A circular automobile racetrack is banked at an angle θ such that no friction between road and tires is required when a car travels at 25.5 m/s. If the radius of the track is 402 m, determine θ.
To determine the angle θ at which the racetrack is banked, we need to consider the forces acting on the car when it is traveling at 25.5 m/s.
When a car is moving in a circular path, there are two main forces acting on it: the gravitational force (mg) and the normal force (N). The normal force is the force exerted by the racetrack surface perpendicular to the car's motion.
The normal force can be resolved into two components: one perpendicular to the track (N⊥) and one parallel to the track (N//). The component N⊥ provides the necessary centripetal force to keep the car in a circular path.
We can find the angle θ using the equation:
tan(θ) = N// / N⊥
At equilibrium, the gravitational force mg and the normal force N both act towards the center of the circular path, perpendicular to each other. Therefore, we can write:
N⊥ = mg
The horizontal component N// can be derived using Newton's second law in the horizontal direction:
N// = m * a
The centripetal acceleration a can be calculated using the formula:
a = v^2 / r
Substituting the given values:
v = 25.5 m/s (velocity of the car)
r = 402 m (radius of the track)
We can calculate the centripetal acceleration:
a = (25.5 m/s)^2 / 402 m = 1.6165 m/s^2
Substituting this value into the equation for N//:
N// = m * a = m * (1.6165 m/s^2)
Since we do not have the mass of the car, we can cancel it out as follows:
m * (1.6165 m/s^2) / m = 1.6165 m/s^2
Now, substituting N⊥ = mg = N⊥, we can rewrite the equation for tan(θ) as:
tan(θ) = N// / N⊥ = 1.6165 m/s^2 / 9.8 m/s^2
Using a scientific calculator or trigonometric table, we can find the value of θ by taking the inverse tangent (arctan) of the ratio:
θ = arctan(1.6165 m/s^2 / 9.8 m/s^2)
Evaluating this expression, we find:
θ ≈ 9.86 degrees