What is the Radius of Curvature for a Banked Curve with Ice on the Road?

[Sabrina2372]

Homework Statement

A civil engineer is asked to design a curved
section of roadway that meets the following
conditions:
With ice on the road, when the coefficient of
static friction between the road and rubber is 0.12, a car at rest must not slide into the ditch
and a car traveling less than 70 km/h must not skid to the outside of the curve.
What is the minimum radius of curvature of the curve? Answer in units of m[/B]

Homework Equations

∑F=mv^2/r

The Attempt at a Solution

FnsinΘ+ FfcosΘ= mv^2 /r
Then I went to the y direction to find Fn and got
Fn=(mg+FfsinΘ/cosΘ)
When I tried to get Ff out of the equation, I ended up subbing it for (μFnsinΘ/cosΘ)
But then I still have Fn in the equation!
This is really confusing me and the homework is due tonight so please help![/B]

 

[mfb]

You have two different car speeds you can use. This allows to solve for r and Θ, all other parameters are known or cancel out.

 

[Sabrina2372]

@mfb I did solve for theta and got the right answer, 6.842773413 however I'm not sure on how to find r. Is my formula correct? What is the next step I need to take, in order to get read of my Forces?

 

[mfb]

You know Fn (as function of m) and you can calculate Ff with your known angle. I guess this is the force from friction?
Then r is the only unknown value in the equation.

If it is still unclear, please post all your calculations so I can see what went wrong.

 

[Sabrina2372]

This is my FBD if that clears anything up at all. Ff is my Friction force, yes. I don't know Fn independent of Ff because there are three forces in the y direction.

 

[mfb]

I would expect to see a body on a tilted plane in that picture, otherwise it is hard to understand.
Friction is not independent of the normal force, they are linked via the coefficient of friction (that equation is in post 1).