17-Apr-2017: Lab 13: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy

Joel Cook

Lynel Ornedo

Eric Chong

Nina Song

Lab performed April 17, 2017

We do not have an equation for the magnetic potential energy that exists between two magnets repelling each other (same polarity). In this lab we will attempt to model the magnetic potential energy between a pair of magnets to verify that energy is conserved in our experiment.

Procedure: In this lab, we used an air glider and air track. By pushing air up through holes in the track, the glider and move along the track with almost no friction. The photos below show the track, glider, and magnets.

With the magnets repelling each other, a force exists as a function of the separation distance between the magnets. The force changes as the separation distance changes. To determine the potential energy, we will need to find the negative integral of the force, F(r), from infinity to r, where r is the separation distance. We will assume that the force is zero when the separation distance goes to infinity.

To find the integral, first we need to determine F(r). We measured the mass of the cart and used calipers to measure the separation distance between the magnets. If we raise the end of the track opposite the magnets, the force exerted by the magnet to separate the magnets is equal to the x component of the force gravity is exerting on the cart. An smart phone application was used to digitally measure the angle at each position.

F = mgsinΘ

The track was raised to various angles. The angles were measured and recorded, as well as the separation distance between the magnets at that angle. The force at each angle was calculated from the previous equation. Next, the a graph was created for Force vs Separation Distance. Using logger pro, a power fit of the line created shows gives F(r).

Now that we have a function for separation force as a function of separation distance, we can verify the conservation of energy in our system. First we positioned a motion sensor at the end of the track  (as shown above) and determined the the relationship between the distance from the motion sensor to the cart and the separation distance. With the air to the track turned off, the distance was measured using the motion sensor and logger pro. Calipers were used to measure the separation distance and a calculated column was created for the actual separation distance based on the distance the motion sensor is reading. 

Separation distance = (Distance from cart to sensor) - (Difference between sensor distance and actual separation distance)

Using separation distance, a calculated column was created for U(r), magnetic potential energy as a function of separation distance, from the integral of F(r).

Next, with the air track leveled, the air was turned on to the air track. The cart was pushed toward the motion sensor and allowed to "bounce" back. The motion sensor was then used to plot the speed of the cart as it travelled on the track. The speed of the cart was used to plot kinetic energy of the cart by creating a calculated column KE = 1/2*mass*speed^2.

We expect that the energy in the system will be conserved. Graphs were plotted for KE, PE, and total energy to compare the values and verify that energy was conserved.

Data:

The following table shows the data for the separation distances that were gathered at various angles:

Θ (degrees)	r (mm)	F (N)
1.7	28.0	0.100
2.2	24.6	0.129
3.5	21.3	0.206
4.2	19.7	0.247
5.1	18.5	0.300
7.2	16.3	0.423
11.6	13.0	0.678

The mass of the cart was measured as 0.344 kg using a digital scale. As previously stated, F was calculated as the following sample shows:

F = mgsinΘ

F = 0.344 * 9.8 * sin(1.7)

F = 0.1000

The following graph was created for Force (N) vs Separation Distance (m).

As the graph shows, a power fit of the graphed data points shows the following function:

F(r) = 1.864 * 10^(-5) r ^ (-2.421)

The uncertainty in the first term was +/- 6.337 * 10^(-6).

The uncertainty in the exponent was +/- 0.08114.

To determine magnetic potential energy as a function of separation distance, the negative integral of the function F(r) from 0 to r was taken as shown:

The photo above shows the three graphs that were created from pushing the cart towards the motion sensor and letting it "bounce" off the other magnet. The first graph shows position (m) vs time (s). 

The second graph shows KE, PE [U(r)], and Total Energy (PE + KE) vs position (m). As the graph shows, as the cart moved toward the end of the track, the potential energy increases until the "collision" and then decreases. The kinetic energy is approximately constant and then decreases until the "collision" and then increases again to a value slightly lower than the initial kinetic energy. The total energy is approximately constant.

The third graph shows KE, PE [U(r)], and Total Energy (PE + KE) vs time (s). As the graph shows, the potential energy increases until the collision at about 1.7 seconds and then decreases. The kinetic energy decreases until the collision (KE is zero at collision) and then increases again as speed increases. The total energy is approximately constant.

Analysis/Conclusions:

As the graph shows, the model we developed for magnetic potential energy as a function of separation distance for our pair of magnets was accurate. The kinetic energy of the system as a function of time, or position, changed in equal magnitude to the potential energy and the total energy was approximately consistent. The conservation of energy in our system proves that our model was accurate. This also confirms that our method for developing force as a function of separation distance was accurate.

Although uncertainty exists in our lab, the experiment was successful. First, it is not possible that our track be completely frictionless. As the glider moves along the track it is cutting streams of air that may affect the speed of the glider. From our graph of velocity vs time it is evident that our glider is losing some speed as it travels. Second, the most accurate data points used to calculate F(r), were the points at a larger angle. This is due to the fact that when the glider collides with the magnet in the second portion, it has a smaller separation distance. Therefore, the larger angles will produce a smaller separation distance and will thus be comparable to the force during collision. Finally, errors and uncertainty exist inherently in our measurements. Angles were measured to the 1/10th of a degree and distance was measured with digital calipers to the 1/10th of a millimeter. As shown in the graph previously, uncertainty in our function of F(r) was calculated by logger pro and reported in the lab earlier. Even though all these errors and uncertainties exist, and possibly others, our results were sufficiently accurate and successful in confirming conservation of energy in our system.