Design a carnival ride on which standing passengers are pressed against the inside curved wall of a rotating vertical cylinder. It is to turn at most at 1/3 revolution per second. Assuming a minimum coefficient of friction of 0.70 between clothing and wall, what diameter should the ride have if we can safely make the floor drop away when it reaches running speed? WHAT IS THE DIAMETER BECAUSE I TYPED IN 3.19 AND IT SAID INCORRECT. PLEASE HELP ASAP.
Hey - You are supposed to follow what I did, not just type any old number I give you!!!!!
I did r = 3.19
You can multiply by 2 to get the diameter but you better decide to understand what I did first.
To determine the diameter of the carnival ride, we can use the concept of centripetal force and the minimum coefficient of friction. We want the standing passengers to be pressed against the inside curved wall of the rotating vertical cylinder.
First, let's analyze the forces acting on a standing passenger:
1. Normal Force (N): This is the force perpendicular to the curved wall, counteracting the weight of the passenger.
2. Friction Force (Ff): This force acts tangentially to the curved wall, preventing the passenger from sliding down.
The maximum value of friction force can be calculated using the coefficient of friction (μ) and the normal force (N):
Ff = μN
In this case, the maximum coefficient of friction is given as 0.70.
Now, let's equate the maximum friction force to the centripetal force acting on the passenger. Centripetal force is given by the equation:
Fc = mv²/r
Where:
m = mass of the passenger
v = velocity of the rotating cylinder (maximum of 1/3 revolution per second)
r = radius of the cylinder's curved wall (we are trying to find the diameter)
Since the mass of the passenger is not given, we will cancel it out by considering the ratio of the normal force to the weight of the passenger:
N/m = g
Where:
g = acceleration due to gravity
Now, combining these equations, we get:
Ff = Fc
μN = mv²/r
μ(mg) = m((2π/3)r)²/r
μg = 4π²r/9
Simplifying further:
r = (9μg) / (4π²)
Let's substitute the given minimum coefficient of friction (0.70) and the acceleration due to gravity (9.8 m/s²) into the equation:
r = (9 * 0.70 * 9.8) / (4 * 3.14²)
r = (62.706) / (39.4784)
r ≈ 1.585 meters
Finally, the diameter of the ride is twice the radius:
Diameter = 2 * r = 2 * 1.585 = 3.17 meters
Therefore, the correct diameter for the ride, rounded to two decimal places, would be approximately 3.17 meters.
Note: The answer you typed in (3.19) is close to the correct answer, but it might be due to rounding errors or using more decimal places in the intermediate calculations.