Lab Partners: Charles, Anthony
Date performed: 10/05/16
In this lab, our goal is to show that the energy in an oscillating mass-spring system is conserved. Our setup includes a vertically-oscillating spring with a mass attached to it. On the floor below the mass and spring is a motion sensor that detects the motion of the mass as it moves up and down.
Setup of mass-spring system with motion sensor on floor
Before we could actually start the physical part of the lab, we had to do some calculations to show that the gravitational potential energy (GPE) of the spring is equal to mg(H+y)/2, H being the height from the very top of the system (extended arm from which the spring and mass hang) to right above the motion sensor, and y being the height from the bottom of the mass (when supported with hand so that the spring is NOT stretched) to right above the motion sensor. This calculus was done by the professor on the board, and as a class our job was to listen and understand to the point that we could do those calculations on our own.
The same was repeated for the equation for kinetic energy of the spring. Using the same concept and selecting a little "representative" piece of the spring, the professor showed us that KE is equal to 1/2(1/3mspring)v^2.
After that, we were able to start on the experiment. First, we had to take measurements for the mass of the spring and the mass hanger. For our lab, instead of using a mass+mass hanger, we just used a 1kg hanging mass. The mass of our spring was 0.065kg. Using the motion sensor and Logger Pro, we measured our y and confirmed that value using a meter stick. We measured 1.36m. Then, we released the hanging mass from the support of our hand and let it hang at rest (not oscillating), and measured that height, which came out to be roughly 0.718m. This is our equilibrium position. Finally, to put our system into motion, we pulled the spring slightly (about 10cm) so that it started oscillating. Then we pressed start on Logger Pro to begin collecting data.
Logger Pro view: Graphs of position and velocity vs. time; calculated column for stretch of spring
The result was a sinusoidal graph for our position vs. time graph, which was expected from the oscillating system because it moves up and down. For our velocity vs. time graph, we also got a sinusoidal graph because the speed at which the spring falls downward is faster than the speed at which it recoils back up. This is due to the constant weight of the mass pulling the spring downward. Since the system is oscillating, this pattern repeats. We moved on to create three more calculated columns, one for each energy (kinetic, gravitational potential, and elastic potential). For each of these energies, we produced two graphs, one of the energy vs. position and another of the energy vs. velocity. Each of our graphs produced fits correctly with what we should expect.
Logger Pro view: Graphs of KE vs. position and KE vs. velocity
Logger Pro view: Graphs of GPE vs. position and GPE vs. velocity
Logger Pro view: Graphs of EPE vs. position and EPE vs. velocity
For our KE vs. position graph, it is a parabola facing downwards because the kinetic energy reaches its maximum at the equilibrium position, which we measured earlier in the lab to be about 0.718m. The graph shows that the top of the parabola is at about 0.7m, so our data is accurate here. For our KE vs. velocity graph, since kinetic energy is directly related to the velocity squared, when the value of velocity is greater, so is the kinetic energy. When the velocity is 0, kinetic energy is also 0.
For our GPE vs. position graph, the GPE increased as the position increased. This is because the GPE of the spring is directly related to the position (y) of the mass above the ground (in our case, the motion sensor), as we showed earlier in the lab. For our GPE vs. velocity graph, it is a repeating ellipse, which makes sense for the same reason as earlier. Because the GPE is directly related to the position, it is bigger when the value of position is bigger. Although this graph doesn't show position, we know that there are two points at which the velocity is 0: at the top of its oscillation and at the bottom of its oscillation. In other words, although the mass+spring system is constantly in oscillation (motion), it stops briefly when it bounces back up (right before it falls again), and again when it is pulled all the way down (right before it bounces back up again). Looking back at the graph, you can see that at velocity=0, GPE is at its maximum AND its minimum, which means that it was at its highest and lowest position, proving that this graph is correct.
Finally, for our EPE vs. position graph, the EPE decreased as the position increased. This is because the system's energy is initially stored in the spring as EPE when it is at rest (mass hanging and spring stretching down). As the mass+spring move up and position increases, energy is converted into kinetic energy, thus decreasing the elastic potential energy. For our EPE vs. velocity graph, the reason for the ellipse is similar to the last explanation for our GPE vs. velocity graph. Instead of being directly related to position, however, the elastic potential energy is directly related to the stretch of the spring.
Moving on to the last part of the lab, we created a calculated column called Esum, which is the sum of the KE, GPE, and EPE. To finish off, we produced two graphs: one of Esum, KE, GPE, and EPE vs. position, and another of Esum, KE, GPE, and EPE vs. time.
Logger Pro view: Graphs of Esum, KE, GPE, and EPE vs. position, and Esum, KE, GPE, and EPE vs. time
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