Lab 21 - Mass-Spring Oscillations Lab

Purpose: To prove that the equation for period(below) is true.
T=2Pi*Sqrt(m/k)

Theory: The period of a mass-spring system can be predicted by a formula that is derived from Newton's 2nd law.

Procedure: Each group was assigned one spring of unknown spring constant. All of the springs had varying spring constants. We were assigned spring #5. We hung a spring off of a rod attached to the edge of the lab table, and measured the equilibrium position with respect to the ground. We then hung a mass on the spring, measured the change in distance and used F=kx to find the spring constant which turned out to be 29.4 N/m. We then took 1/3 of the mass of the spring which was 0.008kg, and the mass of the hook on the spring (0.0037kg), and used that information to find the mass that we needed in order to hang an exact mass of 0.115kg. The entire class did this in order to generate a table and graph for which the mass was held constant in the system, and the spring constants were varied.

0.008+0.0037+m=0.115kg
m=hanging mass needed

We then positioned a laptop right in front of the apparatus and set it to record oscillations of the spring system and we then used logger pro to measure the start and end of 10 oscillations. We then calculated the period for our spring with varying masses. We then plotted out the data.

Period v. Spring Constant for masses constant, varying spring constants

Period vs. Oscillating Mass

Below are percent error calculations for the periods that were measured in lab. The picture below shows the calculations for the first part, where we kept the hanging masses constant. For some reason, our group had the most error out of any other.

After calculating the percent errors for the second part, where we varied the masses, I noticed that we clearly had a systematic error somewhere in our experiment. As it turns out, what I believe that happened is that there was one extra oscillation in our calculations that we did not account for. When I applied a correction of (10/11), or percent errors jumped down by a lot. I do not believe that the error came from anywhere else because we were very careful, but maybe we did not start counting the oscillations from zero, which threw off all of our values for the period.

Here is my derivation for an expression for the period of a mass spring system using Newton's 2nd law.

Conclusion: What I learned in this lab is that the equation for finding the period of an oscillating spring-mass system holds true, whether you change the spring constant or the hanging mass. Also, I learned that it does not matter how much you stretch a spring before you release it into oscillation because the period will not change, the spring will just accelerate faster in order to compensate the extra distance. If the value of k was off by 5%, then the answer would be off by a factor of almost 25%. The period of a spring system should decrease if you increase the spring constant because the spring will be able to provide a larger force to an object. Using F=Ma, we can see that the acceleration will be much larger and thus, a smaller period. The period of a spring system will increase if you increase the oscillating mass by the same formula, because the acceleration will be smaller if you divide a force by a larger mass.