A bicyclist rides 5.0km due east, while the resistive force from the air has a magnitude of 3.0N and points due west. The rider then turns around and rides 5.0km due west, back to her starting point. The resistive force from the air on the return trip has a magnitude of 3.0N and points due east. (a) Find the work done by the resistive force during the round trip. (b) Based on your answer to part (a), is the resistive force a conservative force? Explain.
In part a, I got -3.0*10^4J, but I don't understand what it means by a conservative force. Can anyone please explain to me? Thanks a lot!
Say it were a conservative force like gravity.
As you went down from the top of a hill to the bottom, gravity would push down in your direction and you would get a positive amount of work done by gravity.
When you go back up the hill, gravity is pushing down still but you are going up, so force times distance is negative and the work done by gravity is negative.
By the time you get back to the starting point at the top of the hill, gravity has done just as much negative work coming back up as it did positive work on the way down.
That is NOT what happened with your air resistance. By the time you got back to your starting point, the air resistance force had done non-zero work on you. Therefore it is not conservative.
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In order to find the work done by the resistive force during the round trip, we first need to calculate the work done on each leg of the trip separately.
On the first leg, when the bicyclist rides 5.0km due east, the force from the air has a magnitude of 3.0N and points due west. The work done by this force is given by the equation:
Work = force x distance x cos(theta)
where theta is the angle between the force and the direction of motion. In this case, the angle is 180 degrees as the force and motion are in opposite directions.
Work = -3.0N x 5.0km x cos(180 degrees) = -3.0N x 5000m x (-1) = 15000J
On the return leg, when the cyclist rides 5.0km due west, the force from the air has a magnitude of 3.0N and points due east. Again, using the same equation, we have:
Work = force x distance x cos(theta)
In this case, the angle between the force and the direction of motion is 0 degrees as they are in the same direction.
Work = 3.0N x 5.0km x cos(0 degrees) = 3.0N x 5000m x 1 = 15000J
To find the total work done during the round trip, we add the work done on each leg:
Total work = Work on first leg + Work on return leg
= 15000J + 15000J
= 30000J
So, the work done by the resistive force during the round trip is 30000 Joules (J).
Now, to answer your question about whether the resistive force is a conservative force, a conservative force is one where the total work done by the force depends only on the initial and final positions, and not the path taken. In this case, the work done by the resistive force on the round trip is 30000J, which is non-zero. Therefore, the resistive force is not a conservative force. It depends on the path taken by the bicyclist.
To find the work done by the resistive force during the round trip, you need to understand the concept of work and apply it to this situation.
Work is defined as the product of the force applied on an object and the displacement of that object in the direction of the force. Mathematically, it is given by the equation W = F * d * cos(theta), where W is the work done, F is the force applied, d is the displacement, and theta is the angle between the force and displacement vectors.
Now let's break down the problem into two parts:
1. First Leg of the Trip:
The bicyclist rides 5.0 km due east. The resistive force from the air has a magnitude of 3.0 N and points due west.
- The displacement (d) is 5.0 km since the rider is moving in a straight line.
- The force (F) is 3.0 N since that is the magnitude of the resistive force.
Using the equation for work, we can calculate the work done during the first leg of the trip as:
W1 = F * d * cos(180 degrees) = -3.0 N * 5.0 km * cos(180 degrees) = -15.0 N·km
2. Second Leg of the Trip:
The bicyclist turns around and rides 5.0 km due west, back to her starting point. The resistive force from the air has a magnitude of 3.0 N and points due east.
- The displacement (d) is 5.0 km since the rider is moving in a straight line.
- The force (F) is 3.0 N since that is the magnitude of the resistive force.
Using the equation for work, we can calculate the work done during the second leg of the trip as:
W2 = F * d * cos(0 degrees) = 3.0 N * 5.0 km * cos(0 degrees) = 15.0 N·km
Now, to find the total work done during the round trip, we sum up the work done during each leg:
Total work done = W1 + W2 = -15.0 N·km + 15.0 N·km = 0 N·km
Therefore, the work done by the resistive force during the round trip is 0 N·km.
Regarding part (b), conservative forces are those where the work done is independent of the path taken and only depends on the initial and final positions. In this case, since the total work done by the resistive force during the round trip is 0 N·km, it means that the work done by the resistive force depends on the path taken, specifically the direction of travel. Therefore, the resistive force is not a conservative force.
I hope this explanation helps! Let me know if you have any further questions.