Final answer:
To find the maximum speed at which a car can safely negotiate a horizontal, curved roadbed of radius 74.0 m with a coefficient of static friction of 0.330, use the formula for centripetal force and static friction. After calculations, the maximum safe speed is approximately 15.1 m/s.
Step-by-step explanation:
To calculate the maximum speed with which a car can safely negotiate an unbanked curve without slipping, we can use the formula for centripetal force (FC) and the concept of static friction. The force providing the centripetal acceleration comes from the static frictional force (therefore, FC equals the static frictional force).
The formula for the maximum static frictional force is fmax = μN, where μ is the coefficient of static friction and N is the normal force. In this case, the normal force is equal to the weight of the car, N = mg, where m is the mass of the car and g is the acceleration due to gravity (9.80 m/s^2).
The centripetal force needed to keep the car moving in a circle of radius r at velocity v is given by FC = mv^2/r. Setting the maximum static frictional force equal to the centripetal force, we get μmg = mv^2/r. The mass m cancels out from both sides, giving us v^2 = μgr. Therefore, the maximum speed v is μgr^1/2.
Using the given values, μ = 0.330 and r = 74.0 m, we find:
v = μgr^1/2 = (0.330)(9.80 m/s^2)(74.0 m)^1/2.
After calculation, the maximum speed at which the car can safely negotiate the curve is given by:
v = (0.330)(9.80 m/s^2)(74.0 m)^1/2 ≈ 15.1 m/s.