8/5/2016 - Conservation of Energy -- Mass-Spring System (Lab 12)

PURPOSE:
To determine whether or not energy is conserved in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.

THEORY:
In this experiment, we have a mass hanging from a uniform spring a certain distance above the ground. When the spring is a certain height above the ground, it has GPE. When it is released from this position, some of its GPE is converted into KE. If there are no other forces acting on the system, then the GPE will be converted into KE until the spring is in its equilibrium position. As the spring-mass system reaches its maximum displacement, all of its KE is converted into EPE. After the spring has stretched its maximum distance, the hanging mass will rise as its EPE is converted back into KE. As the spring-mass system oscillates, this energy cycle continues until external forces such as friction retard the spring to a standstill. If energy is conserved, then the total energy of the system will be constant. However, in order to take the mass of the spring into account for our energy calculations, we must derive new formulas for GPE and KE using calculus.

APPARATUS:
We hung a 200 g mass on a hanger for a total mass of 250 g. The bottom end of the spring is holding the mass, while the top side is hanging from a horizontal metal rod, which is fixed to another metal rod that is clamped to the lab table.

We also attached a force sensor to the horizontal metal rod, such that the spring could hang from it with the hanging mass attached. Additionally, we placed a motion sensor beneath the spring-mass system. Lastly, we taped a flashcard to the bottom of the hanging mass to make it easier for the motion sensor to detect.

Apparatus: Part 2

EXPERIMENTAL PROCEDURE:
Part 1: Determine the Spring Constant

1. Calibrate the force sensor.
2. Set up Apparatus: Part 2, as shown. Note: Remember to zero the force sensor when spring is attached.
3. Calibrate motion sensor.
4. Collect Force vs. Time data and Stretch vs. Time data. 
5. Plot a Force vs. Stretch graph to obtain an equation for the force of the spring. (F = kx + F0)
6. Holding the mass+spring system from its equilibrium position, use the motion detector to determine its position relative to the "ground." (i.e. from the front of the motion sensor).
7. Make a new Calculated Column called "Stretch." This equation will be based on the motion sensor reading.

Part 2:

1. Hang 200 g on the mass hanger for a total mass of 250 grams.
2. Pull the spring down about  10 cm and let it go. 
3. Record position and velocity graphs and sketch them in your lab module.
4. Make prediction sketches of KE vs. time, GPE vs. time, and EPE vs. time.
   
   

   

   Energy Prediction Sketches

5. Now, set up Logger Pro to calculate the various energies in New Calculated Columns.
6. Use Logger Pro to produce plots of KE vs. y, KE vs. v, GPE vs. y, GPE vs. v, EPE vs. y, EPE vs. v.

Here are the equations we used for various energies, *Note: We derived them in class using calculus:

KE = (1/2)[mhanging + (1/3)mspring]v^2

GPE = [mhanging + (1/2)mspring]gy

Elastic PE = (1/2)k(stretch)^2

DATA/GRAPHS:
Part 1:
mspring = 86.0 g
mhanging = 350 g
k = 5.794

Part 2:

GPE vs. t - KE vs. t - EPE vs. t 

KE vs. y

KE vs. v

GPE vs. y - EPE vs. y

GPE vs. v - EPE vs. v

GPE vs y -EPE vs. y - Energy Sum (for each)

GPE vs. v - EPE vs. v - Energy Sum (for each)

ANALYSIS:
After analyzing my energy data and comparing it to my predictions, it is apparent that I made a few incorrect assumptions. For example, when I created my sketches, I predicted that GPE would increase at the same rate EPE would decrease and vice versa. I made this prediction because GPE should be at its lowest when EPE is at its highest. This, however, was not the case at all according to our data. In fact, my data illustrates that my prediction is the antithesis of what happens because both the GPE and EPE oscillate together at very similar rates. Another error I made in my prediction was that I took the EPE of the system to be positive when in reality it is negative because it acts in the opposite direction of the spring force (which is in the positive direction for this experiment). In order to verify that our data was consistent with our physical model, we also analyzed our energy vs y and energy vs. v graphs. Our KE vs. v,y graphs made sense because they they were both consistent with the increase in the speed of the system. Our GPE, EPE vs. y graphs were logically sound because they both increased and decreased respectively with respect to the height relative to the motion sensor. Moreover, our GPE, EPE vs. v graphs had warranted values because they are both reach their max when the KE decreases to zero.

CONCLUSION:
Overall, our group was able to successfully demonstrate that the energy of the mass+spring system was conserved because our data showed that the total energy of the system remained nearly constant throughout. However, the lines that Logger Pro displayed were not as straight as we hoped they would be. Therefore, there could have been a few sources of uncertainty in our experiment that we did not account for. One of the many possible sources of error we encountered were external forces acting on the system. This includes, but is not limited to: air resistance caused by the flashcard, damping caused by dissipated heat due to friction, and permanent spring deformation. Another possible source of uncertainty was through human error. For instance, if we inaccurately recorded length measurements, then the preciseness of our initial data would be negatively affected, thereby decreasing the accuracy of our energy calculations.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.