Lab 13: Magnetic Potential Energy

Thus far, we have stuck to energy forms with neat, predefined formulas to calculate using the work-energy theorem. In fact, all kinds of energy can be used in that way. This time, we demonstrate that this is true with magnetic potential energy, whose formula we derive. We do this by experimentally recording magnetic force under different circumstances. Magnetic force which repels two objects will have magnetic potential energy, since the objects, which have the inclination to move towards each other, is being forced apart by a distance. As before, we graph this relation between force and distance and integrate to find the work caused by the energy.

To do this successful, we must first reasonably eliminate friction. Pictured above is an air track with little holes connected to a blower, which pushes air through those holes. The red cart on the air track will approach 0 friction, such that it could be slid across the track unimpeded until it hits the end. At the ends of the air track and the cart are magnets which repel each other, causing the cart to bounce elastically from the end even if the magnitude of velocity overcomes the repelling force. However, first, we'd like to discover the relation between this magnetic force and distance, so to alter the role of gravity, we lift the air track to different angles and record how far the cart is held from the ends. The force of the magnet is, of course, mg sinθ. We label distance x, in meters.

The air track is lifted by placing a number of books under the other end. This rudimentary method to change the angle is counteracted by the advanced cellphone angle measuring apps that put Inspector Gadget out of business. The measured angles should be accurate relative to each other, but we expect an offset due to the geometry of the phone, and that the table isn't exactly flat. There is a ruler taped to the end of the air track, wherein the position of the magnet measured at 527 mm. This aided in measuring the distance, for all we had to do was measure the position of the magnet on the cart in the same way, and find the difference. We recorded 8 data points, although the first or last may not be as accurate, as we later found. The data is as follows:
 

Position (mm)	x (m)	θ (º)	sinθ
476	0.051	1.1	0.0192
487	0.040	2.7	0.0471
492	0.035	4.9	0.0854
497	0.030	6.9	0.1201
500	0.027	8.8	0.1530
502	0.025	10.5	0.1822
503	0.024	12.1	0.2096
506	0.021	14.0	0.2419

 Now that we've gathered the data, we can plot the graph to find the relationship:

We set a curved fit that we expect the data to conform to. The data doesn't exactly fit, but it is reasonably close considering the experiment. Here, we've found our magnetic force equation to be F = 3.168 x 10^-5 r^-2.690. We integrate and flip the signs to get the potential energy:

We can now test our magnetic potential energy against kinetic energy to see whether energy is conserved like we expect it to. If our calculations are close, then we should see the pattern of conservation. To do so, we attach a motion sensor to the end of the air track and measure its offsets from the magnet. We need to know the differences to accurately reflect distance. We also make sure the air track is horizontally flat, such that the cart doesn't move due to gravity.

We measure that the back plate or our cart, which the infra-red beam from the motion sensor would hit, is at position 433 mm. Thus, if we subtract it from the end position at 527 mm, so have 94 mm. This figure will be entered into Logger Pro to calibrate the motion sensor.

The thinking is this: If we give the cart a push, then record its movement with the motion sensor, we should get its position (distance) and velocity, which should be enough to calculate the potential energy and kinetic energy, respectively. Entering new calculated columns:

And summing up total energy, we get:

The cart hitting the end of the air track causes the dip in kinetic energy, and at the same time increases magnetic potential energy since the magnets approach each other. We can see that the energies are inverse of each other, causing the sum to be relatively even compared to the total energy in the middle of the track. The larger spikes in energy can be explained by gaps in the motion capture, causing Logger Pro to interpolate and guess what's there. The smaller spikes are likely explained by the previous inaccurate modeling of the force, possibly due to loose measurements of the angles. Also, despite the air track, friction isn't completely eliminated, nor air resistance.

Here is a snapshot of our data, exported to a CSV file, with the gaps taken out (which have 0.001 J kinetic energy). As you can see, the total energy is even.

Time	Position	Velocity	Acceleration	Kinetic Energy	Potential Energy	Total Energy
1.65	0.625	-0.268	0.109	0.013	0	0.013
1.7	0.608	-0.252	0.216	0.011	0	0.011
1.8	0.587	-0.248	-0.111	0.011	0	0.011
1.85	0.574	-0.254	-0.067	0.011	0	0.012
1.9	0.561	-0.255	0.008	0.011	0	0.012
1.95	0.548	-0.252	0.058	0.011	0	0.011
2	0.536	-0.247	0.058	0.011	0	0.011
2.1	0.513	-0.251	-0.096	0.011	0	0.011
2.15	0.498	-0.259	-0.011	0.012	0	0.012
2.2	0.486	-0.251	0.059	0.011	0	0.012
2.25	0.473	-0.248	0.046	0.011	0.001	0.011
2.3	0.461	-0.247	0.029	0.011	0.001	0.011
2.35	0.449	-0.246	0.027	0.011	0.001	0.011
2.4	0.437	-0.245	0.041	0.01	0.001	0.011
2.45	0.424	-0.243	0.071	0.01	0.001	0.012
2.5	0.412	-0.238	0.117	0.01	0.001	0.011
2.55	0.4	-0.232	0.187	0.009	0.002	0.011
2.6	0.389	-0.221	0.317	0.009	0.003	0.011
2.65	0.378	-0.205	0.573	0.007	0.004	0.012
2.7	0.368	-0.172	0.999	0.005	0.007	0.012
2.75	0.359	-0.108	1.461	0.002	0.011	0.014
2.85	0.358	0.08	1.573	0.001	0.013	0.014
2.9	0.365	0.154	1.15	0.004	0.008	0.012
2.95	0.374	0.196	0.677	0.007	0.005	0.012
3	0.385	0.215	0.332	0.008	0.003	0.011
3.05	0.396	0.221	0.168	0.009	0.002	0.011
3.1	0.407	0.227	0.112	0.009	0.002	0.011
3.15	0.419	0.232	0.076	0.009	0.001	0.011
3.2	0.431	0.235	0.043	0.009	0.001	0.011
3.25	0.443	0.236	0.018	0.009	0.001	0.011
3.3	0.454	0.236	0.007	0.009	0.001	0.011
3.35	0.466	0.237	-0.007	0.009	0.001	0.011
3.4	0.478	0.235	-0.01	0.008	0.001	0.011