Homework Help: A curve of radius 76 m is banked for a design speed of 100 km/h

1. Jun 23, 2015

rperez1

1. The problem statement, all variables and given/known data
If the coefficient of static friction is 0.38 (wet pavement), at what range of speeds can a car safely make the curve? [Hint: Consider the direction of the friction force when the car goes too slow or too fast.]

2. Relevant equations

I am trying to figure out the Vmax

3. The attempt at a solution

I was able to find the Vmin in doing:

ΣFx = ma = mv^2/r
Fnsinθ = mv^2/r (1)

Designed for 100km/h ~ 27.78m/s so theres no friction involved for now. Fn is the normal force.
Fnsinθ/Fncosθ = mv^2/mgr
tanθ = v^2/gr
θ = arctan(v^2/gr) = arctan((27.78)^2/(9.8*76))
= 46.02°

When you go very slow, the car's tendency is to slide down, so the friction acts up the ramp:

ΣFx = mv^2/r
Fnsinθ - fcosθ = mv^2/r (1)

ΣFy = 0
Fncosθ + fsinθ - mg = 0
Fncosθ + fsinθ = mg (2)

To eliminate Fn , f and m, i used the definition of friction:

f = μ*Fn
Fn = f/μ

And replaced Fn by f/μ in both equations:

fsinθ/μ - fcosθ = mv^2/r (1)*
fcosθ/μ + fsinθ = mg (2)*

I divided (1)* by (2)* to eliminate some values:

(fsinθ/μ - fcosθ)/(fcosθ/μ + fsinθ) = mv^2/mgr

(Factor f out and cancel it)
(sinθ/μ - cosθ)/(cosθ/μ + sinθ) = v^2/gr

Plugging in θ = 46.02° I found the min speed:

v = 18.73m/s = 67.43km/h

I have to find the max speed now but this time friction acts down the ramp to prevent car from shooting up. I just can't figure out how?

2. Jun 23, 2015

SammyS

Staff Emeritus
(I see you're new here, but ...) Please state your problem in the body of the Original Post of your thread. Don't post information in the title without including it in the thread, if it's important information.

I assume the "100 k" refers to 100 km/hr. Right ?

3. Jun 23, 2015

SammyS

Staff Emeritus
Wouldn't that simply change the sign you use for the force of friction ?

4. Jun 23, 2015

rperez1

yes it refers to 100 km/hr