A car is able to drive around a curve the is banked at 18° to the horizontal without sliding. In order for this to occur, μs = 0.24 between the wheels and road. What is the radius of curvature of the banked curve? The car is travelling at 27 m/s.
To determine the radius of curvature of the banked curve, we can use the concept of centripetal force.
When a car drives around a curved road, there are two main forces acting on it: the gravitational force (mg) and the friction force (Ff) between the tires and the road.
The gravitational force can be resolved into two components: the normal force (N), perpendicular to the road, and the force component (mg.sinθ) parallel to the road, where θ is the angle of banking.
In order to prevent the car from sliding, the friction force between the tires and the road needs to provide the necessary centripetal force (Fc) to keep the car moving in a circular path.
The centripetal force (Fc) can be calculated using the formula Fc = (m.v^2)/R, where m is the mass of the car, v is its velocity, and R is the radius of curvature.
The friction force (Ff) can be given by the equation Ff = μs.N, where μs is the coefficient of static friction between the tires and the road, and N is the normal force.
The normal force (N) can be determined by balancing the vertical forces, so N = mg.cosθ.
Now we can equate the centripetal force (Fc) with the friction force (Ff) to find the radius of curvature (R).
(m.v^2)/R = μs.N
Substituting the values we have:
(m.v^2)/R = μs.(mg.cosθ)
Simplifying:
R = (m.v^2)/(μs.g.cosθ)
Now, let's substitute the given values:
m = mass of the car
v = 27 m/s
μs = 0.24
g = acceleration due to gravity (approximately 9.8 m/s^2)
θ = 18°
Note: Make sure to convert the angle θ from degrees to radians before calculating.
θ (in radians) = (18° * π) / 180°
Finally, we can calculate the radius of curvature (R).