The purpose of this lab is to verify conservation of energy applies to this system that uses magnets. Since we do not have a formula for magnetic potential energy, one will have to be derived to prove energy is conserved. Fortunately, for any system where there is non constant potential energy , the potential energy can be defined by a force F as the relationship between the two is that that the negative integral of F(r) dr is equal to potential energy. So now, all we have to do to find magnetic potential energy is to find F(r).
Set up:
A glider was placed on an air track. On one end of the air track was a strong magnet. When the glider was closest to the magnet, the glider's kinetic energy was zero and all the energy in the system was transformed into magnetic potential energy. So first we had the air track level to take initial measurements. But for the majority of the time the track was at an angle. We also attached a reflector to the glider so that its motion could be recorded easily by the motion detector at the end of the air track.
Procedure:
1) First, we leveled our air track and measured the height.
2) Next we tilted the air track. We tilted it because when tilting the track, the glider will end up at some equilibrium position where the magnetic repulsion force between the two magnets well equal the gravitational force component of the glider parallel to the track. We again measured the height from and compared it to first measurement of height.
3) We placed a glider with a strong magnet and an aluminum reflector on the air track while it was tilted and measured the distance between the fixed magnet on the air track and the opposing magnet of the glider.
plotted a graph of F vs r. We assumed that the relationship of a power law such that F=A(r^n)
4) We then discovered the values of A and n from a curve fit of our graph. From this we could develop a function of potential energy (U(r)) for the interaction between the magnets.
5) Next we leveled our track again and placed the glider on the air track close to the fixed magnet and ran the motion detector, We set the motion detector to record 30 measurements/second. We then ran the motion detector and gave the cart a gentle push towards the fixed magnet and were able to record the glider's speed and distance from the magnet.
6) With this data we were able to prove energy was conserved.
Results:
These are the results we achieved when we were simply trying to plot a graph of F vs r. We obtained r by measuring the distance between the two magnets, mass of the glider by using a scale, and theta by using an angle measuring device on a classmate's phone. With this information we were able to get the gravitational force on the glider.
Now that we knew Force and r were were able to graph a Force (N) vs r(m) graph:
With our force and distance r measurements we created the above graph and assumed the graph was a power law graph. Therefore we assumed force was defined by the function F=Ar^B and with our power fit we were able to find that A=3.98*10^-5 and B=-2.71.
We can now find U(r) or potential energy using these values:
Since we just derived the formula for magnetic potential energy, we still had to show energy was conserved. We collected data from running the motion detector and pushing the glider towards the magnet while the track was level which included distances from the magnets and the velocity at which the glider approached the magnet. Using these values and the above equation for magnetic potential energy we were able to create graphs that proved energy was conserved:
You'll notice we were able to graph Kinetic Energy (yellow), Potential Energy (purple), and Total Energy (red). The graph of KE was calculated by .5 x (mass of glider) x (velocity recorded by motion detector). PE used the above formula and was calculated by 0.0000233 x ((distance recorded by motion detector)-(distance magnet is from motion detector))^ (-1.711). And lastly TE was calculated by (KE)+ (PE).
Conclusion:
As you can see from the above graph, the dip in KE almost mirrors the hill in PE. And although the graph of TE is not completely level, it does not fluctuate wildly. In general, it can be said energy was conserved. The graph is also effective because when KE= 0, the PE is at its maximum value. At this poing, all KE has been transferred to magnetic potential energy as hypothesized.
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