A race car is making a U-turn at constant speed. The coefficient of friction between the tires and the track is μ= 1.20. If the radius of the curve is 10.0 m, what is the maximum speed at which the car can turn without sliding? Assume that the car is undergoing uniform circular motion.
The maximum speed a car can take a turn without sliding is determined by the maximum force of static friction that the tires can provide. This maximum force of static friction is given by:
f_max = μN
where μ is the coefficient of friction and N is the normal force, which is equal to the weight of the car in this case. Since the car is undergoing uniform circular motion, the centripetal force required to keep it on the curve is given by:
F_c = mv^2/r
where m is the mass of the car, v is its speed, and r is the radius of the curve. This force must be provided by the force of static friction, so we have:
f_max = F_c
Combining these equations, we get:
μN = mv^2/r
Solving for v, we get:
v = sqrt(μgr)
where g is the acceleration due to gravity. Plugging in the values given, we get:
v = sqrt(1.20 x 9.81 x 10.0) = 12.3 m/s
Therefore, the maximum speed at which the car can turn without sliding is 12.3 m/s.