13-Apr-2015: Conservation of Energy_Mass-Spring System

Purpose: To prove that energy is being conserved in a vertically-oscillating mass-spring system.

Procedure: We begin with the set up for the lab, which consisted of a table clamp holding up a vertical rod with a horizontal rod clamped at the top. A spring hangs from the raised horizontal rod and a motion sensor is set up to measure position from under the spring. Before beginning the lab, we had to measure the height of the unstretched spring from the motion sensor and the weight of the spring.

Lab Apparatus

To begin the experiment, we had to solve for the spring constant k of our spring. In order to do so we put a force sensor on the horizontal rod and calibrated it using zero mass and a 1 kg weight, then removed the weight. Then we attached the spring to the sensor and zeroed the sensor. We made sure that our motion sensor was set u to reverse direction in order to get a positive value for position. Next we placed a 50 gram mass hanger on the string but held it at that position where the spring was still unstretched in order to get an initial position. Then we began collecting our data for force vs. time and stretch vs. time and plotted a graph of force vs stretch. The slope of the graph gives us the spring constant k for our spring, which came out to 9.123.

graph of force vs. stretch

Now that we have our spring constant, we have to find the stretch of the spring to use for our elastic potential energy equation. We took off the force sensor and kept the motion sensor on the floor to measure the unstretched spring with hanging mass distance from the sensor and also the stretched distance with an additional 200 grams hanging on the spring. We had to make sure that the the measurements verified our spring constant that we just solved for. Then we made a calculated column with the formula (xunstretched -xstretched) to find the "stretch".

Having calculated the mass of spring, mass hanging, spring constant, stretch from no hanging mass to hanging mass, we can now see if the total energy of the system is constant at any position of oscillation. In order to do so, we must utilize the equations of the various energies involved.

KEmass = 0.5(mhanging)”velocity”^2

GPEmass = (mhanging)(“position”)g

EPEspring­ = 0.5k(stretch)^2

GPEspring itself = 0.5(Mspring)(“position”) - The mass of the spring itself adds to the potential energy of the system.

KEspring itself = 0.5(Mspring/3)(Vend)2 - The kinetic energy of the spring itself adds to the kinetic energy of the system as it oscillates up and down. Below are the calculations for the GPE and KE of the spring itself.

Calculations for deriving GPE spring and KE Spring

With the equations inputed in the calculated columns, we can now begin collecting data again. We hang a 250 gram mass to the mass hanger and pull down about 10 cm and let go. We record the position and velocity vs. time graphs in order to plug in the data to our calculated columns. A 6th calculated column was then created to calculate the total energy of the system by adding all the energies at any given position together. If conservation of energy holds true for this Mass-spring system, then the Total Energy for any given position should be constant.

Graph of each energy in color and total energy in black

Results: Looking at our graph, you can see the separate energies in different colors oscillating as the spring oscillated. The black line indicates the total energy of the system and theoretically should be a straight constant line. Although the line is not perfectly straight, we can say that our experiment verifies the conservation of energy theorem with some degree of error.

Conclusion/Uncertainty: Through experimenting with a vertical mass-spring system, we were able to prove that energy was being conserved throughout the oscillation of the spring and mass. We solved for the spring constant and measured the stretch of the spring when adding a mass to the system. Then we pulled on the string to oscillate the system in order to measure the energy of the system over a period of time. Energy was being transferred from kinetic to potential to elastic, but the total amount of energy in the system remained fairly constant. We can conclude that the amount of kinetic, potential, and elastic energies all depended on the height of the oscillating spring and that total energy was conserved. A few uncertainties were apparent to us when experimenting such as when we pulled the spring and mass down, we could not get a perfectly vertical oscillation which would effect the position over time of the weight. Also some of the measurements including the length of the spring having an uncertainty of +/- 1 cm and mass of spring having +/- 1 gram.