Calculating Minimum Coefficient of Friction for Safe Rotor-Ride at Carnival

[cubejunkies]

In a "Rotor-ride" at a carnival, people are rotated in a cylindrically walled "room". The room radius is 4.6 m, and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down?

Attempt at solution

Force of friction + Centripetal force = Force of gravity
μ_s F_n + ma_r = mg
μ_s g + v²/r = g
μ_s = (g- v²/r)/g
μ_s = (g-4π²r f²)/g
μ_s = (9.8 - 4π²(4.6)(0.5)²)/ 9.8
μ_s = -3.63

This obviously makes no sense because a coefficient of static friction must be less than 1 and positive to begin with and the answer in the back of the book is 0.22

I've rechecked my math several times over so i know my error has to be in my initial step of setting the force of friction and the centripetal force equal to the force of gravity, but this is my best guess as to how the answer could come about

Thank you for any help!
Anthony

 

[zhermes]

This is not true: "Force of friction + Centripetal force = Force of gravity"

For a person to not slip downwards, what forces must be balanced? Draw a free body diagram

 

[cubejunkies]

This is not true: "Force of friction + Centripetal force = Force of gravity"

For a person to not slip downwards, what forces must be balanced? Draw a free body diagram

In a free body diagram there would be a person with centripetal force going inward, force of gravity vertically down and the normal force to gravity straight up, and to not fall would mean the gravity and normal forces being equal to each other...?

 

[zhermes]

Yes, if by "normal force to gravity" you mean "force of friction."

 

[cubejunkies]

so i should try solving F_fr= F_g ?

 

[zhermes]

Exactly. What equation do you get force the force of friction?

 

[cubejunkies]

Is it F_n μ_s = F_fr ?

Okay I think I got it

F_fr=F_g
F_n μ_s = mg
[STRIKE]m[/STRIKE](v²/r) μ_s = [STRIKE]m[/STRIKE]g
4 $$\pi$$²r f² μ_s = g

Then inputting all the given values and solving for μ_s yields the correct answer of 0.22

Thank you so much!

 

[zhermes]

Good work! Happy to help.