A circular curve of highway with radius 200m is designed for traffic moving at 100km/h, and is banked so that cars can make the curve even without friction at that speed.
What is the correct angle of banking for this curve?
Show steps, please.
Horizontal component = m v^2/R
Vertical component = m g
tan angle up from horizontal = v2/Rg
To find the correct angle of banking for the curve, we need to use the concepts of centripetal force and weight force acting on a vehicle as it travels along the curve.
Let's begin by analyzing the forces acting on the vehicle:
1. Centripetal Force (Fc): This force is responsible for keeping the vehicle moving in a curved path. It is directed towards the center of the curve.
2. Weight Force (W): This force is directed vertically downwards towards the ground.
Next, we need to consider the vector components of the Weight Force. We can break down the Weight Force into two components:
1. Perpendicular Component (W⊥): This component acts perpendicular to the surface of the road.
2. Parallel Component (W∥): This component acts parallel to the surface of the road.
Now, let's apply Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration:
Sum of Forces = Net Force
W⊥ - Fc = 0 (since the vehicle is not sliding off the road)
W∥ = 0
Considering the forces acting on the vehicle, we can determine the relationship between these forces:
1. Fc is provided by the friction between the tires and the road surface. Given that the curve is designed to be frictionless, Fc = 0.
2. The Perpendicular Component of the Weight Force (W⊥) provides the necessary centripetal force:
W⊥ = Fc = 0
Now, we can use trigonometry to find the angle of banking (θ). We can do this by considering the right triangle formed by the Perpendicular Component of the Weight Force (W⊥) and the Weight Force (W).
The relationship is given by the following equation:
W⊥ / W = sin(θ)
Since W⊥ = 0, the equation becomes:
0 / W = sin(θ)
Since the value of sin(θ) = 0, this implies that θ = 0 degrees.
Therefore, in this case, the correct angle of banking for this curve would be 0 degrees.