Lab 12 Conservation of Energy

Student: Xavier Lomeli
Partners: Matthew Ibarra, Billy Justin
Date of Lab: 5 October 2016 

Lab 12 Conservation of Energy

Mission Statement:To look at  the energy in a vertically-oscillating mass-spring system, where the spring is a non-negligible mass.

Theory:Since we must now consider the spring into our calculations, calculus is required to identify the show that the GPE of the spring is mg(H+y)/2 and that the KE of the moving spring is described formulaically by (1/2)((1/3)m_spring)*v^2. If energy is truly conserved, then the graphs of each energy involved (GPE, KE, and EPE) should reflect this conservation.
Experimental Procedure:
For this experiment, we hung a 200 g mass onto a hanger so that the total hanging mass was 250 g, shown below.

Directly below the hanging mass, on the floor, we placed a motion sensor which could detect the movement of the hanging mass as it osscilated back and forth, shown below.

We proceeded to pull the spring down about 10 cm before releasing and recording the position and velocity v. time graphs, aligning them appropriately. However, before we did this, we drew sketches of how we predicted each energy graph to look like, together with the aforementioned position and velocity graphs, shown below.

Since the hanging mass is initially displaced some distance, we predicted that the position graph would be some variant of the cosine function, and since the velocity is initially zero, we assumed its graph would be some variant of the sine function. Likewise, since the velocity is initially zero, we predicted  that the kinetic energy graph would likewise start at y = 0, but would possess a period twice as rapid since the spring stretches twice during each oscillation. Conversely, the gravitational potential energy would be substantial in the beginning, but diminish down to zero as the hanging mass reached the lowest point of its stretch.

Shown below, we proceeded to plot the KE(purple), GPE(orange), and EPE(red) graphs together,  to see if our predictions were accurate.

 As we can see from the plot above, our predictions were quite accurate indeed, with the general shape of each energy graph corresponding well with our predictions except for the odd discrepancy between the amplitude of the KE graph and the other two graphs, which is most likely due to the paper taped to the falling mass briefly falling out of the sight range of the motion detector.
Next we used LoggerPro to produce plots of KE, GPE, and EPE verses distance and KE, GPE, and EPE verses time, respectively. However, before doing this, we drew sketches of our predictions, shown immediately down below.

With these predictions in mind, we found the graphs themselves below.

Above we see the respective GPE and EPE verses position graphs, and the trends correspond with our predictions, which indicated an inverse relationship between the two.

With regards to the kinetic energy verses position, we received exactly what we expected, increasing up until the equilibrium position of the string is reached, then slows down until the sting is fully stretched with the given hanging mass.

With regards to the GPE and EPE verses velocity graphs, we see that they correspond directly, which is to be expected.

Next, above, we see our graph of the kinetic energy verses velocity, which is the opposite of the graph of the velocity verses position, as expected.
Below we made a new column to contain the sum of all the energies, known as E_sum, and we found the relationship between E_sum(blue) and time and space.

Above, we see how the sum of the energies is a horizontal line, which corresponds with the notion that energy is conserved.

Lastly, above we see the sum of the energies with respect to velocity, which seemingly yields a horizontal circle, again coherent with the idea of energy conservation.

Conclusion- The most obvious source of error in our experiment unquestionably came from how the paper attatched to the bottom of the hanging mass to be detected by the motion sensor as it either ascended or descended sometimes would stray away from the line of sight of the motion sensor. Furthermore, the paper itself was not flat against the hanging mass, but rather only attatched at a point, with the edges of the paper dangling considerably. It  is entirely possible that the senor detected these dangling edges rather than the flat middle during different trials, thus introducing another source of error.
Even so, the premise of the lab was satisfied in the sense that we obtained graphs whose respective qualities were correctly related, either inversely or directly. We showed how energy is conserved within the system, thus validating our theory.