13-March-2015: Conservation of Energy: Mass-Spring System

The purpose of this lab was to see if Energy was conserved in a vertically-oscillating mass-spring system.

Setup:

Fig. 1

In order to measure the total energy in this spring mass system, we needed to measure the system's potential energy, spring potential energy, and the kinetic energy. To set up the system, we needed: a spring, a mass, a ring stand, clamps, a laptop, and our LoggerPro equipment. We attached the ring stand to our lab bench using the clamps. Once attached, we hung the spring via a bar connecting to the ring stand. We used a motion sensor and LoggerPro in order to find the velocity and position of the oscillating mass and spring (see Fig. 1).

Finding Spring Constant "k":
We needed to calculate the spring constant "k" since LoggerPro could not measure it for us. To do this, we planned to measure the "stretch" of the spring as we hung different masses from it. Since we knew that the spring constant was a ratio between force and length (N/m), if we graphed enough points of stretch and weight, we could use the slope of this graph to find the spring constant "k".
First, we began by measuring the length of the relaxed spring. Once we had this, we hung a 100g mass to the end of the spring and measured the new length of the spring, We subtracted the length of the relaxed spring to this in order to find the displacement of the spring. We repeated this twice, adding 100g each time. Once we had this, we created a data table in LoggerPro that found us the force in newtons of the hanging masses. We then graphed force vs displacement and found our slope to be 11.01. This meant that the spring constant "k" of our spring was 11.01 N/m. ( see Fig. 2-1 & 2-2)

Fig. 2-2
The slope of the graph was the value of our spring constant.

Fig. 2-1
Using a ruler, we measured the new length of
the spring every time we added more mass
to it. 

How to Express Kinetic, Gravitational Potential, and Spring Potential Energy:
Because the spring had mass, we needed to account for its kinetic and gravitational potential energy. The total energy of the system would consist of the kinetic energy of the spring and mass, the gravitational energy of the spring and mass, and the elastic potential energy of the spring. To find the kinetic energy of the spring itself, we needed to consider it as a large amount of littler pieces of mass(dm), each piece having its own velocity. Each piece of the spring had its own kinetic energy. since we had a ratio of the mass and length of the spring, we could assume that dm/dy, where dy was a little length of the spring, was equal to M/L (M being the mass of the spring and L being it's relaxed length.). We then integrated to find our spring's kinetic energy (see Fig. 3 for calculations).

Fig. 3
We found our final equation for the kinetic energy of our spring to be KE=1/2(M/3)v^2

In order to find the gravitational potential energy of the spring, we again needed to find a representation for dm (small piece of the spring.) Since we knew that the length of the spring would vary, we needed to come up with a function to represent it. We used "h" to be the length from the top of the spring to the motion sensor, "y(0)" to be the length from the end of the spring to the motion sensor. Thus, the length of the spring, L, would be "h"-"y(0)".  We integrated dm once we found it to be M/(h-y(0))*dy (see Fig. 4 for calculations).

Fig. 4
We found our final equation for the gravitational potential energy of our spring to be GPE=(M/2)gy

Once we had these two expression, we were able to find an overall expression for the total kinetic energy of our system. The total kinetic energy was the sum of the spring's and masses kinetic energy. The total gravitational potential energy was the sum of the spring's and masses gravitational energy. The total energy was the sum of all of these along with the spring potential energy (see Fig. 5).

Fig. 5

The Experiment:

Once we had all of our expressions, it was time to actually do the lab. We inputted our expression for total KE, total GPE, and EPE into LoggerPro. We used "position" as our y-variable for GPE, "velocity" as our v-variable for KE, and (0.887-"position") as our y-variable for EPE. (this was because we needed to square the spring's stretch, which was the difference in initial relaxed height and "position") Once we had all of this, we began to collect data as we set the spring into an oscillating motion. All of the graph took a sinusoidal type of shape. This was because at different points of oscillation, each energy was become larger and smaller. At high points, GPE was large, and at low points it was small. The velocity continuously increased and decreased, changing the KE. And the stretch in the spring continuously changed, changing the EPE. If energy was to be conserved, then the total energy of this experiment should have remained constant. To test this, we created a new total energy column, which was the sum of the KE, GPE, and EPE. What we found was that our total energy was relatively consistent. Ranging from 1.82 Joules to 1.88 Joules justified that energy was conserved throughout the oscillation. (see Fig. 6)

Fig. 6
The total energy graph remained relatively consistent for both a function of time and position.

 Conclusion:

Considering that our environment for the lab was imperfect, we gained a very pleasing result to our test of conservation of energy. The were a few possible sources of error within our experiment. For one, our measurement for our spring constant had some uncertainty due to our measurements using a ruler. This in turn affected our function of spring potential energy. Along with this, when we set the spring into an oscillating motion we had multiple trials to try to get it moving strictly vertically. Considering that the spring could sway in the horizontal direction, our values for kinetic and gravitational potential energy could have been affected. However, considering all of our uncertainty, we were able to conclude that energy was conserved throughout this motion.