Magnetic Potential Energy Lab

Objective: 

• To come up with an equation to calculate magnetic potential energy
• To verify that the theory of conservation of energy applies to this system.

Part 1:

• Finding an equation for magnetic potential energy

Set-up:

1. Let an air track sit on the horizontal table.
2. Place the glider on the air track, make sure the air track is leveled which means that the glider stays still unless we give it a push.
3. Attach a magnet at the end of glider, and another magnet with the same polarity at the end of air track as well.
4. Setting up a motion sensor behind the magnet on a vertical rod, make sure the sensor can read as the glider moves. To make it easier for the sensor to read, we ask a square on top of the glider.
5. Connect the other end of air track to the air source which is already connected to the power.

The apparatus of the experiment is shown in figure 1

Figure 1: The apparatus of the experiment.

Performing the experiment:

• We use books to raise one end of the air track and turn on the air source to let the glider move on the air track, then give the glider a push (Figure 2)
  
  Figure 2: Using books to tilt one end of the air track.
  
  
• A glider will move toward the magnet and when the glider is at the position of closet approach to the fixed magnet, the glider KE is momentarily zero (we also call it equilibrium position).
• At equilibrium position, all of the energy of the system is stored in the magnetic field as magnetic potential energy, and this energy equals the gravitational force component on the glider parallel to the track.
• We measure the angle using our phone or the angle measurer in the lab (Figure 3)

Figure 3: Measuring the angle.

Approach.

• Because we don't know an equation of magnetic potential energy, we call magnetic potential energy U(r) as a function of separation distance r.
• We will calculate U(r) by integrating the force F(r) in term of r which is the separation distance between two magnets.
• To find the function of force in term of r, we will use Logger Pro. We thus will enter the data for force and r (after calculation and measurement), graph them, and obtain the equation.
• We predict that force F(r) and r is related to each other though this equation : F(r) = Arⁿ , while force is equal the gravitational force component on the glider parallel to the track.

Calculating force F and measure the separation distance r:

• We first draw a free body diagram

Figure 4: Drawing a free body diagram and figure out the force of magnet.

• Since we derive F=mgsin⦶, we then need to measure the angle and the mass of glider.
• Calculating force is shown below

Figure 5: Calculating the force.

• Measuring the separation distance r.
  
  Figure 6: Measuring the separation distance between two magnets.

Working with Logger Pro

1. We first enter the value of force and separation distance that we just got above into Logger Pro.

Figure 7: Data for force and distance.

1. We then curve fit the data and obtain the graph of force F vs. separation distance r.

Figure 8: Curve fit the graph and analyze the equation

       Now we have a function of force in term of separation distance r: F(r) = 0.0002038 * r^-1.871, we then integrate to get the magnetic potential energy. The work is shown below

Figure 9: Integrating F(r) to obtain an equation for U(r).

U(r) = 0.000234 * r^-0.871

Part 2: 

      Verify the theory of the conservation of energy applies to this experiment.

Approach:

• From part 1, we know that when the glider moves toward the magnet, all of kinetic energy is transferred to magnetic potential energy.
• To prove the conservation of energy, we need to find the initial kinetic energy and the final magnetic energy.
• Our prediction is that KE_intital = MPE_final or total energy = constant number.
• To find kinetic energy, we need mass and velocity, while to find magnetic energy we need the separation distance between two magnets.

Procedure:

1. We use the same set-up as part 1, however, we add a motion sensor behind the magnet to record the velocity of the glider.
2. With the air source turned off, place the glider on the air track, reasonably close to fixed magnet. Run the motion detector. Determine the relationship between the distance the motion detector reads and the separation distance between the magnets.
3. Using books to tilt the track at various angles, measure the angle, then give the glider a gently push so that it moves toward the magnet.
4. Set up the Logger Pro, create New Calculated Column that will let us get the separation between the magnets from the position as measured by the motion detector. 
5. Create New Calculated Columns to calculate the kinetic energy, magnetic potential energy, and total energy of the system as a function of time.

Data Collection

• Following the procedure above, we will use Logger Pro to collect the velocity of the glider. After giving the glider a push, hit the Collect button, and the data appears on Logger Pro like shown below.
  
  Figure 10: Data collecting for time, position, and velocity.

Data Analysis:

• We first calculate the initial distance 

Figure 11: Calculating the initial distance.

Initial distance = 0.3045m

• Calculating the separation distance r

Figure 12: Calculating separation distance.

• Calculating kinetic energy

Figure 13: Calculating kinetic energy.

• Calculating magnetic potential energy

Figure 14: Calculating magnetic potential energy.

• Calculating total energy

Figure 15: Calculating the total energy.

      When we're done calculation, we got the result as shown below

Figure 16: Result table after calculation.

     We then plot the graph of kinetic energy, magnetic potential energy, and total energy as a function of time in Logger Pro.

Figure 17: Graph of kinetic energy, magnetic potential energy, and total energy.

      What we expected was that the total energy was supposed to be a horizontal line in order to prove the theory of the conservation of energy. However, what we obtain is not actually a horizontal line, thus there are some possible errors happening during the experiment. These are:

• The angle we measure has an uncertainty (± 0.1) as our phone can't give us the exact number
• The separation distance also has an uncertainty (± 0.05) based on our ruler.
• When measuring the velocity of the glider using Logger Pro, we may press Collect button slower than the actual time the glider gets moving, thus the value we get for velocity may affect our kinetic energy.
• Since separation distance and the angle both have uncertainty, these lead to the uncertainty in magnetic potential energy because the equation is derived from F(r) while F is a function of ⦶ (F=mgsin⦶)
• We also assume that the air track is ideal frictionless; however, it is not perfectly smooth, thus this condition may affect our experiment.

Conclusion:

      In this lab, we perform experiments in order to obtain two goals:

1. To come up with the equation of magnetic potential energy.
2. To prove that the theory of the conservation of energy

      Our result for part 1 is : U(r) = 0.000234 * r^-0.871 

      For part 2, we're close to obtain the horizontal line for total energy. However, we explain reasons why we don't get a perfectly horizontal line. Actually, there are many errors that we hardly avoid which may affect our experiment. But overall, our result is "pretty good" to prove the theory of the conservation of energy in this experiment.