Driving in your car with a constant speed of v = 11.7 m/s, you encounter a bump in the road that has a circular cross section, as indicated in the figure below. If the radius of curvature of the bump is r = 34.4 m, calculate the apparent weight of a 65.9 kg person in your car as you pass over the top of the bump.
I honestly don't even know where to start with this question. I am having a very hard time with apparent weight in general so please don't criticize me. Thank you for any and all help!
Assume that car tires and shock absorbers do not "cushion" the bump, so that the passenger's center of mass follows the profile of the bump.
The apparent weight is the force that the seat applies to the person to keep him or her in place. At the top of the bump, less force is required because there is a downward centripetal acceleration.
The person's actual weight is M g = 645.8 N
That gets reduced by
M V^2/R = 262.2 N
temporarily, due to the bump
The apparent weight is then 383.6 N
The centripetal acceleration is down, toward the center of the circle
Ac = v^2/R = 11.7^2/34.4 = 3.98 m/s^2
The gravity force down on driver = m g = 65.9*9.8 = 646 Newtons
The force up on the driver from the car is F, the weight a scale would measure if placed on the seat.
so
total force down on driver = m (Ac)
646 - F = 65.9 (3.98)
F = 646 - 262
F = 384 Newtons
No problem! I'll guide you through the steps to solve this problem.
To determine the apparent weight of the person as they pass over the bump, we need to consider the forces acting on them. The apparent weight is the normal force exerted on the person by the car seat.
Here's how you can approach this:
1. Identify the forces:
The person in the car experiences three forces:
- The person's weight (mg), where m is the mass and g is the acceleration due to gravity (9.8 m/s²).
- The normal force (N) exerted by the car seat on the person.
- The centripetal force (Fc) caused by the circular motion of the car over the bump.
2. Analyze the motion:
The person is moving at a constant speed in a circular path, so there must be a net force acting towards the center of the circle.
3. Relate forces:
The centripetal force is provided by the normal force. The net force is the difference between the centripetal force and the weight.
Therefore, Fc = N - mg.
4. Find the centripetal force:
The centripetal force can be calculated using the formula Fc = mv²/r, where m is the mass, v is the velocity, and r is the radius of curvature of the bump.
Plug in the values:
- Mass (m) = 65.9 kg
- Velocity (v) = 11.7 m/s
- Radius (r) = 34.4 m
Calculate Fc = (65.9 kg)(11.7 m/s)² / 34.4 m.
5. Substitute the centripetal force into the equation:
We can now substitute the centripetal force (Fc) into the equation N - mg = Fc.
Now we have: N - (65.9 kg)(9.8 m/s²) = (65.9 kg)(11.7 m/s)² / 34.4 m.
6. Solve for the normal force:
Rearrange the equation to solve for N, the normal force exerted on the person.
N = (65.9 kg)(9.8 m/s²) + (65.9 kg)(11.7 m/s)² / 34.4 m.
7. Calculate the normal force:
Plug in the given values and evaluate the answer.