Halliday, Resnick & Walker — Force and Motion–II: Problem 68

Engineering a highway curve. If a car goes through a curve too fast, the car tends to slide out of the curve. For a banked curve with friction, a frictional force acts on a fast car to oppose the tendency to slide out of the curve; the force is directed down the bank (in the direction water would drain). Consider a circular curve of radius \(R = 200\text{ m}\) and bank angle \(\theta\), where the coefficient of static friction between tires and pavement is \(\mu_s\). A car (without negative lift) is driven around the curve. (a) Find an expression for the car speed \(v_{\text{max}}\) that puts the car on the verge of sliding out. (b) On the same graph, plot \(v_{\text{max}}\) versus angle \(\theta\) for the range \(0^\circ\) to \(50^\circ\), first for \(\mu_s = 0.60\) (dry pavement) and then for \(\mu_s = 0.050\) (wet or icy pavement). In kilometers per hour, evaluate \(v_{\text{max}}\) for a bank angle of \(\theta = 10^\circ\) and for (c) \(\mu_s = 0.60\) and (d) \(\mu_s = 0.050\). (Now you can see why accidents occur in highway curves when icy conditions are not obvious to drivers, who tend to drive at normal speeds.)