I would love for someone to please verify my solution for this problem:
A former amusement park attraction consisted of a huge vertical cylinder rotating rapidly around its axis. The rotation speed should be sufficient for the people found inside to be stuck to the wall when the floor was removed. what should be the minimum speed of rotation for this to happen? The coefficient of static friction between the people and the wall of the cylinder is 0.300 and the cylinders radius is 4 meters.
Solution:
F = mv^2\r
fs = mv^2\r
us(mg) = mv^2\g
us = v^2\g
0.300 * 9,8 m\s^2 = v^2\4m.
2.94 m\s^2 * 4m = v^2
square root (11.76 m^2\s^2) = v
v = 3.43 m\s
Thank you
To verify the solution, we can use the concept of centripetal force and static friction.
First, let's understand the forces acting on the people when they are stuck to the wall of the cylinder. The only force bonding them to the wall is the static friction force.
The equation for centripetal force is given by F = mv^2/r, where F is the centripetal force, m is the mass of an object, v is the velocity of the object, and r is the radius of the circular path.
In this case, the centripetal force is provided by the static friction force (fs). The equation for the static friction force is given by fs = μsN, where μs is the coefficient of static friction and N is the normal force.
We can express N in terms of mg, where m is the mass of a person and g is the acceleration due to gravity.
So, μs(mg) = mv^2/r.
Now we can plug in the given values:
Coefficient of static friction, μs = 0.300.
Radius of the cylinder, r = 4m.
Acceleration due to gravity, g = 9.8 m/s^2.
Substituting these values into the equation, we have:
0.300(mg) = mv^2/4.
Simplifying further, we get:
0.300 * (9.8 m/s^2) = v^2/4.
2.94 m/s^2 = v^2/4.
Multiplying both sides by 4, we have:
v^2 = 2.94 m/s^2 * 4m.
v^2 = 11.76 m^2/s^2.
Now, take the square root of both sides to find the velocity v:
v = √(11.76 m^2/s^2).
Calculating the square root, we get:
v ≈ 3.43 m/s.
Therefore, the calculated minimum speed of rotation is approximately 3.43 m/s.
So, your solution of v = 3.43 m/s is correct.