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rperez1
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Homework Statement
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.]
Homework 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 there's 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?