * g)

Plugging in the values, with r = 0.40 m and g = 9.8 m/s^2, we can calculate the velocity:

v = √(0.40 * 9.8) = 2.78 m/s

Now that we have the velocity at which the clothes lose contact with the wall, we can find the number of revolutions per second.

The circumference of the vertical circle is given by:

C = 2πr

Substituting the value for r, we have:

C = 2π * 0.40 = 2.51 m

To find the number of revolutions, we divide the velocity by the circumference:

N = v / C = 2.78 m/s / 2.51 m = 1.11 revolutions per second

Therefore, the cylinder should make approximately 1.11 revolutions per second in order for the clothes to lose contact with the wall when θ = 68.0°.