PCS-125-Lab 1 (2)

In the lab the main equation used was ( . This equation shows how the period of an 𝑇 = 2π ? ? ) object in simple harmonic motions changes when the mass and the spring constant are changed. The derivation of this equation is shown below. It is known that to describe and understand simple harmonic motion this equation is used. ???𝑎?𝑖?? 1: ?(?) = 𝐴𝑐??(?? + Φ) It is also known that since cos(x) is a periodic function then the following equation is true. ???𝑎?𝑖?? 2: 𝑐??(?) = 𝑐??(? + 2π) Combining equation 2 and 3 will allow us to derive the period equation. ???𝑎?𝑖?? 3 : 𝐴𝑐??(?? + Φ) = 𝐴𝑐??(?(? + 𝑇) + Φ) ???𝑎?𝑖?? 4: ? = ?? + Φ ???𝑎?𝑖?? 5: 𝑐??(?? + Φ) = 𝑐??(?? + ?? + Φ) Sub equation 4 into equation 5 Equation 6: 𝑐??(?) = 𝑐??(? + ?𝑇) Comparing equation 6 and 2 we can see that ?𝑇 = 2π Simply solve for T and we get the period equation to be 𝑇 = 2π ? = 2π ? ? I predict that for experiment one the period will not change since in experiment one the only thing that we are changing is the amplitude and as seen from the equation the amplitude will not affect the period. In experiment two I predict that the period will increase as the mass increases. And for the third experiment, I predict that the displacement will increase as the mass increases. Producer: Experiment 1:

Table 1: Amplitude and Period For each trial the time for ten oscillations was measured instead of just one. This is because it reduces the risk of human error (stopping the stopwatch) involved with the experiment. As expected the measured periods for different amplitudes are more or less the same. This is because the equation for the period ( does not include the amplitude therefore it is 𝑇 = 2π ? ? ) inferred and shown that changing the amplitude will not have any effect on the value of the period. The uncertainty for the period will be 土 0.005 Experiment 2: Mass vs Period Amplitud is fixed at 0.2m. K is fixed at the second increment. Mass(kg) Time For Ten Oscillations Trial 1 (s) Time For Ten Oscillations Trial 2 (s) Time For Ten Oscillations Trial 3 (s) AverageTim e For Ten Oscillations (s) Average Period (s) Average Period^2 (s) 0.05 6.32 6.33 6.35 6.33 0.633 土 0.005 0.400 土 0.005 0.1 8.92 8.90 8.90 8.91 0.891 土 0.005 0.0.794 土 0.005 0.15 10.92 10.93 10.88 10.91 1.09 土 0.005 1.19 土 0.005 0.2 12.62 12.65 12.63 12.63 1.263 土 0.005 1.59 土 0.005 0.25 14.10 14.13 14.11 14.11 1.411 土 0.005 1.99 土 0.005

Graph 2: Force Of Gravity vs Displacement Uncertainty was calculated by taking the smallest increment on the measuring device(ruler) which was 1mm and dividing it by 2 which resulted in an uncertainty of 0.5mm. A total of 6 masses were used since that was the amount used in experiment 2. Calculation for the spring constant: There are two ways of calculating the spring constant in this experiment. The first is to calculate the slope of the graph (force of gravity vs displacement). The second is to use Hooke's law to isolate for the spring constant. Both methods will be shown below. First method: Slope = k ????𝑒 = (?2−?1) (?2−?1)