Part one- Finding a Magnetic Potential Energy Function:
The first part of this experiment was to determine the potential energy function between two magnets. Since this was unknown, we had to derive it using the given formula F= -du/dr. To do this we set up our experiment as such (see Fig. 1):
- We placed an air track (attached with a magnet) on our lab bench with an air track glider (attached with a different magnet) on top of it.
- Connected to the air track was a vacuum which would slightly push the air track glider off of the air track, making it relatively friction-less.
- We had books on the side that would allow us to put the air track at an incline.
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| Fig. 1 |
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| Fig. 2 Using this free-body diagram, we could conclude that F=mgsinθ. |
Since we had to find a function of force relative to position, we had to put the air track at an angle to get the glider moving. If we created a free-body diagram of the system, we would find that raising the air track, (increasing it's angle θ with the horizontal) would continuously change that magnitude of this force, increasing it each time (see Fig. 2). This meant that we would have to measure multiple angles of inclination, measure the distance between the two magnets at each incline, and calculate the force exerted at each distance. If we graphed the force vs distance in LoggerPro, we would have our force as a function of distance which would then allow us to derive an expression for the magnetic potential energy.
To collect data, we had to increase θ a number of times while recording the distance between the two magnets. To do this, we placed books at the end of the air track, causing it to incline. Using our phones, we measured the angle at which it was inclining. Once the magnet attached to the glider was at its maximum distance from the magnet attached to the air track, we measured how far apart they were with a ruler. We did this five times, which was sufficient enough to use in LoggerPro. (see Fig. 3 and Fig. 4)
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| Fig. 4 We had LoggerPro calculate the force. |
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| Fig. 3 Using a ruler, we measured the distance between the two magnets at each incline. |
Once we had our data table set, we plotted force vs distance, which we curve fit and found an equation of F = 0.0002036r^-1.871 (see Fig. 5). Once we had this function, we plugged it into F = -du/dr and derived it to find our magnetic potential energy function of U(r)= 0.0002338r^-0.871 (see Fig. 6).
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| Fig. 5 Using this graph, we were able to find a the magnetic force as a function of distance between the two magnets. |
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| Fig. 7 Since we knew that F=-du/dr, we were able to find our function for U using integration. |
Once we had our magnetic potential energy function, we had to test if it actually worked. To do this, we would set up the air track horizontally with the glider on top as before. With an aluminum reflector attached to the glider, we set up a motion sensor that would measure position and velocity of the glider. If the glider slid across the air track at a constant speed, LoggerPro would be able to calculate our kinetic and magnetic potential energy. Since energy was to be conserved, the sum of kinetic energy and magnetic potential energy throughout the sliding of the glider should remain constant. That is, the total energy should remain constant. This would verify that our magnetic potential energy function was good.
The setup is relatively simple (see Fig. 7):
- Place the air track vertical on the lab bench, again with the air glider on top.
- Place the motion sensor facing the glider at the end of the air track which had the magnet.
- Slightly push the glider across the friction-less surface and allow it to reach the other magnet and repel back.
- Using the data obtained from the motion sensor, calculate KE, U(r), and total energy.
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| Fig. 7 |
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| Fig. 8 Since the motion sensor could not calculate the separation between the two magnets directly, we had to find an expression that could describe it. |
When recording the data, we created a column that would calculate KE using 1/2m"velocity"^2, a column that would calculate the separation distance between the magnets (r) using "position"-0.3045, a column that would calculate U using 0.0002338r^-0.871, and a column that would calculate total energy by adding KE to U.
Once we had our columns ready, we gave the glider a slight push and began collecting data. We found that total energy was relatively horizontal toward the beginning and end of the run. However, once the magnet stopped the glider and pushed it in the opposite direction, there was a spike in the total energy. This meant that there was some uncertainty in our lab (see Fig. 9 and Fig. 10).
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| Fig. 9 The total energy rises as the velocity turns from negative to positive (the turn around point). |
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| Fig. 10 Our energy graphs. |
There were a few sources of uncertainty in this lab:
- In our function of force, we had uncertainty in both our measurement for theta (phone) and distance (ruler)
- Since the force function had uncertainty, the magnetic potential energy function U(r) had some uncertainty.
- Also, the value of "r" had some uncertainty since we measured it with a ruler.
- Most importantly, however, we assumed that the air track was friction-less. Since energy was not completely conserved, we can also conclude that the air track did indeed have some friction acting on it.
Conclusion:
Considering the imperfection of our environment, I would most certainly consider this lab a success. We were able to find a relatively decent function for magnetic potential energy, and although it didn't seem as if energy was conserved, this in fact makes sense. Considering that there must have been some friction acting on the system, it makes sense that the total energy spiked up when the glider was in its turn around period. The upward slope the total energy graph makes as it reaches this point suggests the presence of possible a frictional force acting on the glider.
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