An amusement park ride consists of a large vertical cylinder that spins about its axis fast enough that any person inside is held up against the wall when the floor drops away (Fig. P6.65). The coefficient of static friction between person and wall is μs, and the radius of the cylinder is R. (a) Show that the maximum period of revolution necessary to keep the person from falling is T = (4π2Rμs/g)1/2. (b) Obtain a numerical value for T if R = 4.00 m and μs = 0.400. How many revolutions per minute does the cylinder make?

To determine the maximum period of revolution necessary to keep the person from falling, we can start by considering the forces acting on the person. There are two forces to consider: the normal force (N) exerted towards the center of the cylinder by the wall, and the force of gravity (mg) acting downward.

Let's break this problem down into steps:

Step 1: Identify the forces in the system:
- The normal force (N): This force acts perpendicular to the wall and towards the center of the cylinder, providing the centripetal force.
- The force of gravity (mg): This force acts downward, opposing the normal force.

Step 2: Determine the maximum static friction necessary to keep the person from falling:
Since the person is held up against the wall, the maximum static friction would be required. The maximum static friction force (fs) can be calculated using the equation fs = μsN, where μs is the coefficient of static friction.

Step 3: Equate the maximum static friction force with the centripetal force:
The centripetal force required to keep the person against the wall is equal to the maximum static friction force. Hence, we can write fs = mω²R, where m is the mass of the person and ω is the angular velocity (ω = 2π / T, where T is the period of revolution).

Step 4: Substitute the equations and solve for T:
Substituting fs = mω²R and fs = μsN into the equation, we get mω²R = μsN. Since N = mg, we can write mω²R = μsmg. Rearranging the equation, we have ω² = μsg/R. Substituting ω² = (2π / T)², we get (2π / T)² = μsg/R. Solving for T, we obtain T = (4π²Rμs / g)^(1/2).

(a) We have derived the expression for T as T = (4π²Rμs / g)^(1/2).

(b) Now, let's substitute the given numerical values into the formula to find T:
R = 4.00 m and μs = 0.400.

T = (4π²(4.00 m)(0.400) / 9.8 m/s²)^(1/2)
T = (4π²(1.60) / 9.8)^(1/2)
T = (64π² / 9.8)^(1/2)
T ≈ 8.012 s

To find the number of revolutions per minute, we can use the formula:
Revolutions per minute = (1 / T) * (60 sec / 1 min)

Revolutions per minute = (1 / 8.012 s) * (60 sec / 1 min)
Revolutions per minute ≈ 7.493 rpm

Therefore, the cylinder makes approximately 7.493 revolutions per minute.