“Rotor� is a ride found in many amusement parks that consists of a hollow cylindrical room (roughly 8 ft in radius) that rotates around a central vertical axis. Riders enter the room and stand against the canvas-covered wall. The room begins to rotate, and when a certain speed is reached, the floor of the room drops away, revealing a deep pit. The riders do not fall, though: they are supported by a static friction force exerted by the persons contact interaction with wall. Estimate the rate at which the room should rotate (in revolutions per minute) to safely pin the riders to the wall. (You may have to make some estimates.)
Perhaps T or TR can help you. They seem to be taking the same course.
To estimate the rate at which the room should rotate to safely pin the riders to the wall, we need to consider the forces involved.
The force of static friction between the riders and the wall of the room provides the necessary centripetal force to keep the riders pinned. The centripetal force is given by the equation:
F = m * ω^2 * r
Where:
F is the centripetal force
m is the mass of the rider
ω is the angular velocity (in radians per second)
r is the radius of the room
Assuming the average mass of a rider is 70 kg, and the radius of the room is approximately 8 ft, which is roughly 2.44 meters, we can use these values to estimate the required angular velocity.
First, let's convert the radius to meters: 8 ft * 0.3048 m/ft = 2.44 m.
Now, rearranging the formula, we get:
ω = sqrt(F / (m * r))
To determine the centripetal force, we need to consider the force of static friction. The maximum static friction force can be calculated using the equation:
F_friction = μ * N
Where:
F_friction is the maximum static friction force
μ is the coefficient of static friction between the riders and the wall
N is the normal force on the rider
Since the riders are pressed against the wall, the normal force is equal to their weight, which can be calculated as:
N = m * g
Where:
g is the acceleration due to gravity, approximately 9.8 m/s^2.
Assuming a coefficient of static friction (μ) of 0.7, we can calculate the maximum static friction force:
F_friction = 0.7 * (m * g)
Now, we can substitute this value into the equation for the centripetal force to solve for ω:
ω = sqrt((0.7 * m * g) / (m * r))
Simplifying:
ω = sqrt(0.7 * g / r)
Now, we can substitute the known values:
ω = sqrt(0.7 * 9.8 / 2.44)
Calculating this expression, we find:
ω ≈ 3.5 radians/second
Finally, to get the rate in revolutions per minute, we need to convert radians per second to revolutions per minute. There are 2π radians in one revolution, and 60 seconds in one minute. So, the conversion factor is:
1 revolution/minute = (1 / 2π) radians/second
Converting ω to revolutions per minute:
ω_rev = ω * (1 / 2π) * 60
Substituting ω = 3.5 radians/second:
ω_rev ≈ 3.5 * (1 / 2π) * 60
Calculating this expression, we find:
ω_rev ≈ 33 revolutions/minute
Therefore, to safely pin the riders to the wall, the room should rotate at approximately 33 revolutions per minute.