Final answer:
The minimum static coefficient of friction required for a car to travel around a 363.0 m radius curve at a constant speed of 30.0 m/s without sliding is calculated using the formula μ_s = v²/(rg) and is found to be approximately 0.225.
Step-by-step explanation:
To find the minimum static coefficient of friction that allows a car to travel at a given speed without sliding on a flat curve, we can use the following steps. Since the car is moving in a circle, there is a need for a centripetal force to keep the car moving in that circle. This force is provided by the frictional force between the tires and the road.
The formula for centripetal force (Fc) is given by Fc = mv²/r, where m is the car's mass, v is the velocity, and r is the radius of the curve.
Since we're looking for the coefficient of friction (let's denote it as μs), we set the centripetal force equal to the maximum static frictional force, which is Ff = μs · N, with N being the normal force.
For a flat curve, N is equal to the gravitational force (mg), so we have Fc = μs · mg. After substituting Fc with mv²/r, we get mv²/r = μs mg, and after canceling m and g on both sides, we solve for μs to get μs = v²/(rg).
Therefore, for a car traveling at 30.0 m/s around a 363.0 m radius curve with g = 9.80 m/s², the minimum static coefficient of friction is calculated as μs = (30.0 m/s)² / (363.0 m · 9.80 m/s²), which equals approximately 0.225.
This value is the minimum required for the car to negotiate the curve without sliding.