Suppose a 1800-kg car passes over a bump in a roadway that follows the arc of a circle of radius 18.4 m as in the figure shown below.
(a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 9.24 m/s? (Neglect any friction that may occur.)
magnitude N
direction
(b) What is the maximum speed the car can have without losing contact with the road as it passes this highest point?
m/s
To answer these questions, we need to analyze the forces acting on the car at the highest point of the bump.
(a) The force exerted by the road on the car can be determined using Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. At the highest point of the bump, the car is momentarily at rest, so its acceleration is zero. Therefore, the net force acting on the car must be zero as well.
The two forces acting on the car at the highest point are its weight (mg) and the normal force (N) exerted by the road. The weight acts vertically downwards and has a magnitude of mg = 1800 kg * 9.8 m/s^2 = 17640 N.
Since the car is not moving vertically at the highest point, the normal force must have the same magnitude as the weight but in the opposite direction, to balance it out. So, the magnitude of the normal force is also 17640 N.
(b) The maximum speed the car can have without losing contact with the road at the highest point is when the normal force becomes zero, i.e., when it is just about to lose contact. This happens when the net force acting on the car becomes equal to zero.
The net force at the highest point is the centrifugal force, directed towards the center of the circular path. Therefore, we equate the magnitude of the centrifugal force (mv^2 / r) to the weight of the car:
mv^2 / r = mg
Simplifying and solving for the velocity v:
v = sqrt(gr)
where g is the acceleration due to gravity (9.8 m/s^2) and r is the radius of the circular path (18.4 m).
Plugging in the values:
v = sqrt(9.8 m/s^2 * 18.4 m) ≈ 13.4 m/s
So, the maximum speed the car can have without losing contact with the road at the highest point is approximately 13.4 m/s.