Final answer:
To prevent riders from dropping in an amusement park ride, we used the coefficient of static friction and the radius of the spinning cylinder to determine the necessary centripetal force. We calculated an angular velocity and converted it to RPM, finding that approximately 15.6 RPM is needed. As this is not an available option, the closest and safest choice given is 18 RPM.
Step-by-step explanation:
Calculating Minimum Revolutions Per Minute
To find the minimum number of revolutions per minute (RPM) needed to prevent riders from dropping on an amusement park ride, we must consider the force of friction that acts on the riders. This force must be equal to or greater than the gravitational force pulling the riders toward the ground. We utilize the coefficient of static friction and the radius of the spinning cylinder to determine this.
The force of friction (f) can be calculated by using the formula: f = μmg, where μ is the coefficient of static friction, m is the mass of the rider, and g is the acceleration due to gravity (9.81 m/s²). The centripetal force (Fc) required to keep the rider against the wall is Fc = mv²/r, where v is the velocity of the rider, and r is the radius of the cylinder.
Since f = Fc, we can set these two equations equal to each other: μmg = mv²/r. Solving for v, we get v = sqrt(μr*g). As the ride is spinning in a circular motion, we need to find the angular velocity (ω) where v = ω*r. Therefore, substituting the value of v in ω = v/r, we obtain ω = sqrt(μ*g/r). Finally, to get RPM, we convert ω from radians per second to RPM through the relationship 1 RPM = 2π/60 radians/s.
Plugging in the given numbers: r = 4.2m and μ = 0.35, we get: ω = sqrt(0.35*9.81/4.2), which calculates to approximately 1.63 rad/s. Converting to RPM gives us RPM = (1.63*60)/(2π) which is about 15.6 RPM. The closest answer that is above this figure and hence safe enough to not let the riders fall is 16 RPM, which is not provided among the options a) 12 RPM b) 15 RPM c) 18 RPM d) 20 RPM. Therefore, either there is a mistake in the question, or the provided options are incorrect. For safety, we choose option c) 18 RPM which is the minimum acceptable RPM from the given choices.