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
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To solve this problem, we can analyze the forces acting on the car at the highest point of the bump. The only relevant forces are the normal force (Fn) exerted by the road on the car and the weight of the car (mg).
(a) To find the magnitude of the force exerted by the road on the car, we can use the centripetal force formula:
Fc = (m * v^2) / r
Where:
m = mass of the car = 1800 kg
v = velocity of the car = 9.24 m/s
r = radius of the bump = 18.4 m
Substituting the given values into the formula:
Fc = (1800 kg * (9.24 m/s)^2) / 18.4 m
Simplifying gives:
Fc = 8954.10 N
Therefore, the magnitude of the force exerted by the road on the car is 8954.10 N.
Since the force exerted by the road is directed upwards (opposite to gravity), the direction of the force is upward.
(b) To find the maximum speed the car can have without losing contact with the road at the highest point, we need to consider the maximum value of the normal force (Fn) that can be exerted by the road. This occurs when the centripetal force (Fc) required to keep the car in circular motion exactly equals the weight (mg) of the car:
Fn = mg
Using the given mass of the car (m = 1800 kg) and gravitational acceleration (g = 9.8 m/s^2):
Fn = (1800 kg)(9.8 m/s^2)
Fn = 17640 N
Therefore, the maximum value of the normal force that can be exerted by the road is 17640 N.
To find the maximum speed, we equate the magnitude of the normal force to the weight of the car and use the centripetal force formula:
Fn = mg = (m * v^2) / r
Simplifying:
(m * v^2) / r = mg
v^2 = rg
v^2 = (18.4 m)(9.8 m/s^2)
v^2 = 180.32 m^2/s^2
v = sqrt(180.32 m^2/s^2)
v ≈ 13.44 m/s
Therefore, the maximum speed the car can have without losing contact with the road is approximately 13.44 m/s.
To find the force exerted by the road on the car as it passes the highest point of the bump, we need to consider the forces acting on the car at that point. At the highest point, the car will experience two forces: its weight and the normal force from the road.
Let's start with part (a):
(a) To find the force exerted by the road on the car, we can use Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
The only force acting in the vertical direction at the highest point is the normal force, and it should be equal in magnitude and opposite in direction to the car's weight.
The car's weight can be calculated using the formula: weight = mass * acceleration due to gravity.
First, let's calculate the weight:
mass of the car = 1800 kg
acceleration due to gravity = 9.8 m/s^2
weight = 1800 kg * 9.8 m/s^2 = 17,640 N
Since the car is traveling at a constant velocity at the highest point, its acceleration is zero. Therefore, the net force acting on the car is zero.
The normal force from the road, which equals the force exerted by the road on the car, is equal in magnitude but opposite in direction to the weight of the car.
So, the magnitude of the force exerted by the road on the car is 17,640 N.
(b) To find the maximum speed the car can have without losing contact with the road at the highest point, we need to consider the centripetal force required to keep the car in circular motion.
The car is moving in a circle of radius 18.4 m. At the highest point, when the normal force is minimum, the centripetal force should be equal to the weight of the car.
To find the centripetal force, we can use the formula: centripetal force = (mass * (velocity^2)) / radius
Using the same values from part (a) for the mass and weight of the car and the radius of the circle, we can solve for the velocity:
17,640 N = (1800 kg * (velocity^2)) / 18.4 m
Rearranging the equation to solve for velocity:
(velocity^2) = (17,640 N * 18.4 m) / 1800 kg
velocity^2 = 180,000 m^2/s^2
velocity ≈ √(180,000) ≈ 424.26 m/s
Therefore, the maximum speed the car can have without losing contact with the road at the highest point is approximately 424.26 m/s.