13-April-2015: Magnetic potential energy

     The purpose of this experiment was to verify that energy was conserved for a magnetic system. In this experiment, a cart has a magnet on one end of it and approaches another stationary magnet. When the cart gets close to the magnet, the energy goes all into magnetic potential energy, which repels the cart back the way it came. The problem is that there is no equation for magnetic potential energy (MPE) like there is for gravitational or elastic potential energy. So, one needed to be found.
     There exists a relationship between force and potential energy, and it is that potential energy is the negative integral of force. For instance, we know that the force of a spring is defined as F=-kx. If the negative integral is taken you get (1/2)kx^2, which we know to be the equation for elastic potential energy. The same approach was taken to find the MPE. First, lets look at the setup.
     The setup was an air track and a glider cart. The track worked by blowing air through it so that the air comes out of tiny holes, lifting the cart, and essentially making the system frictionless.

setup of glider cart on air track

     The glider cart had a piece of square metal attached to the top so a motion sensor could detect it. It also had a magnet attached to one end (the small circle with tape over it).

     At the very end of the air track was another magnet with the same polarity. A motion sensor was set up on the same side as this magnet (shown below).

     The idea was to come up with some force function that could be integrated to give potential energy. One way to find the force of the magnets was to lift up the end of the track that did not have the magnet and let gravity push the cart to approach the magnet. The cart will not touch the magnet since the two magnets are repelling each other, but the cart will eventually come to rest at an equilibrium point some distance, r, from the stationary magnet. At that point the only forces acting on the cart are gravity, which pushes the cart toward the magnet, and the force of the magnet pushing back on the cart. Since the cart is in equilibrium, the two forces equal each other.
     By tilting the cart at various angles, a set of distances between the magnets were recorded as well as the force from them. The force was calculated by the equation mgsin(theta) that could be found from drawing a free-body diagram of the tilted cart and track. The Force and distance (r) data was put into Logger Pro and graphed. Below is that graph.

The graph curved much like an exponential graph, so a power fit was done. This means the equation for the force of the magnets was going to be

The power fit provided constants for A and n. They can be seen in the data set below.

Taking the negative integral of the force function gave a MPE function of

     A way needed to be found to measure the speed of the cart and the distance between the magnets. The motion sensor was used for this step. The cart was placed so that the distance between the two magnets was 10 cm. The motion sensor was activated and the distance, d, from the motion sensor to the piece of metal on the cart was recorded. That distance, d, minus the 10 cm would be used to calculate r in the MPE function above. So, r was defined as the position, x, of the cart at any given time minus (d-0.1m), so r=x-(d-0.1). Note that (d-0.1) is a constant. Since the motion sensor can measure the distance, it can measure velocity to find the kinetic energy (KE) of the cart. The mass of the cart was found with an electronic balance.
     All equations were found to figure out if energy is conserved. All that was left was to put the equations in new calculated columns and then to do a trial. The columns in Logger Pro are below.

TE is the total energy of the system.

     The motion sensor was activated and the cart was given a gentle push toward the stationary magnet. The cart got close to the magnet, momentarily stopped so that kinetic energy was zero, and was then repelled back by the magnet. You can see from the graph below that as kinetic energy approaches zero, magnetic potential energy goes up to its highest point.

     Adding these two energies gives a line for Total Energy, which was straight across. It was not perfectly straight due to errors and uncertainty but it was close enough to tell us that energy was indeed conserved for the magnetic system. Sources for uncertainty included the measurements made for the angle at which the track was lifted, the measurements with a ruler for the distance between magnets, and the subsequent calculations for force of the magnets. The air track was also not perfectly frictionless, which was an assumption made for the experiment.