Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (called the "Holly hump") and had it installed. Suppose a 1800-kg car passes over a hump in a roadway that follows the arc of a circle of radius 19.4 m as in the figure below.
A car traveling to the right over a hump on the road at a velocity vector v.
(a) If the car travels at 23.6 km/h what force does the road exert on the car as the car passes the highest point of the hump?
magnitude
N
(b) What is the maximum speed the car can have without losing contact with the road as it passes this highest point?
km/h
(a) To find the force exerted by the road on the car at the highest point of the hump, we need to consider the forces acting on the car at that point. The only horizontal force present is the normal force exerted by the road on the car, as there is no net force in the horizontal direction.
The force of gravity acting downwards on the car is given by Fg = mg, where m is the mass of the car and g is the acceleration due to gravity (taken as 9.81 m/s^2).
At the highest point of the hump, the force of gravity can be split into two components: one acting vertically downward and one acting horizontally towards the center of the circle. The normal force exerted by the road on the car is equal in magnitude but opposite in direction to the horizontal component of the force of gravity.
Using Newton's second law, we can set up the equation:
N = mv^2 / r - mg
Where N is the normal force, m is the mass of the car, v is the speed of the car at the highest point, and r is the radius of the circle.
Plugging in the values, we get:
N = (1800 kg)(23.6 km/h)^2 / (19.4 m) - (1800 kg)(9.81 m/s^2)
N = 800 N
Therefore, the force exerted by the road on the car at the highest point of the hump is 800 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 minimum value of the normal force at that point. The normal force must be greater than or equal to zero for the car to stay on the road.
At the highest point of the hump, the normal force can be set to zero to find the maximum speed without losing contact with the road:
0 = mv^2 / r - mg
Solving for v, we get:
v = sqrt(gr)
Plugging in the values, we get:
v = sqrt((9.81 m/s^2)(19.4 m))
v = 13.4 m/s
Converting to km/h, we get:
v = 13.4 m/s * (3600 s/1hr) * (1 km / 1000 m)
v = 48.2 km/h
Therefore, the maximum speed the car can have without losing contact with the road at the highest point is 48.2 km/h.