Final answer:
The minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/s is calculated using the equation for centripetal force, resulting in a value that may be unreasonably high for average road conditions.
Step-by-step explanation:
To calculate the minimum coefficient of friction required for a car to safely negotiate a curve, we use the principles of circular motion. The centripetal force needed to keep the car in a circular path is provided by the static friction between the tires and the road. This force can be expressed by the equation Fc = mv²/r, where m is the mass of the car, v is the velocity, and r is the radius of the curve.
Since static friction (f) equals the coefficient of friction (μ) times the normal force (N), and because the normal force here is equal to the weight of the car (mg), we can set the centripetal force equal to the maximum static frictional force. So, mv²/r = μ mg. We can cancel the mass (m) from both sides of the equation and solve for the coefficient of friction as μ = v²/(rg).
Substituting the given values from our question, μ = (30.0 m/s)² / (50.0 m * 9.8 m/s²) = 1.83.
However, this result is greater than the typical values of friction coefficient encountered in everyday scenarios; this suggests either the car should drive at a lower speed or the premise of needing such high friction is unreasonable. A more realistic coefficient would allow the car to navigate the turn at a safe speed without relying on an unusually high frictional force that may not be present in real-life conditions, especially on slippery surfaces.