A stunt driver is driving a rigid car in a horizontal circle of radius $R$ on the inner surface of a

Question

A stunt driver is driving a rigid car in a horizontal circle of radius $R$ on the inner surface of a perfectly vertical cylindrical wall. The car's tires are separated by a track width $d$. The center of mass is at a perpendicular distance $h$ from the wall, positioned symmetrically midway between the upper and lower tires. The coefficient of static friction is $\mu$. Let $g$ be gravity. Which statements are correct?

Professor PC Thomas Classes · Answer

To prevent sliding: $mg \le \mu \frac{mv^2}{R} \implies v \ge \sqrt{\frac{gR}{\mu}}$. To prevent toppling about lower tire: limiting case is normal force on upper tire is zero. Balancing torques: $mgh = \frac{mv^2}{R} \frac{d}{2} \implies v \ge \sqrt{\frac{2gRh}{d}}$. Toppling dominates if $\sqrt{\frac{2gRh}{d}} > \sqrt{\frac{gR}{\mu}} \implies \mu > \frac{d}{2h}$.