17-April-2017: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy Lab

17-April-2017
Idea: Model the conservation of energy for a cart-track system with a magnetic potential energy, with an equation we do not yet have. 

Summary:

1. After leveling an air track like the one below, where a cart on an air track opposes a magnet attached to the other end, I tilt the track at various angles (recorded with a smartphone), and I obtain a Force vs Position graph by plotting these test cases. In Logger Pro, I use a power curve to fit this graph, and record the given uncertainties, and reasonably ascertain that this graph gives me the force exerted by the magnet.
   
2. Based on the values given by the curve I determine an approximate function for Force, which refers to the magnet. Its integral would give me the PE of the magnet.
3. Given this function, I now attach an aluminum reflector to the cart, level the track again, weigh my cart, and run the motion detector whilst fixed near to the magnet.
4. From the distance that the motion detector records, I would subtract the distance between the sensor and the magnet in order to obtain the distance between the cart and the track. This is important in order to calculate the Potential Energy. PE = F*x
5. I create a new calculated column given this distance. I then turn on the air track, and give the cart a gentle push.
6. I am then able to obtain a single graph showing KE, PE, and total energy (PE + KE) of the system as functions of time.

Analysis/Explanations:

1) From the first part of the lab wherein we recorded the force at various angles of the incline air track, we obtained a force vs position power fit graph that gave us a value of f(r) as a function of y: 4.281*10^-5 * r^-2.282.

• By graphing Force vs Position, we could obtain it as a function of position along the track. This, we could reasonably assume, was the only force present in the track-system, assuming that friction was zero and loss of energy in general was zero. Therefore, we could further infer that this force must reference the force exerted against the cart by the magnet. This is why it was important that we record record the force using our force sensor.

2) By taking the integral of this function of force, we could obtain the PE of this magnetic force over the distance along the track. Doing so gave me a value for PE of 3.34*10^-1.282 * r^-1.282. From our base assumption, that in this case we could find the PE of the magnet by measuring force and taking its integral, we should see that its change in energy should be equal and opposite to KE. Furthermore, we expect the sums of PE and KE to result in a straight line, because as the KE of the cart decreases, we expect the PE of the magnet to increase. 

3) Therefore, we graph KE, PE, and their sum in order to visually demonstrate this idea.

After graphing the following: KE, PE, and KE+PE vs time, we get a visual representation of the two and if our assumptions were true, then we expect KE to be the opposite of PE, and their sums should result in a straight line.

4) Judging from the graphs that we got, we see that this is accurate: the PE of the magnet peaks just as the cart's KE is at its minimum, and their sum, while not exactly a straight line, demonstrates our base assumption as aforementioned.

Measured Data:

Distance between the magnets at various angles, Theta

Calculated Data:

Calculating the distance(s) between the cart and the magnet for each case.

Integration of function of Force, resulting in a PE of 3.34*10^-5 * r^-1.282

Graphs:

Graph of Force vs Position. The power fit curve as aforementioned gives a function of force as 4.281*10^-5 * r^-2.282.

Graph of KE, PE, and KE+PE.

Conclusion:

We were able to model conservation of energy for a cart-track system and extrapolate from this the equation of PE for our magnet, something we did not know beforehand, because our graphs showed that the change in energy of KE and PE were largely equal and opposite. Therefore, their sum resulted in a graph where we could see this occur. While the graph of their sums isn't a perfect line by any means, it adequately demonstrates this idea with the fact that the graph of KE reaches its absolute minimum just as PE reaches its maximum.

• With sources of uncertainty, there were the usual culprits like the scale, which has an uncertainty to about 0.1 grams. It would damage the graph of KE, as it is the product of a-half, mass, and velocity squared. However, I'd imagine that it didn't hurt the accuracy of my graph very much because this lab was worked entirely in meters, kilograms, etc, units larger upwards to a magnitude of a thousand.
• The other comes from the 'protractor' apps on our phones, as the accuracy of its readings depend on:
• The hardware. For a lot of smartphones the G sensor isn't meant to be fine tuned for accurate scientific measurements.
• The App itself. Various apps had wildly varying levels of accuracy. With the app we used, we determined it had an uncertainty of .1 degrees. Overall, while this would not directly damage the accuracy of our calculations later on, it does create an inaccurate representation of the system.
• The Force sensors and motion tracker. The former was inaccurate to at least.001 N and as a result it would lead to my integration being inaccurate, which means an inaccurate value of PE. The latter, I would not be aware of any errors, but if there were, it would of course lead to an inaccurate KE graph.
• Friction between the cart and the track. Although with the track running, we can reasonably assume friction to be zero, we realize that it isn't a perfect system and there must be, to a certain extent, a loss of energy to such forces. This would affect our calculations for the PE due to force readings that rely on how much the cart accelerates on its way across the track.