In a popular amusement park ride, a rotating cylinder of radius 4.00 m is set in rotation at an angular speed of 5.00 rad/s. The floor then drops away, leaving the riders suspended against the wall in a vertical position. What minimum coefficient of friction between a rider's clothing and the wall is needed to keep the rider from slipping? (Hint: Recall that the magnitude of the maximum force of static friction is equal to µn, where n is the normal force - in this case, the force causing the centripetal acceleration.)
To find the minimum coefficient of friction needed to keep the rider from slipping, we need to consider the forces acting on the rider.
The only force acting horizontally is the static friction force between the rider's clothing and the wall. This force provides the centripetal acceleration required to keep the rider moving in a circle.
The centripetal force required is provided by the normal force acting vertically. The normal force is equal to the rider's weight, which can be calculated using the formula:
Weight = mass × gravity
Let's assume the mass of the rider is m.
Weight = m × g
The centripetal force is given by the formula:
Centripetal Force = mass × centripetal acceleration
The centripetal acceleration can be calculated using the formula:
Centripetal Acceleration = (angular speed)^2 × radius
Substituting the given values:
Centripetal Acceleration = (5.00 rad/s)^2 × 4.00 m
Now, equating the centripetal force and the maximum force of static friction:
Centripetal Force = Maximum Force of Static Friction
mass × centripetal acceleration = µ × normal force
mass × (angular speed)^2 × radius = µ × (mass × gravity)
Mass cancels out, leaving:
(angular speed)^2 × radius = µ × gravity
We can rearrange the equation to solve for the minimum coefficient of friction:
µ = (angular speed)^2 × radius / gravity
Plugging in the values:
µ = (5.00 rad/s)^2 × 4.00 m / 9.8 m/s^2
To determine the minimum coefficient of friction needed to keep the rider from slipping, we need to analyze the forces acting on the rider.
First, let's consider the forces acting on the rider when they are in a vertical position against the wall:
1. Normal force (n): This force is exerted by the wall perpendicular to its surface. It is equal in magnitude and opposite in direction to the force that the rider exerts on the wall. The normal force provides the centripetal force required to keep the rider moving in a circular path.
2. Gravitational force (mg): This force acts vertically downward, pulling the rider toward the center of the circular path.
3. Frictional force (f): This is the force of static friction acting between the rider's clothing and the wall. It acts parallel to the wall's surface and prevents the rider from slipping.
Now, let's derive the equation relating these forces:
The centripetal force required to keep the rider moving in a circular path is provided by the normal force (n). It can be calculated using the formula:
Fc = m * a
Where Fc is the centripetal force, m is the mass of the rider, and a is the centripetal acceleration.
For a rotating object, the centripetal acceleration is given by:
a = r * ω^2
Where r is the radius of the circular path and ω is the angular speed.
Substituting the equations, we get:
Fc = m * r * ω^2
Since the centripetal force is provided by the normal force (n), we can equate these two forces:
n = Fc = m * r * ω^2
The maximum static friction force is given by:
f = μ * n
Where μ is the coefficient of static friction.
Setting the equations for n and f equal to each other:
μ * n = m * r * ω^2
Dividing both sides by n:
μ = (m * r * ω^2) / n
Now we have an expression for the minimum coefficient of friction (μ) needed to prevent slipping. We need the value of the normal force (n) to calculate it.
The normal force can be determined by considering the forces in the vertical direction:
n - mg = 0
Solving for n:
n = mg
Substituting this value back into our equation for μ:
μ = (m * r * ω^2) / mg
Simplifying the equation:
μ = (r * ω^2) / g
Now we can compute the minimum coefficient of friction (μ) given the provided values for the radius (r) and angular speed (ω). Simply substitute the values into the equation and evaluate it.
The friction force M R w^2 * u must equal or exceed the weight M g.
Therefore u > g/(R w^2)
u is the coefficient of friction and w is the angular velocity. R is the radius of the cylinder.