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°?
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To determine the number of revolutions per second the cylinder should make, we can use the concept of angular velocity.
The angular velocity (ω) is defined as the change in angle (θ) per unit of time (t). It is given by the equation:
ω = Δθ / Δt
In this case, we want to find the angular velocity (ω) at an angle of 72.0°. Let's assume that the initial angle is 0°.
The angle at which the clothes lose contact with the wall is when they are above the horizontal, which is 90°. So, the change in angle (Δθ) is 90° - 0° = 90°.
The time (Δt) taken to move from 0° to 90° depends on the revolution per second of the cylinder.
To find the time taken, we need to know the circumference of the circle that the clothes move in. The circumference of a circle is given by the formula:
C = 2πr
where r is the radius of the circle. In this case, the radius is given as 0.39 m.
Let's assume the number of revolutions per second is "n" (unknown).
In one revolution, the clothes move through a distance equal to the circumference of the circle. So, the time taken to complete one revolution (Δt) is:
Δt = C / v
where v is the linear velocity of the clothes on the edge of the cylinder.
The linear velocity (v) can be calculated using the formula:
v = ωr
Substituting the value of ω as n revolutions per second and r as 0.39 m, we can rewrite the equation as:
Δt = C / (n * r)
Now, let's substitute the value of C using the given radius of 0.39 m:
Δt = (2π * 0.39) / (n * 0.39)
Simplifying:
Δt = 2π / n
Now, we can substitute the values of Δθ = 90° and Δt = 2π / n into the formula for angular velocity, and solve for n:
ω = Δθ / Δt
ω = 90° / (2π / n)
Converting 90° to radians (since angular velocity is typically measured in radians per second):
ω = (90° * π / 180°) / (2π / n)
ω = n / 2
Now, we can substitute the given angle of 72.0° into the equation and solve for n:
72.0° = n / 2
Multiplying both sides by 2:
144.0° = n
Therefore, the cylinder should make approximately 144 revolutions per second in order for the clothes to lose contact with the wall when the angle is 72.0°.