In a popular amusement park ride, a rotating cylinder of radius 3.0 m is set in rotation at an angular speed of 0.80 revolutions per second. The floor then drops away, leaving the riders suspended against the wall in a vertical position.
(a) What minimum coefficient of friction between a rider’s clothing and the wall is needed to keep the rider from slipping?
(b) Be sure the coefficient of friction makes sense… compare the coefficient of friction you derive to one that might be typical for amusement park riders against a wall.
To find the minimum coefficient of friction needed to keep the rider from slipping against the wall, we can start by analyzing the forces acting on the rider.
(a) The key force responsible for preventing slipping is the frictional force between the rider's clothing and the wall. When the floor drops away, the only two forces acting on the rider are the gravitational force pulling downward and the normal force pushing inward (perpendicular to the wall). The frictional force acts parallel to the wall in an opposing direction to prevent slipping.
Let's calculate the magnitude of the normal force first. The normal force is equal to the gravitational force in the radial direction, which can be determined using the centrifugal force:
Centrifugal force = m * ω^2 * r
where m is the mass of the rider, ω is the angular speed in radians per second, and r is the radius of the cylinder.
Given:
Angular speed, ω = 0.80 revolutions per second
Radius, r = 3.0 m
Converting the angular speed to radians per second:
ω = 0.80 revolutions/second * 2π radians/revolution = 1.60π radians/second
The magnitude of the normal force is determined by the gravitational force:
Centrifugal force = m * ω^2 * r = m * (1.60π)^2 * 3.0
The gravitational force is given by:
Gravitational force = m * g
where g is the acceleration due to gravity. The normal force is equal to the gravitational force, so:
m * g = m * (1.60π)^2 * 3.0
We can cancel the mass (m) from both sides of the equation:
g = (1.60π)^2 * 3.0
Now we can calculate the minimum coefficient of friction needed to keep the rider from slipping. The frictional force is given by:
Frictional force = coefficient of friction * normal force
The minimum coefficient of friction required will be equal to the tangent of the angle between the normal force and the gravitational force:
μ(minimum) = tan(θ) = (magnitude of the gravitational force) / (magnitude of the normal force) = g / (1.60π)^2 * 3.0
Calculating μ(minimum) will give us the answer for part (a).
(b) To determine whether the coefficient of friction we derive makes sense, we can compare it to typical values for amusement park riders against a wall. Typical coefficients of friction for this scenario are usually between 0.5 and 1.0. If our calculated coefficient of friction falls within this range, it can be considered reasonable for an amusement park ride.
To determine the minimum coefficient of friction required to keep the rider from slipping, we need to consider the forces acting on the rider.
First, let's look at the normal force (N) acting on the rider. The normal force is the force exerted by the wall perpendicularly to the surface. In this case, it is equal to the rider's weight, which can be calculated using the formula:
N = mg
where m is the mass of the rider and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Next, let's analyze the friction force (f) acting between the rider and the wall. Since the rider is in a vertical position, the friction force is responsible for preventing the rider from slipping down. The friction force can be calculated using the formula:
f = μN
where μ is the coefficient of friction.
To prevent slipping, the maximum friction force available should be equal to or greater than the force trying to make the rider slip, which in this case is the rider's weight (mg). Therefore,
f ≥ mg
Now, we can substitute the formulas for the friction force and normal force into the inequality:
μN ≥ mg
Substituting the formulas for N and f:
μmg ≥ mg
The mass cancels out on both sides, leaving us with:
μ ≥ 1
From this equation, we can see that the minimum coefficient of friction required to keep the rider from slipping is 1.
Now, let's consider the coefficient of friction in the context of amusement park riders against a wall. Typical coefficients of friction for dry surfaces range from 0.3 to 0.8. Since the minimum coefficient of friction required in this case is 1, it is higher than typical values. This suggests that the coefficient of friction needed for the riders to stay in place against the wall is quite high and not typical for amusement park rides. Additional safety measures, such as safety harnesses or restraints, would likely be necessary to ensure the riders' safety.