Minimum Coefficient of Friction for a Banked Curve

[72_Cutlass_S]

Hello, I'm towards the end of my first semester of Physics 1 (cal based) and I've had this homework problem that has been making want to pull my hair out. I know that the coeffeicent of friction is between 0 and 1 and I have worked out the problem like my professor did but keep getting the wrong answer. Thank you for any help.

Michael

Homework Statement

A banked circular highway curve is designed for traffic moving at 70 km/h. The radius of the curve is 215 m. Traffic is moving along the highway at 40 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the turn without sliding off the road?

Homework Equations

f-mg*sin(x)=ma*cos(x)
N-mg*cos(x)=ma*sin(x)

tan(x)=v^2/Rg

where:
m= mass = cancels out
g= gravity = 9.8 m/s^2
a= acceleration = v^2/R = 11.11^2/215 = .574 m/s^2
x= angle
v= velocity = 40 km/h = 11.11 m/s
R= radius = 215m

The Attempt at a Solution

coefficent of friction = (g*sin(x)-a*cos(x))/(g*cos(x)+a*sin(x))

x= tan^-1(11.11^2/(215*9.8))
x= 3.353

f = (9.8*sin(3.353)-.574*cos(3.353))
(9.8*cos(3.353)-.574*sin(3.353))

f = -1.01*10^-15

 

[alphysicist]

Hi 72_Cutlass_S,

I think there is a problem with your angle. When the problem says that the highway is designed for a speed of 70km/h, that means that at that speed friction is not needed to carry the car around the curve. The formula $\tan\theta = v^2/(rg)$ applies to that special speed. So I think you put in the wrong speed.

There is also a typo in your first equation (f-mg*sin(x)=ma*cos(x)); the right hand side needs a negative sign to be consistent with the left hand side. However, the minus sign appears later on, which is why I think it you just accidently left it out.

The next to the last line also has a minus sign in the denominator which I don't think should be there.

 

[Moetasim]

First of all, great job on working through the problem and showing your work! It's important to understand the concepts and equations behind the problem rather than just trying to plug in numbers and get the right answer.

Now, to address your issue with getting the wrong answer. I noticed that you used 9.8 m/s^2 as the value for acceleration due to gravity, but in this problem, we are dealing with a circular motion, so the acceleration is given by v^2/R, not just 9.8 m/s^2. So, your value of a = 0.574 m/s^2 is correct, but you should use this value in your calculations instead of 9.8 m/s^2.

Also, I would suggest using the formula for the minimum coefficient of friction, which is μ = tan(x), where x is the angle of the banked curve. In this case, x = tan^-1(11.11^2/(215*0.574)) = 3.36 degrees. Using this value for x, you should get a coefficient of friction of μ = 0.059.

I hope this helps and keep up the good work! Don't hesitate to ask for help if you're still having trouble with this problem.