Lab 12: Conservation of Energy

An oscillating mass-spring is a relatively closed system, so the total energy should be preserved even if kinetic and potential energies alternate. For this lab, we look at how to use calculus to derive a formula for all the energies in play, and then measure them in Logger Pro to see that total energy has, in fact, been preserved!

Although the math is long, we will discuss it a little bit to provide a context. The above diagram shows a spring of arbitrary length L, hanging on some apparatus at height h above ground. The mass of the spring weighs to be m = .100 kg. When we do physics problems, we tend to represent these object with a single point, but in real life the spring takes up quite a bit of space. Therefore, since gravitational potential energy depends on height, the potential energy of the spring actually varies from top to bottom. To solve this problem, we see that it is theoretically possible to slice this spring horizontally into a number of pieces, each with a portion of the mass of the entire spring, we call dm. Each dm slice is also the overall mass m divided by overall length L, times the height of the slice, or change in height dy. We can sum up all the slices to get an approximation of potential energy, with it being more accurate as the slices approach infinity, such that:

Note that this is the same as integrating, which we can do to solve for gravitation potential energy:

In class, we had derived the kinetic energy in this manner as well, but for the sake of brevity (and my precious time), I present to you the formulas. Ta-da!

Now, the set up is easy: We hang a lengthy weighted (weights are .600 kg) spring on a stand, over a motion sensor on the ground. The motion sensor is, of course, hooked up to our trusty Macbook running Logger Pro, so that we could record the position, and the corresponding velocity, of the spring, by pulling it and letting it go. The hardest part was aligning the motion detector, and making sure the spring stays over it, and making sure the weights don't fall and damage the motion detector, causing a lawsuit!

What was really helpful was taping a piece of paper under the weights so that it makes it easier for the motion sensor to pick it up. Here's out data:

All those wavy lines... The only thing missing now is the spring constant k, and the unstretched height of the spring. Although we installed a force sensor to our spring, we didn't actually use it. Finding k was simpler measuring the spring at rest without any mass (96 cm from ground), and the spring at rest with the mass (72 cm from ground), and finding the difference, such that:

kx = m_weightsg

k = m_weightsg / x

k = (0.6 kg * 9.8 m/s²) / 0.24 m
k = 24.5 N/m

We also do a run in Logger Pro with the spring at rest with mass, just to see how far the motion detector figures that height to be. Since we didn't zero or calibrate the motion sensor, there is expected to be an offset. This will help us find elastic potential energy, which requires that we find the stretch of the spring. With all the variables known, we start making calculated columns in Logger Pro with the formulas we've discovered, as shown above. We substitute y for the position data, and v for the velocity data. Once completed, we sum up the potential energies of the mass and spring, both elastic and gravity. We also sum up the kinetic energy of the mass and spring. Finally, we get the total between the two as well. If theory stands, this final total should be relatively stable, since all the variable energy should be conserved at any given time.

And we see, in light blue, that the total is conserved, that despite the peaks and valleys of the other forces, the sum is relatively constant. The slight error could be accounted for by inaccurate spring constant measurement, that the height isn't exactly calibrated with the motion sensor, and the spring itself isn't perfect (it could have flopped during the experiment). But in all honesty, this is as good as we should expect.