You are driving your car over a circular-shaped bump in the road that has a radius of curvature of 75.9 m.
A)If the car is traveling at a constant speed of 18.3 m/s, calculate the apparent weight of your 56.1 kg passenger as you pass over the top of the bump.
B)What is the maximum speed that you can drive the car over the top of the bump without losing contact with the road?
To calculate the apparent weight of the passenger as you pass over the top of the bump, we need to consider the forces acting on the passenger. At the top of the bump, there are two forces to consider: the normal force (N) and the gravitational force (mg).
A) The apparent weight of the passenger is the force the passenger feels due to the combination of these forces. We can calculate this using Newton's second law:
Apparent Weight = N - mg
To find the normal force, we need to consider the net force acting on the car at the top of the bump. The net force is the centripetal force, which is provided by the normal force:
Net Force = Centripetal Force = (mass × velocity^2) / radius
Therefore, the normal force is equal to the centripetal force:
N = (mass × velocity^2) / radius
Substituting the given values, we have:
N = (56.1 kg × (18.3 m/s)^2) / 75.9 m
Calculating this, we find:
N ≈ 249.7 N
Now, we can calculate the apparent weight:
Apparent Weight = N - mg = 249.7 N - (56.1 kg × 9.8 m/s^2)
Calculating this, we find:
Apparent Weight ≈ 195.7 N
Therefore, the apparent weight of your passenger as you pass over the top of the bump is approximately 195.7 N.
B) The maximum speed at which you can drive the car over the top of the bump without losing contact with the road can be determined by considering the minimum normal force required.
At the top of the bump, the normal force should be equal to or greater than zero to maintain contact with the road. Therefore, the minimum normal force (Nmin) occurs when the net force is zero:
Net Force = Centripetal Force = (mass × velocity^2) / radius = Nmin
Solving for the velocity, we get:
velocity^2 = (Nmin × radius) / mass
Rearranging, we find:
velocity = sqrt((Nmin × radius) / mass)
Substituting the given values, we have:
velocity = sqrt((0 × 75.9 m) / 56.1 kg)
Calculating this, we find:
velocity = 0 m/s
Therefore, the maximum speed you can drive the car over the top of the bump without losing contact with the road is zero. Any speed greater than zero would result in the car losing contact with the road at the top of the bump.