Purpose: To show the conservation of energy in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.
Setup:
As seen above, a spring is hung from the force sensor. H and y will be used to determine the spring's unstretched position and the GPE of the spring.
Pre-Lab: As a class, we identified the types of energy that will be present during the experiment and we derived equations for each of them. We will need this equations in order to determine the total energy of the system.
- KE(hanging mass) : (1/2)m*v^2
- GPE (hanging mass): Position (measured by motion sensor)*m*g
- EPE: (1/2)k* stretch^2
- GPE (spring): (Position*Mspring*g)/2
- KE (spring): (Mspring*v^2)/6
How we determined GPE Spring:
First, we chose a representative piece (dm) of the spring. Next, we wrote an expression for the GPE of that piece. Then, we summed the GPE of all the pieces of the spring from y to H. Finally, we solved the integral to find our equation for the GPE Spring.
The same process was used to find the KE Spring:
Essentially, our job for this lab is to calculate all of the energies shown above. Once we achieve this, we then graph the sum of all the energies and based on this graph,we determine if energy is conserved.
Procedure: In the first part of the lab, we found the spring constant of the spring.
- Mass of spring (measured) = 64 +/-.1 g
- Unstretched position of spring (measured)= 48.5+/- .1 cm
Determining the Spring Constant
First, we calibrated the force sensor with a 1 kg mass and reversed the direction of the motion sensor. We then hung the spring on the force sensor, and zeroed it. Next, we attached a 50g mass to the spring and pulled on it, while collecting data. We did this to verify that the sensors were setup properly and that LoggerPro was able to plot the data.
Next, we followed the same steps and recorded data. We collected Force vs. Time and Stretch vs. Time data for the oscillating spring with an additional 50g mass. These graphs are needed in order to calculate the spring constant of the spring.
Graph 1:
- x-axis: Force (provided by force sensor)
- y-axis: Time
Graph 2:
- x-axis: Stretch (Position measured by motion sensor - Unstretched position)
- y-axis: Time
We then graphed Force vs Stretch to determine the spring constant of the spring. The spring constant would be the slope of the Force vs Stretch graph.
- x-axis: Force (provided by force sensor)
- y-axis: Stretch (Position measured by motion sensor - Unstretched position)
- Calculated Spring Constant: 14.86 N/m
In the second part of the lab, we added mass to the spring and recorded data while it was oscillating. Then, we calculated all of the energies involved and determined if energy was conserved.
Conservation of Energy
Energies involved:
- KE(hanging mass) : (1/2)m*v^2
- GPE (hanging mass): Position (measured by motion sensor)*m*g
- EPE: (1/2)k* stretch^2
- GPE (spring): (Position*Mspring*g)/2
- KE (spring): (Mspring*v^2)/6
Next, we added 250 grams to the spring. The hanging mass is now 300g. Then, we followed the same procedure; we pulled on the spring and recorded data. The equations shown above were applied to calculated columns on LoggerPro and were then plotted. We also created a new column to include the sum of KE, GPE, EPE.
- Note: When KE=0 and EPE=0, all energy is found in GPE
From Left to Right: Time, Force, Position, Velocity, Acceleration, Stretch, KE, EPE, GPE, KE Spring, GPE Spring, Total Energy
The graphs for all the energies can be seen below:
Finally, we ran the experiment one last time. This time we only graphed Total Energy vs. Time
 |
| Bottom Graph: Energy Total vs. Time |
As you can see the graph has a constant oscillation. Ultimately, this means that energy is conserved since no energy was lost or gained.
Conclusions:
In this lab, we proved the conservation of energy in a mass-spring system. First, we determined the spring constant of the spring by graphing Force vs. Stretch. Then, we calculated the various energies of the hanging mass and the spring. Finally, we graphed the energies and noticed small patterns. The main one being that the sum of the energies has a constant oscillation. This proves that energy is conserved within the system, since no energy was lost or gained.
Uncertainties: This experiment heavily relies upon the use of force and motion sensors. This means that our values our affected by the precision of the instruments. Furthermore, human error also comes into play when measuring the unstretched position of the spring. In this case, the uncertainty is +/-.1 cm. This could have an effect on the value of the spring constant. Human error can also be at fault, when analyzing the data on LoggerPro. Ultimately, the concept behind the lab is sound and the uncertainties were negligible to the point that it did not drastically affect the final result.
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