4/15/2015: Magnetic Potential Energy Lab 13

Purpose: Demonstrate conservation of energy for a magnetic system.

Set up / Procedure:

The set up consists of a level air track with a cart that has a strong magnet on one end and fixed magnet on the track of the same polarity. The track has tiny holes along its surface and an air source shoots air throughout the track, making the cart frictionless. 

For this experiment, we will be determining an equation for magnetic potential energy and verifying that conservation of energy holds true for this system. 

Part 1: Equation for magnetic potential energy

We begin by analyzing a cart when it is approaching the fixed magnet at some speed v. When the cart is closest to the magnet, the kinetic energy the cart had is momentarily converted and stored in the magnetic field as magnetic potential energy. The cart rebounds back and the magnetic potential energy is converted back into kinetic energy. The challenge is that we don't have an equation for magnetic potential energy, but we can find one by finding an equation for the force between magnets as a function of their separation, which we will can F(r) and then integrating F(r) to find U(r), which will be our equation for magnetic potential energy.

In order to find F(r), we will raise one end of the track using books so that the cart will end up at its position closest to the fixed magnet. At this equilibrium position, the magnetic repulsion force between the two magnets will equal the component force of gravity parallel to the track. We will vary ø, measure r (separation distance), and calculate F magnetic. We will then use logger pro to plot F vs. r in order to find a function of force in terms of r and integrate that function to get U(r).

We can calculate F magnetic by drawing a free body diagram when the cart is in its equilibrium position.

Below is a data table for F magnetic and r. We will plot these value in logger pro and use a power fit to find F(r).

As observed F(r) = Ar^n, where A = .0001092 n = -2.003

We can integrate this function to get a function for magnetic potential energy. Below is the derivation.

Part 2: Verifying conservation of energy for this system.

We will be using the same set up as in part 1, but adding a motion detector so we can determine the velocity of the glider. Level the track and with the air turned off, place the cart a reasonably close distance to the fixed magnet and run the motion detector. Determine the relationship between the distance the motion detector reads and the separation distance between the magnets. Below is an example how to calculate the separation. Next we set the motion detector to record 30 measurements per second. Finally, we can start with the cart at the far end of the track, turn the air back on, and hit collect on logger pro and give the cart a gentle push. Logger pro will generate data for position and velocity vs. time. 

Because the we want to verify conservation of energy, we can model our system by saying the KE(cart) + U(magnet) =  Total energy of the system.

We will create four calculated columns in logger pro for our system
mass of cart = .354 kg
distance from motion detector 0.341 m
distance between magnets 0.1 m

1. Seperation
2. KE of the cart
3. U magnetic
4. Total energy 

Below are calculated columns of how we entered the separation, KE, U magnetic, and total energy. 

We will need the position and velocity data logger pro generated when we gave the cart a gentle push in order to solve our four calculated columns. 

We can now verify conservation of energy for the time before, during, and after the collision. Below is our final data table showing how the energy remains almost constant throughout the carts motion. 

Below is a single graph showing KE, PE, and Total energy as a function of time. 

By analyzing our graph, we should see that the total energy should remain constant, therefore the total energy should be a horizontal line. What we generated was not horizontal, so there is some error in our experiment. The errors that could have deviated our total energy from being a horizontal line could have been: The uncertainty when measuring magnet separation, uncertainty when using our phones to measure the angle the track made with the table, there was an uncertainty in our fit for F vs r as shown in the image of our graph at the beginning, the air track may not be entirely frictionless, and the square plate on top of the glider could have contributed some air resistance during the carts motion towards the sensor. 

Conclusion: 

Our main purpose was to verify that conservation of energy holds true for a magnetic system. The challenge was that we didn't have a function for magnetic potential energy, but by using the connection between force and potential energy, we can say F = -du/dx, which means, the direction of force is opposite to the sign of du/dx. With this connection, we were able to determine the force between magnets as a function of their separation and then integrate that function to get the magnetic potential energy as a function of their separation. With a function for U magnetic, we can then verify conservation of energy for this system. Although our final graph was not perfect, we can still see the theory holds true.