15-April-2015: Magnetic Potential Energy Lab

Lab 12: Magnetic Potential Energy

Purpose: To verify that the conservation of energy exists within a magnetic system.

Apparatus:  For this experiment,we used a glider on an air track. The glider had a magnet attached to it and another magnet was fixed to one end of the track, we positioned a motion sensor at this point.

Air track with vacuum.

 Procedure:  In order to prove the conservation of energy in this system, we must calculate the KE of the glider and compare it to the magnetic PE between two repelling magnets. Since an equation for PE between two magnets does not exist, we must find one. We achieved this by conducting several runs, where we raised the track to different angles. From here, the glider would travel along the track and stop at distance away from the fixed magnet. This is due to the magnet attached to the glider. The the glider would stop due to the repelling forces between the magnets. At this point, we measured the angle of elevation (theta) and the separation distance between the magnets. We gathered a sufficient set of data so that we could derive an expression for the PE between two magnets. The rest of the lab was fairly simple. We leveled the track and calculated the KE of the glider with the aid of the motion sensor. Then we determined if energy was conserved by graphing the KE of the glider, our expression of PE between the magnetics, and the sum of the energies. If energy is conserved, the sum of the energies should be constant.

Run 1

Finding PE Magnetic 

In order to find PE Magnetic, we must derive an expression for the force acting on the glider based on the separation distance between the magnets F(r).

 

First, we found the forces acting on the glider by drawing a FBD.

• Fx (along slope): F=mgsin(theta)
• Fy: N=mgcos(theta)

Note: We assume the track is frictionless.

 

As mentioned above, several runs were conducted at different angles. We collected theta and separation distance(r) for each run.

 

Mass of cart (measured) = .347 kg

Theta was measured with phones.

For F, theta was converted into radians.

 

Run 1

• Theta = 3 +/- 0.1 degrees
• r = .036 m
• F =  0.178 N

Run 2

• Theta = 6.8 +/- 0.1 degrees
• r = .018 m
• F =  0.403 N

Run 3

• Theta = 11.6 +/- 0.1 degrees
• r = .036 m
• F =  0.684 N

Run 4

• Theta = 14.7 +/- 0.1 degrees
• r = .011 m
• F =  0.863 N

Run 5

• Theta = 19.0 +/- 0.1 degrees
• r = .0085 m
• F = 1.107 N

Run 6

• Theta =25.0 +/- 0.1 degrees
• r = .0065 m
• F = 1.437 N
   

F was calculated in LoggerPro, under a calculated column using F=mgsin(theta)

 

From here, we found an expression for F(r) by graphing  Force vs r on LoggerPro.

• x-axis: Force (N) = mgsin(theta)
• y-axis: r = separation distance

We plotted the points and took a power fit of the graph to find F(r). 

 

As you can see, our function is:  F(r) = .004344r^-1.157

Finally, we can find a function for the PE between the magnets [U(r)] simply by taking the integral of F(r)*dr.

• U(r) = 0.02766879r^-0.157

 

Now, we can finally verify the conservation of energy within the system.

First, we leveled the track and then placed a motion sensor near the fixed magnet at the end of the track. Then we ran a test run to determine the relationship between the distance the motion sensor reads and the separation distance between the magnets.
Next, we recorded and calculated the following on LoggerPro:

• Time
• Position of cart
• Velocity of cart
• Separation Distance: "Position"- 0.25
• KE of cart: (1/2)m*v^2         m=.347 kg (measured)
• Umag: U(r) = 0.02766879r^-0.157    where r = separation distance
• Total Energy: KE + Umag

We predict that the energy graphs will mirror each other. This means that KE will decrease to zero and then increase to its original value; while Umag should be nonexistent until KE = 0. Therefore, if we graph Total Energy, it should look like a straight line.

Results:

Position and Velocity graphs

KE=Orange, Umag=Red,Total Energy=Blue

As you can see, our graphs are slightly off. The error seems to originate from Umag, since KE is constant and reaches zero. However, we still proved that energy is conserved within the system.

Conclusion:
In this lab, we verified the conservation of energy within the system to a certain degree. First, we determined a function for the forces acting on the cart based on the separation distance of the magnets. Then, we calculated the PE, U(r), within the magnets and the KE of the cart. We compared the graphs of U and KE over time. Finally, we calculated the Total Energy of the system and determined whether energy was conserved. Unfortunately, our U graph was slightly off from the predicted result. This is due to the uncertainty in the power fit of the F vs. r graph. Furthermore, there is also uncertainty in our measurements of magnet separation. Lastly, there could also be a systematic error since we assumed that the air track was frictionless.