Suppose you have a car traveling on a banked curve with friction. The curve makes an angle $\theta$ with the horizontal and has coefficient of static friction $\mu$.
Suppose we wish to calculate the minimum speed at which the car can travel so that it does not slip down the curve towards the center of the circular racetrack (of radius) $R$.
I know how to calculate this speed, but the following argument seems to reach a contradiction, and I cant tell what I'm doing wrong.
The force of friction keeping the car from slipping down the curve acts opposite the component of gravity parallel to the track. Thus $F_g\sin\theta=F_f$. Also, since the car is on the verge of slipping, $F_f=\mu F_N$ where $F_N$ is the normal force. Since the net force in the direction perpindicular to the car is 0, $F_N=F_g\cos\theta$. Thus $F_g\sin\theta=F_f=\mu F_g\cos\theta$, so $\mu=\tan\theta$.
This makes no sense since it implies the initial conditions must have satisfied this constraint. There is obviously something wrong with this reasoning, and I think I made the incorrect assumption that $F_N=mg\cos\theta$, because the correct method of calculating the minimum speed has a different result for $F_n$.
Please identify the flaw, thank you.