17-April-2017: LAB 13: PHYS 4A--Magnetic Potential Energy Lab

Eric Chong

LAB 13: PHYS 4A--Magnetic Potential Energy Lab

Lab Partners: Lynel Ornedo, Nina Song, and Joel Cook

Purpose: The goal of this lab is to verify that the conservation of energy applies to this system and to determine an appropriate function U(r) for the interaction between the magnets.

Introduction/Theory: The conservation of energy is a concept that says that energy is neither created nor destroyed. Energy is instead converted to other forms, such as gravitational potential energy, kinetic energy, spring potential energy, and magnetic potential energy. For the case of magnetic potential energy, there is no given equation that determines the amount of magnetic potential energy, unlike gravitational or elastic potential energy. Thus, given what we know, that for any system that has a non-constant potential energy, the potential energy is determined by the interaction of the force. By taking the integral of a force equation, we can find the equation for magnetic potential energy. Therefore, the challenge is to first find the force equation through observations and analysis of the relationship between the force of gravity and the magnetic force going against it. We can then take the integral of that equation from infinity to r, where r is the separation distance, and determine the equation of magnetic potential energy. Note that this equation will be applicable only to this type of magnet. Any other magnets would most likely have different equations. Finally, the last part of the lab is to verify that the conservation of energy is applied to this system. To do so, we can analyze the kinetic energy of the glider and observing its motion when a magnetic force is applied to the cart to reverse its motion.

Procedure: For the setup of the lab, we used a glider on an air track with a magnet attached to its end. Another magnet is also attached to one end of the air track, so the magnet on the glider will repel the magnet on the end of air track. A motion sensor is also attached to the same end of the air track for the second part of the lab. Here is what the setup looks like:





To determine the equation of U(r), we first measured the mass of the glider, and then raised the end of the track without the motion sensor and magnet to a certain height and measured the angle the track makes with the table. We also measured the distance the glider is from the magnet at the end of the track, since the magnets repel each other and the glider cannot enter a certain distance. We kept raising the track and measuring the angle and distance of the glider until we had 7 data points. Afterwards, we calculated the force of the magnet by setting the component of the gravitational force parallel to the track equal to the force of the magnet for all of the data points. Here are our data measurements:



We then plotted the force vs. distance graph, with the force being the magnetic force, and the distance being the distance between the glider and the magnet. Here is what the graph looks like:



We expected that the graph would have an equation of F=Ar^n, and as expected, we see that the data points created a power curve, with A being 1.864*10^-5 +/- 6.337*10^-6, and n being -2.421 +/- 0.08114. With this, we determined the magnetic force equation. Now we can determine the equation U(r) of the magnet by taking the integral from infinity to r. Here is the calculation:

Notice that the integral is negative, and this is because the force is going in the opposite direction of motion. Also, the infinity portion turns to zero because the exponent of little "a," when the numbers are plugged in, is negative, and this turns it to the denominator, and the limit when "a" goes to infinity is zero.

Now that we have the equation for U(r), we can verify that the conservation of energy. We first setup the glider at a reasonably close distance to the magnet. Then, we set up the motion detector to track the glider's movement. Then, we determined the relationship between the distance the motion detector reads and the separation distance between the magnets. From here, we now have a way to measure both the speed of the cart and the separation between the magnets at the same time. We then placed the glider at the other end of the track and made sure the motion sensor could detect it. Afterwards, we gave the glider a gentle push and recorded its movements on LoggerPro. On LoggerPro, we made a single graph showing Kinetic Energy, Potential Energy, and the total energy of the system as a function of time. We also included a graph with the position as the x-axis, instead of time, in order to see if there is a loss of energy as we gain distance. Here are what the graphs looks like:

As expected, the third graph showcases the magnetic potential energy rising as the glider gets close to the magnet, and then dropping once the glider starts moving away, and the kinetic energy dropping as it gets to the magnet, and then rising as it starts moving in the other direction, all while the total energy being relatively the same. However, from the second graph, we can see that there is a slight loss of total energy (the blue line), and this could be due to friction and air resistance.

Conclusion: From the experiment, there are a lot of uncertainties that could be found. For instance, the calculations to get the force equation is very delicate. Since we measured the distance the magnets are from each other in millimeters, a slight change in the numbers could make the uncertainty very high, and further make the rest of the calculations for U(r) to be off from what we expected. Also, in the second part of the experiment, we observed that we lost some of the total energy, and we mentioned that this was due to friction and air resistance. The friction could come from the uneven surface of the track, and the air resistance could come from the air that comes blowing from the track and onto the metal plate attached to the glider, and thereby reducing the glider's speed.

Overall, the experiment taught us not only the concept of magnetic potential energy, but also the steps to solve for the potential energy equations. We first observed what we could use to derive an equation, and in this case it is the component of gravitational force that is parallel to the track. We then took the integral from infinity to r, in order to find the equation of U(r). From here, we could verify that the conservation of energy is applied to the system. This experiment has a lot of practical value in many other experiments, and from our learning this procedure, the experiment was a success.