4/13/15: Conservation of Energy Mass-Spring System Lab 12

Purpose: To verify that conservation of energy holds true for a vertically oscillating mass-spring system.

Procedure / Set up:

We will be performing two experiments: determining the spring constant and oscillating the spring to sum up the various energies involved in our system in order to find the total mechanical energy. 

Determining k

Mount a table clamp with a vertical rod to the table and to the vertical rod, mount a horizontal rod. Attach a force sensor to the horizontal rod and calibrate it using 1kg. Once calibrated, attach a spring to the force sensor and attach a hanging mass to the spring. Hit collect on Logger Pro and slowly pull down the mass so that the spring stretches. Logger pro will generate a graph of force vs. position and the slope of the graph will give us the value for k.

Our spring constant k was 8.057 N/m.

Stretch

We will need this value for the second part of this experiment because one of the various energies within our system is elastic potential energy in the spring, which includes the "stretch" as a variable. Set up a motion detector on the floor so it is right below the mass. Attach a notecard to the bottom of the mass so the motion detector can get an accurate reading. Adjust the spring so it is un-stretched and hit collect on logger pro to record the position of the mass. Our un-stretched value was 0.723 m

The stretch of the the spring = (un-stretched position - stretch position).

We will also need the mass of the spring which was .089 kg and the mass of the hanging weight which was 0.1 kg. These values will be used for the second part as well.

Total Mechanical Energy of the System

Part 2 will consist of the same set up as in part 1, only this time we will pull down the spring with the hanging mass attached and let it freely oscillate. We will verify that the total energy in the system is constant at any position. 

Various energies within the system, which will be input into logger pro as calculated columns:

EPE_spring­ = 0.5k(stretch)²

GPE_mass = (m_hanging)(“position”)g

KE_mass = 0.5(m_hanging)”velocity”²

GPE_{sping itself} = 0.5(M_spring)(“position”) We must include this as one of the energies in the system because the mass of the spring itself contributes to the potential energy of the system. Below is a derivation of how we can calculate this.

KE_{spring itself} = 0.5(M_spring/3)(V_end)² We must include the kinetic energy of the spring itself because the bottom end of the spring contributes to the kinetic energy of the system as it oscillates up and down. Using the same steps as in the image above, we can derive an equation for the KE of the spring itself. 

Now that we know our system will have 5 various energies, we can run the experiment and input our derived equations into calculated columns. We simply hung 250 grams of mass on the spring, pulled the spring down 10 cm hit collect and let it go. We made sure to record the position and velocity vs. time graphs because we will need those values for input into our calculated energy columns.

The main column we want to observe is the very last column, which is the total mechanical energy of the system.

TE = "EPE "+"GPE MASS"+"KE MASS"+"GPE Spring"+"KE Spring"

For the entire motion of the oscillating spring, the total energy should be constant.

We can verify this as well by summing the various energies at any point.

If we add up each energy at .58 seconds, the total energy is 0.985 J.

Below is a graph of the total energy.

In theory, the total energy graph should be a straight line, so it can be concluded that we have some error within our experiment. 

Possible sources of error:

The directions in the lab manual stated to hang 250 grams of mass when oscillating the spring, we didn't follow directions and hung 100 grams instead and only realized it until the end. 

As a result, the spring went past its equilibrium point when oscillating. Our equilibrium was 0.723 m, and the spring highest position above the ground was 0.732 m. Another source of error could have been made when measuring the equilibrium value of the spring. We measured the position with logger pro and then with a meter stick. Both values did not match. 

Conclusion:

The big picture of this lab was to verify that the conservation of energy theorem for a system holds true. To do this we stretched a spring over a distance and analyzed the graph logger pro generated in order to find the spring constant k. Our system included five various energies and by vertically oscillating a spring, we can see how these energies are transferred from one form to another throughout the motion of our system. We can also conclude that when the the mass reaches its lowest point, the elastic potential energy of the spring it at its max value. And when the mass reaches its highest point, the elastic potential energy of the spring is at its minimum value. We can also say that when the mass reaches its maximum and minimum height, the mass itself has no kinetic energy at that moment in time. Even though our results are not what was expected, we can conclude that all energy in the system was accounted for.