How Do You Calculate Rider Speed and Safety on a Centrifuge Carnival Ride?

[ahrog]

Homework Statement

A ride has a diameter of 4.8 N and makes 2 complete rotations in 5.0s.
a) Determine the speed of the rider.
b)What is the centripetal acceleration of the rider?
c) What is the coefficient of friction required to keep the rider from slipping downwards when the floor drops down?

Homework Equations

v= (2$$\pi$$r)/T
Fc=MAc
coefficient of friction=F/N

The Attempt at a Solution

a) v= (2$$\pi$$2.4)/2.5s
=6.0
b) Ac= (6s)²/2.4m
=15
c) I don't get how to get this...

 

[LowlyPion]

Draw a force diagram.

You have the weight down = m*g

You have the centripetal force = m*v²/r that is radially outward pressing the rider into the ride.

Now you should also know that the Maximum force of friction is μ * the normal force which is your centripetal force in this case.

 

[thomate1]

I would first like to clarify the units used in the given information. The diameter given is 4.8 N, which is a unit of force, and therefore cannot be used as a measure of length. I will assume that the diameter is actually 4.8 meters (m) and the time taken for 2 rotations is 5.0 seconds (s).

a) Using the equation v = (2πr)/T, where r is the radius of the ride, we can calculate the speed of the rider as follows:
v = (2π(4.8m))/5.0s
= 9.6π/5.0 m/s
= 6.09 m/s

b) The centripetal acceleration of the rider can be calculated using the equation Ac = v^2/r, where v is the speed and r is the radius. Therefore, the centripetal acceleration is:
Ac = (6.09 m/s)^2/4.8m
= 7.69 m/s^2

c) To calculate the coefficient of friction required to keep the rider from slipping downwards when the floor drops down, we need to consider the forces acting on the rider. The rider's weight (mg) will be acting downwards, and the normal force (N) from the floor will be acting upwards. The coefficient of friction (μ) is defined as the ratio of the frictional force (F) to the normal force (N). Therefore, we can use the equation μ = F/N to calculate the coefficient of friction.

In this case, the maximum frictional force that the floor can provide is equal to the centripetal force required to keep the rider moving in a circular motion. This can be calculated using the equation Fc = mac, where m is the mass of the rider and ac is the centripetal acceleration calculated in part b. Therefore, the coefficient of friction can be calculated as follows:
μ = F/mac
= (mg)/mac
= (mg)/(mac)
= (mg)/(mv^2/r)
= g/v^2
= (9.8 m/s^2)/(6.09 m/s)^2
= 0.274

Therefore, a coefficient of friction of at least 0.274 is required to keep the rider from slipping downwards when the floor drops down.