Lab 11 Analysis and Conclusion The goals of this lab where to determine the how mass affects the oscillations of a spring demonstrating simple harmonic motion First we had to determine whether or not our spring obeyed Hooke s Law in order to calculate the spring constant throughout our experiment To do this we recorded the force it took in Newtons to displace the spring from equilibrium We recorded the force in 1 cm increments of displacement for a total of 7 cm then graphed the results The graphed results were linear indicating that fore is proportional to displacement and therefore Hooke s Law is obeyed Hooke s law is F kx where F is restoring force x is displacement from equilibrium and k is the spring constant Given this equation and the data collected we can find the spring constant at each increment of displacement by using k F x Displacement m 1 10 2 m 2 10 2 m 3 10 2 m 4 10 2 m 5 10 2 m 6 10 2 m 7 10 2 m K constant 0 89 N 1 10 2m 89 N m 0 99 N 2 10 2m 49 5 N m 1 12 N 3 10 2m 37 33 N m 1 20 N 4 10 2m 30 N m 1 33 N 5 10 2m 26 6 N m 1 42 N 6 10 2 m 23 7 N m 1 58 N 7 10 2m 22 57 N m Since the spring force is indeed directly proportional to the displacement from equilibrium we know that the spring we are using is ideal and well described by Hooke s law Now that we have determined that we are able to start oscillating mass on the spring This will create a sinusoid graph with a spring force Fspring that will be expressed in terms of the sine function F spring t A sin t Where corresponds to the angular frequency of oscillation A is amplitude of oscillation and is the phase For all of our graphs the fitted sinusoidal function and best fit line all matched are actual data very accurately indicating the sine function produced is true to the date collected The value of the period of the oscillation is equal to 2 so it can be calculated using the sine function produced The periods of oscillation calculated in Lab 11 Data and Observations were all reasonable values which further indicates that the curve fit is accurate to the data collected We tested the effects of dropping the spring from different heights above equilibrium 5 cm and 20 cm to see how amplitude effected the period of oscillation and found that it did not have an effect on the oscillation period Lab 6 which used a simple pendulum as an oscillator produced a similar result to this where we found that drop angle did not affect the oscillation period From these to experiments we can assume that displacement does not affect the oscillation period For the next part of this experiment we tested how various hanging masses effected the oscillation period of our spring We tested three different masses 0 25 kg 1 kg and 1 2 kg and from the data collected we were able to calculate k using the equation k m and the oscillation period Mass kg 0 25 kg 1 kg 1 2 kg The resulting k values should be the slope for a graph of mass as a function of 1 2 however clearly there is an error with our data collected from 0 25 kg Looking back at that graph s motion it is evident that more than one type of motion was recorded which would result in the errors seen above To Oscillation period 2 2 m k 0 49 1 19 1 30 1 2 0 006 0 036 0 043 40 96 27 67 27 99 1 2 0 25 0 043 0 006 mitigate that we used the 1 2 25 676 which is almost exactly the same as the slope of our line The slope value of our line and the k constants of our 1 kg graph and 1 2 kg graph do agree within the uncertainty limits of each with the spring constants calculated at the beginning of this experiment when displacement is 5 10 3 m Our graph also has a y intercept of 0 0949 0 018 with the units kg This represents the effective mass of the spring however due to the error mentioned before the magnitude of our intercept value is not aligned with 1 3 of our spring mass Finally we test the spring mass oscillations with damping which means that frictional forces are intentionally applied to the system This will cause the amplitude of oscillations to decrease significantly during the experiment because of the damping effect of the air on the cardboard Additionally during this portion of the lab we has a difficult time producing a spring motion that was exclusively up and down Each attempt produced slight spinning which can be seen in our graph After graphing our data with as minimal spinning as we could manage function Ae Bt sin t C was fitted to the data along with a fitted curve to show that the amplitude decreases decays exponentially to the power of Bt The fitted curve matched our data extremely well indicating that the function produced was accurate The oscillation period of the none damped experiment was much larger than that of the damped experiment