What is the safe speed range for a car to make a 71 m radius banked curve at a coefficient of static friction of 0.30?
The car can safely make the curve within a speed range of approximately 59 km/h to 176 km/h considering the coefficient of static friction of 0.30 and a curve radius of 71 m.
Understanding the Factors for Safe Speed on a Banked Curve
Banked curves are designed to allow vehicles to safely navigate turns at higher speeds by utilizing the friction force between the tires and the road surface. The key concept to consider is that the friction force between the car's tires and the road surface provides the centripetal force required to keep the car moving in a curved path. This force acts inward and is influenced by the coefficient of static friction (μs) and the normal force (N).
When a car goes too slow, the friction force alone may not provide enough centripetal force, causing the car to slip outward. In this scenario, the gravitational force component perpendicular to the surface helps provide the remaining centripetal force. Conversely, when a car goes too fast, the friction force is not needed to provide centripetal force, leading to the car slipping inward.
To calculate the safe speed range for a banked curve, we consider the maximum and minimum speeds at which the car can safely navigate the curve. The maximum speed (Omax) occurs when the friction force reaches its maximum value, which is determined by the static friction coefficient, normal force, and the angle of banking. On the other hand, the minimum speed (Omin) occurs when the friction force is at its minimum value, which is zero, as the car is on the verge of losing contact with the road surface.
By applying the appropriate formulas and considering the given values for static friction coefficient, curve radius, and other factors, we can determine the safe speed range for a car to navigate a banked curve. In this case, the safe speed range is approximately 59 km/h to 176 km/h for a 71 m radius banked curve with a coefficient of static friction of 0.30.
It is essential for drivers to understand the dynamics of friction and centripetal force when approaching banked curves to ensure safe travel at varying speeds.