08-April-2015: Conservation of energy using a mass-spring system

     The purpose of this experiment was to determine if energy was conserved in an oscillating mass-spring system. Normally when approaching an energy problem with a spring, the mass of the spring is ignored. For this experiment, however, the mass of the spring will not be ignored.
     Before starting the lab, an equation for gravitational potential energy (GPE) of the spring needed to be found. Since the spring was hanging, the GPE of the spring varied with its y-position above the ground. The length of the spring (y) was defined as the height from the ground to the very top of the spring minus the y-position of the bottom of the spring. So, y=H-y0. A small piece of the mass of the spring was represented by dm, and a small length of the spring was represented by dy. A relationship between mass and length was found to be

dm/M=dy/y 

Rearranging this to solve for dm gave an equation of 

dm=(M/(H-y0))*dy

 GPE is defined as mass*gravity*height, so plugging in dm for mass gives an equation of 

Integrating the above equation from the end of the spring to the very top sums up the GPE of all points on the spring. So, after integrating, an equation that can be used for the GPE of the spring is

Since the first part of that equation is a constant, it was ignored for this experiment. Therefore, the final equation used was

The exact same process was used to find the sum of the kinetic energy (KE) of all the points in the spring, since they spring is not moving uniformly. The final equation used for KE of the spring was

Vend represents the velocity of the bottom of the spring.

     After the equations were derived, the setup started. A force sensor was attached to a horizontal rod, some height above ground. The spring was attached to the force sensor and allowed to hang vertically. A motion sensor was placed on the floor, directly underneath the spring. A hanging mass, with a small piece of paper on the bottom of it, was attached to the bottom of the spring. The piece of paper was there so that the motion sensor was able to identify something, since the spring was thin.

Setup of spring, force sensor, and hanging mass. Motion sensor is out of frame.

     The first step was to determine the spring constant associated with this particular spring. The forcer sensor and motion sensor were setup with logger pro to provide force vs time data and stretch vs time data. This was so a graph of force vs stretch could be plotted, so that finding the slope of that graph would be the spring constant. The collect data button was pressed and the hanging mass was slowly pushed down toward the motion sensor. Here are the resultant graphs.

A linear fit of the force vs position graph showed the slope to be 8.057 N/m, the spring constant!

     The next step was to find an equation that can be used to calculate the stretch or compression of the spring while oscillating. This was needed to calculate the elastic potential energy (EPE) of the spring. To do this, the force sensor was disconnected and just the motion sensor was used. The hanging mass was held on the spring so that it was unstretched. The motion sensor was activated and the position relative to it was recorded. The stretch or compression was found by the unstretched position minus the position at any given time. This formula was input in a new calculated column in Logger Pro.
     All three energies were now able to be calculated. To recap, the energies involved for the whole system are: EPE, GPE of the spring, KE of the spring, GPE of the hanging mass, and KE of the hanging mass. It was time to do the final experiment. First, new calculated columns were made for each energy listed above. The equations derived in the beginning were used in their respective energy's calculated columns. Below are the columns in Logger Pro. The last column on the right is Total Energy of the System. That column is calculated by adding all the other energy columns. It was predicted that the Total Energy of the System would remain constant, as energy would be conserved.

The experiment could now start. A lot of work for an experiment that lasted no more than 10 seconds!
     The hanging mass was pulled down about 10 cm. The motion sensor was activated so data could begin collecting, and the mass was let go. The motion sensor produced velocity and position graphs which were needed to calculate the energies. After the experiment ended, a graph of Total KE/GPE/EPE and the sum of those energies vs position/time were produced. Below are the graphs.

   
     The first graph shows roughly straight lines for each total energy. This tells us that the initial energies were roughly equal to the final energies, so energy was somewhat conserved. The second graph shows the oscillation. The energies are opposing each other, producing a not-so straight line for the total energy of the system(the pink line). The total energy of the system varied between 0.9 and 1.0, not a huge variation but enough to know there was error somewhere in the experiment. Upon closer look at the position data, it showed that the hanging mass was too light, causing it to oscillate past its equilibrium point and therefore sending EPE to zero.