The Wall of Death in an amusement park is comprised of a vertical cylinder that can spin around the vertical axis. Riders stand against the wall of the spinning cylinder and the floor falls away leaving the riders held up by friction. The radius of the cylinder is 3.8 m and the coefficient of static friction between the rider and the wall is 0.49. Find the minimum number of revolutions per minute necessary so that the riders do not slip down the wall.
I have no clue which equation/steps I need to do for this. Can I please get some help?
The maximum frictional force that the wall can exert must equal the person's weight at the minimum required rotational speed. Call that angular speed, in radians per second), w
The centripetal force is M R w^2 and the maximum frictional force is
M R w^2*0.49
Set that equal to M g and solve for w. The mass M cancels out.
w = sqrt[g/(0.49 R)]= ?
Multiply w in rad/s by
60 s/min/(2 pi rad/rev) to get r.p.m.
To find the minimum number of revolutions per minute needed so that the riders do not slip down the wall, we need to use the concept of centripetal force and static friction.
Let's denote the mass of the rider as m, the radius of the cylinder as r, and the angular velocity (in radians per second) as ω. We can first convert the given angular velocity to revolutions per minute.
The centripetal force required to keep the rider against the wall is provided by the static friction force. The formula for static friction force is given by:
Ffriction = μs * mg
where μs is the coefficient of static friction and mg is the weight of the rider (mass * acceleration due to gravity).
The centripetal force is given by:
Fcentripetal = m * (ω^2) * r
Since the maximum static friction force is equal to the centripetal force, we can set these two forces equal to each other:
μs * mg = m * (ω^2) * r
Dividing both sides by m, we get:
μs * g = (ω^2) * r
Rearranging the equation to solve for ω^2, we have:
(ω^2) = (μs * g) / r
Substituting the given values:
μs = 0.49
g = 9.8 m/s^2 (acceleration due to gravity)
r = 3.8 m
By evaluating the equation, we can find ω^2:
(ω^2) = (0.49 * 9.8) / 3.8
Now, we can take the square root of ω^2 to find ω:
ω = √[(0.49 * 9.8) / 3.8]
Next, we need to convert ω to revolutions per minute. Since there are 2π radians in one revolution and 60 seconds in one minute, we have:
1 revolution = 2π radians
1 minute = 60 seconds
To find ω in revolutions per minute, we can use the following conversion factor:
1 radian/second = (1 revolution / 2π radians) * (60 seconds / 1 minute)
Multiply ω by the conversion factor:
ω_revolutions_per_minute = ω * [(1 revolution / 2π radians) * (60 seconds / 1 minute)]
Finally, evaluate the expression for ω_revolutions_per_minute to find the minimum number of revolutions per minute needed so that the riders do not slip down the wall.
Of course! Let's break down the problem step by step. To find the minimum number of revolutions per minute necessary so that the riders do not slip down the wall, we need to consider the forces acting on the riders.
1. First, let's analyze the forces acting on an individual rider. Since the rider does not slip down the wall, the frictional force between the rider and the wall must be equal to or greater than the gravitational force pulling the rider downward.
2. The frictional force can be calculated using the equation: frictional force = coefficient of static friction * normal force. In this case, the normal force is equal to the gravitational force acting on the rider, which is equal to the rider's weight.
3. Now let's calculate the gravitational force acting on the rider. The formula for gravitational force is: gravitational force = mass * acceleration due to gravity. In this case, we are given the weight, which is the force of gravity acting on the rider.
4. The formula for weight is: weight = mass * acceleration due to gravity. Since the mass of the rider is not given, let's assume a general value of 70 kg for an average rider.
5. The acceleration due to gravity is a constant value on Earth, approximately 9.8 m/s^2.
6. Once we know the weight, we can calculate the normal force and then the frictional force.
7. Now, let's consider the centripetal force required to keep the rider against the wall. The centripetal force is given by the equation: centripetal force = (mass * velocity^2) / radius. In this case, we need to find the velocity at which the riders do not slip down the wall.
8. By equating the frictional force to the centripetal force, we can solve for the minimum velocity required using the calculated values of the frictional force, mass, and radius.
9. Finally, we can convert the minimum velocity into revolutions per minute by realizing that 1 revolution is equal to the circumference of the wall.
By following these steps and applying the relevant equations, you should be able to find the minimum number of revolutions per minute necessary so that the riders do not slip down the wall.