A car, treated as a point particle, enters a turn of radius $R$ on a road banked at an angle $ heta

Question

A car, treated as a point particle, enters a turn of radius $R$ on a road banked at an angle $	heta$. The coefficient of static friction between the tires and the road is $\mu$. Let $v_{\min}$ and $v_{\max}$ be the minimum and maximum speeds to stay on the road without skidding sideways. Which of the following statements is/are correct?

Professor PC Thomas Classes · Answer

Let $f$ be friction pointing up the incline. The force equations are $N \cos	heta + f \sin	heta = mg$ and $N \sin	heta - f \cos	heta = mv^2/R$. For minimum speed $v_{\min}$, friction must prevent sliding down, so $f = +N\mu$ (up the incline), making A correct. If $\mu > 	an	heta$, static friction is sufficient to balance gravity's component down the incline even at rest, so $v_{\min} = 0$, making B correct (the formula gives an imaginary root). For maximum speed, $f = -N\mu$ (down the incline). The formula is $v_{\max}^2 = gR \frac{\sin	heta + \mu\cos	heta}{\cos	heta - \mu\sin	heta}$. If $\mu > \cot	heta$, the denominator becomes negative, indicating that the car will never skid upwards ($v_{\max} 	o \infty$), making C correct. From the vertical force equation, $N = \frac{mg - f\sin	heta}{\cos	heta}$. Since $f$ varies from $+N\mu$ to $-N\mu$ as speed increases, $N$ varies with speed, making D incorrect.