The purpose of this experiment was to derive an equation that models magnetic potential energy by relating potential energy to force. This model was then used to prove that the conservation of energy principle applies to the system used in the experiment.
Apparatus:
Figure 1: Apparatus of this experiment
The apparatus of this experiment consisted of a metal track shaped as a wide triangle that is hollow inside and contains many holes in it. These holes are to allow air to run through the track and out of it so as to minimize almost all friction forces between the track and a gliding object. It is also shaped as a triangle to keep a gliding object from going off course. A small magnet was fixed in place onto the track. An air track glider containing a magnet and a metal sheet was also used in order to derive the magnetic potential energy function and determine if conservation of energy applies to this system. An air pump was also used to run air through the air track. Textbooks were used to change the angle of the air track and a digital protractor was used to measure that angle. Lastly, a motion detector and a computer with Logger Pro were needed to provide data for the calculations of the energies of the system.
Procedure (Part 1):
Once the apparatus was set up, the air track was set at an angle and the track glider was allowed to come close to the fixed magnetic and come to rest while the air track was on. Then, the relationship between the magnetic potential energy and force was identified in this system by noticing that the force applied by the glider's weight parallel to the track is equal to the magnetic force applied on the glider, or mgsin(theta) = Fmag. Then, using this force relationship, a graph of magnetic force vs. the separation distance between the magnet on the cart and the one fixed on the air track was created. The data used to plot this graph was found by changing the angle the air track is sloped on, since, looking at mgsin(theta), the only value that can be changed is theta. Once the slope of the track was changed, the angle was measured via the digital protractor and the distance between the faces of the magnets was measured through the use of a ruler. The mass of the glider also had to be measured, which was found to be 0.343 kg. The data gathered along with the graph created by that data is shown below:
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Angle (degrees)
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Force (N)
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1.30
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0.0763
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2.80
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0.164
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5.90
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0.346
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10.5
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0.613
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Figure 2: Graph of force vs separation distance of magnets (r)
Discussion (Part 1):
A sample calculation for the force based on the angle is shown. Using theta = 10.5 degrees, the force was found to be F = (0.343 kg) (9.80 m/s/s) (sin(10.5)) = 0.613 N.
The last two data points had to be crossed out since they resulted in a large deviation from the power fit. The force as a function of r equation was assumed to be a power law based on the slope of the curve created by the data points. The force of the magnets was assumed to take the form of F = Ar^n, where A and n are obtained from the power fit. The value of A was measured to be 8.464 x 10^-7 +/- 6.325 x 10^-7. The value for n was found to be -3.213 +/- 0.1808. The uncertainty in B (5.6%) is reasonable compared to the errors and uncertainties in this system. These uncertainties/errors include assuming that no friction is present between the track and glider when in fact there was (just not so large), uncertainty in the measurement of the angles, and uncertainty in the measurement of the separation distances between the magnets. On the other hand, as can be seen, the uncertainty in the value for A (74.7%) is very large. This is primarily caused by the fact that A is a very small value (10^-7!). It is also caused by simply the use of inconsistent data, which is why the last two data points had to be crossed out. Taking the rest of the errors/uncertainties into account, these resulted in a large uncertainty in the value of A.
The equation for the magnetic force was found to be F(r) = (8.464 x 10^-7) r^-3.213. Taking the negative integral of the force function with respect to r, the potential energy function between the magnets can be found. The potential energy function was derived to be U(r) = (3.825 x 10^-7) r^-2.213.
Procedure (Part 2):
Next, the track was made so that it is flat (0 degrees) and the glider does not slide when left alone. Then, a motion detector was applied to the side of the track with the fixed magnet to be able to measure the position and speed of the glider. With the air track off, the glider was placed close enough to the fixed magnet so that the relationship between the separation distance between the magnets and the distance between the glider and motion detector were found. The distance between the motion detector and glider was measured to be 0.702 m and the separation distance of the magnets was found to be 0.149 m. Subtracting the two, it was found that the relationship was 0.553 m.
Then, the glider was moved to the far end of the track and given a small push. The motion detector was run, and the position and velocity of the glider were measured and collected. Then, the KE of the cart was found using the measured velocity and the magnetic potential energy was found by using the derived function, where r equals the position data from the motion detector minus the relationship between the separation distance of the magnets and the separation between the motion detector and glider (0.553 m). For example, at 1.20 seconds, the kinetic energy of the cart is 0.5 (0.343 kg) (-0.259 m/s)^2 = 0.0115 J. The magnetic potential energy is (3.825 x 10^-7) (0.791-0.553 m)^-2.313 = 0.00 J. The data recorded and calculated as well as the graph are shown below:
Discussion (Part 2):
Figure 3: Recorded and calculated data of the system
Figure 4: KE, U magnetic, and total energy of the system
The kinetic energy and magnetic potential energy graphs were expected. At the point of "collision" the kinetic energy would drop down to zero and then come back up. In addition, it was expected that U would be almost zero, increase up dramatically during the "collision", then fall back down to nearly zero, as seen in the graph. However, what was not expected was that the total energy would decrease greatly in such a small amount of time (from 0.012 to 0.008 J). Again, this is probably caused by the fact that the energy is so small and since there is some friction and air resistance, some of the energy is lost over time. In addition, the magnets are not ideal, resulting in not all of the energy in the magnets being conserved and then thrown back into kinetic energy of the glider. However, due to the fact that there was a large uncertainty in A, it was surprising to find that the energy graph was a fairly accurate representation of the conservation of energy principle.
Conclusion:
As was seen, although there was high uncertainty for the value of A for the potential energy function, the magnetic potential energy function turned out to be a fairly accurate model of the system. In addition, it was proved that the conservation of energy principle also applies to this system. The largest reason for the uncertainty and error in this experiment is not that the uncertainty and error was large, but the fact that the values were very small, so that any uncertainty would have a large effect on the error in the values. These errors include the assumption that the track is frictionless and that there is no air resistance, when in fact there is both of some degree. In addition, there was also uncertainty in the measurement of the angle when the track was sloped and uncertainty in the measurement of the separation distance between the magnets. In addition, the assumption that the magnets are ideal in that all kinetic energy becomes potential magnetic energy and then all back to kinetic energy is not a reasonable assumption, resulting in error in the energy values. However, if non-conservative work was taken into account, it would be expected that the conservation of energy principle would be better proven and the total energy would remain constant.
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