PURPOSE: To verify the conservation of energy for a vertically-oscillating mass-spring system, taking into account the spring's mass.
We calculate 3 energies for this system, and show that they are conserved (the sum of energies remains constant) while the spring oscillates. Before we got started with the experiment, we derived equations for each of the energies: kinetic energy, gravitational potential energy, and elastic potential energy. However, for kinetic and gravitational potential energy, there are two equations for each, one for the mass and one for the spring.
1) Kinetic energy for the mass is (1/2)mv^2
2) Kinetic energy for the spring was derived from the following:
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| We set a proportion of dm (small piece of mass) over total mass, M, equal to a dy (small length of spring) over total length, L. We can solve for dm to replace for "m" in the equation 1/2mv^2. To find velocity of the small piece, we formed an equation relative to the velocity of the end of the spring (v piece = y/L * v end). Plugged both into 1/2mv^2 and integrated from 0 to L (length of spring). |
4) Gravitational Potential Energy for the spring was derived from the following:
CONCLUSION:
In both graphs, we can see that the total sum of energies looks graphically more like a straight line than the rest of the lines. In the graph of energies vs. position, the standard deviation for the sum of energies line is 0.016 and in the graph of energies vs time, the standard deviation for the sum of energies is also 0.016. Because it is close to zero, this means that our energies were conserved over position and over time, supporting the conservation of energy theory.
Reasons for our error and uncertainty could have been due to the system not being set up directly over the motion detector, or the force sensor not being zeroed more precisely. If the hanging masses were moving a little while we were measuring position to calculate spring constant, this could have altered our data. The spring could also not have a uniform spring constant, meaning it oscillates faster in certain areas, or is tighter or bent in certain parts. During the trial where we pulled down the mass and let it oscillate, if it was not pulled directly down it could have affected velocity or position.
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| This process was similar to finding the kinetic energy of the spring itself. We wrote dm in terms of dy, and substituted that into the equation dGPE = mgy. We then integrated this from the end of the spring (yend) to height H. |
5) Elastic Potential Energy in the spring is 1/2kx^2. To find x, we created a function of (unstretched position - "position").
Once we had our equations ready, we ran the experiment. The force sensor is attached to a rod, with a spring hanging from its hook, and a motion detector directly below facing up (see photo below). We took the mass of the spring and calibrated the force sensor. The first process was to find the spring constant, k. We did this by collecting and plotting force vs. stretch at two different positions of the spring, and doing a linear fit to that graph. The slope of that line was the spring constant, k.
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Next, we zeroed the motion sensor at the position when a 50 gram mass hung from the spring. We created 5 new calculated colums: (1) stretch, (2) elastic potential energy, (3) gravitational potential energy of mass + gravitational potential energy of spring, (4) kinetic energy of mass + kinetic energy of spring, and (5) total energy of the system and entered corresponding equations for each. For mass of hanging object we used 250 g and for the mass of spring we used 87 g. We then added 200 grams, and ran the experiment by pulling down on the mass and letting the spring oscillate. This yielded the following graphs of KE, GPE, EPE, and Esum vs. position, and of KE, GPE, EPE, and Esum vs. time.
CONCLUSION:
In both graphs, we can see that the total sum of energies looks graphically more like a straight line than the rest of the lines. In the graph of energies vs. position, the standard deviation for the sum of energies line is 0.016 and in the graph of energies vs time, the standard deviation for the sum of energies is also 0.016. Because it is close to zero, this means that our energies were conserved over position and over time, supporting the conservation of energy theory.
Reasons for our error and uncertainty could have been due to the system not being set up directly over the motion detector, or the force sensor not being zeroed more precisely. If the hanging masses were moving a little while we were measuring position to calculate spring constant, this could have altered our data. The spring could also not have a uniform spring constant, meaning it oscillates faster in certain areas, or is tighter or bent in certain parts. During the trial where we pulled down the mass and let it oscillate, if it was not pulled directly down it could have affected velocity or position.
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