Driving in your car with a constant speed of v= 22 m/s, you encounter a bump in the road that has a circular cross-section, as indicated in the figure.
If the radius of curvature of the bump is 52 m, find the apparent weight of a 66-kg person in your car as you pass over the top of the bump.
m= 66kg
v= 22 m/s
r= 52m
ΣF (c) = m*a (c) = m*v^2/r = m*g-n
n=m*g - m*v^2/r
n= m (g - v^2/r)
= 66kg [ 9.81 m/s^2 - (22 m/s)^2 / 52m]
=66kg [9.81 m/s^2 - (484 m^2/ s^2) / 52m]
=66kg (9.81 m/s^2 - 9.31 m/s^2)
=66 kg (0.5 m/s^2)
=33 N
To find the apparent weight of a person in the car as it passes over the bump, we need to consider the net force acting on the person at the top of the bump.
At the top of the bump, the person is in circular motion due to the curvature of the bump. The net force acting on the person is the centripetal force required to keep the person in circular motion.
The centripetal force is given by the equation:
Fc = m * (v^2 / r)
Where:
Fc = centripetal force
m = mass of the person (66 kg)
v = velocity of the car (22 m/s)
r = radius of curvature of the bump (52 m)
Plugging in the values:
Fc = 66 kg * (22 m/s)^2 / 52 m
Calculating:
Fc = 649.62 N
Therefore, the apparent weight of the person in the car as it passes over the top of the bump is approximately 649.62 N.
To find the apparent weight of the person in the car as it passes over the top of the bump, we need to consider the forces acting on the person.
When the car is on the top of the bump, the person experiences two main forces: the gravitational force (weight) and the normal force.
The gravitational force acting on the person is given by the equation:
Weight = mass * acceleration due to gravity
Weight = 66 kg * 9.8 m/s^2 (acceleration due to gravity)
Weight = 646.8 N
Now, let's consider the normal force. The normal force is perpendicular to the surface and prevents the person from sinking into the bump.
To find the normal force, we need to consider the acceleration of the person at the top of the bump. Since the car is moving with a constant speed and encountering a circular bump, the person experiences an inward acceleration towards the center of the circular path.
The centripetal acceleration is given by the equation:
Acceleration = (velocity)^2 / radius of curvature
Acceleration = (22 m/s)^2 / 52 m
Acceleration = 484 m^2/s^2 / 52 m
Acceleration ≈ 9.31 m/s^2
Now, let's find the net force acting on the person at the top of the bump:
Net force = mass * acceleration
Net force = 66 kg * 9.31 m/s^2
Net force = 614.46 N
Since the person is not sinking into the bump, the normal force must be equal and opposite of the net force. Thus, the normal force acting on the person is 614.46 N.
Now, to find the apparent weight of the person, we need to consider the difference between the gravitational force and the normal force:
Apparent weight = Weight - Normal force
Apparent weight = 646.8 N - 614.46 N
Apparent weight ≈ 32.34 N
Therefore, the apparent weight of the 66 kg person in the car as it passes over the top of the bump is approximately 32.34 N.