17-April-2017: Magnetic Potential Energy

In our daily lives, we can observe that when we are pushing a magnet to another, we are doing work on it. In the meanwhile, the other magnet is giving us a larger and larger force while approaching. But how exactly the work is done during the process? Today, we did a lab on magnetic potential energy and detected whether the energy inside magnets were conservative or not.

Purpose

Find out the relationship between distance and potential energy, and verify whether conservation of energy applies to this system.

Plan

Because we do not know what is the relationship between force and distance, we have to detect that first. Thus, we divided the lab into two parts. 

Part I:

Find the relationship between magnetic force and distance by the apparatus (We used a glider on an air track with a magnet attached to its end. Another magnet is also attached to one end of the air track) as following (by the air track, we can lower the friction force).

Then, place the track in different angle. When we turned the air on, the cart would be balanced somewhere eventually. Draw the body diagram,

By the function,

we can know the value of magnetic through the angle.
Using the caliber, we could measure the distance between two magnets.
Then, after we collected a few data, we could roughly know the relationship between magnetic force and distance by logger pro. 

Use the integral to calculate the relationship between magnetic potential energy and distance.

Part II:
Level the track.
Then, add a motion detector in the front to detect the distance and velocity. (But because it is hard to detect directly the distance between two magnets, we used the target into top instead of that.)
Collect the data, use

to calculate kinetic energy and magnetic potential energy.
Compare the energy to see whether it is conservative.

Set up & Data

Part I:
Set up a frictionless cart with a strong magnet on one end approaches a forced magnet on one end approaches a fixed magnet of the same polarity as following.

Rising the left side by adding some books.
Then, we collected the angle by phone and turned on the air track.

After the cart came to rest, we turned off the air track and measure the distance between.
Repeat the step for several time to get different data. Eventually, we took 7 data because the beginning five did not seem to be perfect.

Part II:
We leveled the track first (by detect whether the cart can keep the constant speed).
Then, add a motion detector on the right side, and turn it on.
After a slightly push, we collected the data of velocity and distance.

Analyze

Part I:

Through plugging all the data we got to Logger Pro and doing the power fit, 

we got the relationship between distance and magnetic force was 

Then, after doing the integral, we got the function of potential energy was

Part II:

Using the functions

We can get the graph of U(t) and KE(t), and we sum the up to get the graph as below. 

From the graph, we can know that the total energy always staying around some certain number. So we could make a conclusion that the system of kinetic energy and magnetic potential energy is almost conservative. 

Conclusion

Firstly, there are many factor that we do not know about magnet (since many groups got different answer for the relationship), so we can use our experiment's data to calculate the relationship of force and distance, which is

Then, we could get the U(x) by integral.
Secondly, because the total energy is almost the same, so we could say that the energy between kinetic energy and magnetic potential energy is conservative.
However, we can easily observe that the total energy is actually going down. Maybe it is caused by the air resistance or some other friction that made some part of kinetic energy into heat, and the heat is also the reason why would the cart will stop eventually. (My group's kinetic energy is just reducing too fast that maybe some other factors are also influenced.)