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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Well, if we want the clothes to lose contact with the wall at an angle of 72.0°, we need some centrifugal force to kick in. Let's do some math!
First, we need to find the speed at which the clothes will lose contact with the wall. We can use the formula for centrifugal force:
F = mv²/r
Where F is the centrifugal force, m is the mass of the clothes, v is the speed, and r is the radius.
Now, when the clothes lose contact with the wall, the only force acting on them will be gravity. So we can set the centrifugal force equal to the force of gravity:
mv²/r = mg
The mass m cancels out, so we have:
v²/r = g
Now, we can solve for v:
v = √(gr)
The speed v is equal to the circumference of the circle divided by the time it takes to make one revolution (T). So:
v = 2πr / T
Now, let's combine these two equations:
2πr / T = √(gr)
To make things easier, let's square both sides:
(2πr / T)² = gr
Now, we just solve for T:
T = 2π√(r/g)
Plug in the values: r = 0.39 m and g = 9.8 m/s²
T = 2π√(0.39/9.8)
T ≈ 2.47 seconds
Since we want to find the number of revolutions per second, we just take the reciprocal:
Revolutions per second = 1/2.47 ≈ 0.405
So, the cylinder should make approximately 0.405 revolutions per second for the clothes to lose contact with the wall at an angle of 72.0°.
But hey, don't forget to check if the clothes need a bit of a cha-cha-cha dance too! Maybe they want to show off their moves!
To determine the number of revolutions per second the cylinder should make, we can use the concept of centripetal force and centripetal acceleration.
The centripetal force acting on the clothes can be provided by the gravitational force acting on them, which is given by the equation:
F_gravity = m * g
where m is the mass of the clothes and g is the acceleration due to gravity.
The centripetal acceleration of the clothes as they move in a vertical circle can be calculated using the equation:
a_c = (v^2) / r
where v is the linear velocity of the clothes and r is the radius of the circle.
At the point where the clothes lose contact with the wall, the centripetal force is equal to the gravitational force:
F_gravity = F_c
m * g = m * (v^2) / r
We can cancel out the mass m and solve for the linear velocity v:
g = (v^2) / r
v^2 = g * r
v = √(g * r)
To convert the linear velocity to angular velocity, we can use the equation:
v = ω * r
where ω is the angular velocity.
Substituting the value of v, we get:
√(g * r) = ω * r
Solving for ω (angular velocity):
ω = √(g/r)
Now, we know that ω = 2πf, where f is the frequency (revolutions per second).
So, we have:
2πf = √(g/r)
Solving for f:
f = √(g/r) / (2π)
Given that θ = 72°, we can convert it to radians by multiplying it with π/180:
θ = 72° * π/180 = π/5 radians
Substituting the value of θ into the equation for f:
f = √(g/r) / (2π) = √((9.8 m/s^2)/0.39 m) / (2π)
Calculating the value of f:
f ≈ √(25.128) / (2π) ≈ 0.792 / (2π) ≈ 0.126 rev/s
Therefore, the cylinder should make approximately 0.126 revolutions per second in order for the clothes to lose contact with the wall at an angle of 72.0°.
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 given angle, we can use the following steps:
1. Start by finding the angular speed (ω) at which the cylinder should rotate. Angular speed is measured in radians per second and can be calculated using the equation:
ω = Δθ / Δt
where Δθ is the change in angle and Δt is the change in time.
2. Convert the given angle from degrees to radians. Since 1 revolution is equal to 2π radians, we can convert the given angle (72.0°) to radians by multiplying it by (2π/360).
θ_radians = 72.0° * (2π/360)
3. Calculate the change in time (Δt) required for the clothes to lose contact with the wall. This can be found using the equation:
Δt = (2π) / ω
4. Substitute the given angle (θ_radians) and solve for ω in terms of revolutions per second:
(2π) / ω = θ_radians
ω = (2π) / θ_radians
5. Finally, calculate the number of revolutions per second by dividing ω by 2π:
rev per sec = ω / (2π)
By following these steps, you should be able 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 given angle of 72.0°.