What Should the Banking Angle Be to Prevent a Car from Sliding?

[AHinkle]

Homework Statement

Homework Equations

$$\Sigma$$F=ma
a_c=(v^2/r)
f = $$\mu$$N

The Attempt at a Solution

$$\Sigma$$F_radial= (radial-coordinate of normal force) + (radial component of friction) = ((mass)(velocity^2)/(radius))
$$\Sigma$$F_y= (y-component of normal force) - (y-component of friction) = (mass)(gravity)

$$\Sigma$$F_radial= Nsin$$\theta$$+$$\mu$$Ncos$$\theta$$ = (mv^2/r)
$$\Sigma$$F_y=Ncos$$\theta$$ - $$\mu$$Nsin$$\theta$$ = (mg)

I divided the equations for F_radial by the equation for F_y
and it yielded...

tan$$\theta$$ = (v^2-$$\mu$$rg)/(rg+$$\mu$$v^2)

so in order to find theta which I am looking for

$$\theta$$= arctan (v^2-$$\mu$$rg)/(rg+$$\mu$$v^2)

but I have 2 unknowns...
we know
V_max = 100km/h which is approx 27.78 m/s
g = 9.81 m/s^2 (this is given in the problem)
r= ?
$$\theta$$ = ?
$$\mu$$= 0.22

also I am not sure how to conceptualize the part of the problem where i need to find out what theta needs to be to keep the car from sliding in the ditch.
I feel i can find the upper limit but not the lower. I thought about subbing something in for r to find theta but I'm stumped.

 

[dacruick]

what do you have so far? If nothing mathematical, what concepts are you thinking about?

 

[AHinkle]

im trying to get it to work, and the symbol stuff is screwing up.. how do i delete the post and start over?

 

[dacruick]

Just start A new post I suppose.

 

[AHinkle]

okay i fixed everything, please help me out. The homework is already turned in, I just want the knowledge..

 

[dacruick]

think about the x-direction of the normal force from the bank.