8/10/2016 - Magnetic Potential Energy (Lab 13)

PURPOSE:
To verify that conservation of energy applies to a system with magnetic potential energy (MPE).

THEORY:
Consider a cart with a strong magnet sliding along a level air track with another magnet of the same polarity fixed at the other end. As the cart moves along the track, it will have some KE. However, as the cart approached the end of the track, its KE will reduce to zero and all of the energy will be stored in the magnetic field as MPE, then it rebounds back. Since the MPE is not constant, we will use the relationship U(r) = -∫ [r,∞] F(r)dr to relate our MPE to the separation distance, r.

APPARATUS:
We used a glider on a frictionless track. The glider has a strong magnet attached to one end and at the end of the track there is another magnet with the same polarity. On top of the glider, there is a thin aluminum plate that is used to facilitate the collection of position data with a motion sensor, which is located at the same end of the track as the magnet.

EXPERIMENTAL PROCEDURE:
Part 1: Force Equation

1. Prepare apparatus as shown.
2. Weigh cart+reflector.
3. Connect vacuum hose to air track.
4. Calibrate and connect motion detector with Lab/LoggerPro.
5. Tilt the air track at various angles in order to create a relationship between the magnetic force, F and separation distance, r.
6. For a given angle θ, use calipers to record the separation distance, r between the two magnets.
7. Plot a graph of F vs. r. We assume that their relationship is a power type equation of the form: F = Arⁿ. 
8. Get the A and n values from the curve fit of your F vs. r graph. 
9. Integrate the force function to get your equation for the MPE. 

Part 2: Verification

1. With the air turned off, place the cart+reflector reasonably close to the fixed magnet at the end of the track. Run the motion detector. Determine the relationship between the the distance the motion detector reads and the separation distance between the two magnets. Assuming that the distance between the the reflector and magnet of the cart is negligible, the distance r = P - k.
2. Use the motion detector and LoggerPro to measure the speed and the separation between the two magnets.
3. Start the cart at the far end of the track, turn on the air track, and give the cart a gentle push.
4. Record data necessary to to verify that energy is conserved.
5. Create a graph of KE, MPE, and total energy as a function of time.

DATA/GRAPHS:

Data from Part 1

Sample Calculation of Fmag (Part 1)

Curve Fit of Force vs. Time

Derivation of Magnetic Potential Energy Function

KE-MPE-TE vs. Time Graph

ANALYSIS:
In order to verify that energy was conserved for this experiment, all of the KE of the cart must be transferred into MPE when the cart is just about to rebound. As the cart travels down the air track at a constant speed, the kinetic energy remains constant. Moreover, as the cart approaches its minimum distance between the two magnets, the KE exponentially decreases as the MPE increases at an identical rate. Since it appears that the KE reaches zero at the same time the MPE reaches its maximum, it is safe to conclude that energy was conserved for this experiment. Another way we know that energy was conserved is if the total energy (TE) of the system remains constant. If the TE of the system decreased over time, then this would imply that a net external force acted on the system, causing some of the initial KE to be lost in the process. However, since our line for the total system energy stayed relatively constant throughout, it is safe for us to conclude that energy is conserved.

CONCLUSION:
Our group was able to successfully create an equation for the MPE of the system that showcased the energy of the system was relatively conserved. However, there is some uncertainty in our calculations because our line for the total energy of the system is not as straight as we had hoped. This tells us that there was most likely a net external force acting on the system from sources we did not take into account. For example, if the track was unleveled or was not uniformly frictionless, then the speed of the cart could have been hindered by friction or gravity. Furthermore, the inaccuracies in our energy calculations could have also resulted from the imprecise measurement of the separation distance, r, the distance between the motion detector and the magnet, the angle of incline, or even the mass of the  cart+reflector. If we used more precise equipment, a perfectly level and frictionless surface, and minimal human error, then we could expect to see a more straight TE line, indicating that the energy of our system was indeed fully conserved.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.