Lab group 69

Faculty of Science Department of Physics Laboratory Report Cover Page Course Number PCS 125 Course Title Physics: Waves and Fields Semester/Year Spring 2022

Introduction The purpose of this experiment is to investigate the impact of different masses and amplitudes on the period of a spring-mass system, and to use Hooke’s law to determine the spring constant of the system. Theory In this section, talk about hook law, newtons law (F=ma) and the theory of SMH. Starting with the equation: F = ma = m ( d 2 y d t 2 ) =− k ∆ y We can substitute the equation y ( t ) = A coscos ( ωt + φ ) and setting y 0 =0 to solve for ω . Taking the first derivative of y ( t ) = A coscos ( ωt + φ ) , we get dy dt =− ωA sin sin ( ωt + φ ) . Taking the second derivative results in the equation d 2 y dt 2 =− ω 2 A cos cos ( ωt + φ ) . Substituting these equations and cancelling like terms, we can solve for k : − mω 2 A cos cos ( ωt + φ ) =− kA coscos ( ωt + φ ) . m ω 2 = k Procedure A spring was attached to a horizontal rod that was connected to a stand while hanging a 50 g hanger to the end of the spring. Two springs were available for this experiment, and the red one was chosen. In the first part of this experiment, increasing masses of 50 g (up to 250 g) were placed on the hanger and the distance from the top of the table to the bottom of the hanger were measured. In the second portion of the experiment, a total of 200g was loaded onto the spring while a motion detector was placed below the mass. The motion detector was protected by a metal cage. The motion connector was connected to

0.30 0.35 0.40 0.45 0.50 0.55 0.00 0.50 1.00 1.50 2.00 2.50 3.00 f(x) = − 9.87 x + 5.53 R² = 1 Spring Force VS Position Position (m) Spring Force (N) Figure 1: Plot showing the relationship between the spring force and the position used to determine the spring constant of the red spring. Table 1: Summary table showing the calculated spring force and position that was used to plot the graph in figure 1. Spring Force (N) Position (m) 0.512 0.4905 0.46 0.981 0.411 1.4715 0.361 1.962 0.313 2.4525 Table 2: Summary table which includes the mass, spring constant, and the phase angle for the last run that was performed in the experiment. Mass (g) Spring constant (N/m) Amplitude (m) Phase Angle (rad) Angular Velocity (rad/s) 300 6.37 0.0742 -0.0985 5.76

Table 3: Summary table showing the data that was collected from the best fit sine curve for the last trial run of this experiment. A B C D 0.067 5.645 1.618 0.00710 The letter C from the sine function was related to the phase angle. The phase angle that was calculated was not consistent with the curve fit that was manually generated on the computer. As a matter of fact, it was completely different. The relative error for the two values was found by the following: ℜ= ( − 0.328 ) − 1.618 1.618 =− 204 The spring constant that was found in part one of the experiment was consistent with the spring constant found from the fitted curve. Discussion and Conclusion There were various observations made during this lab. Firstly, when the spring constant and mass were kept the same, changing the amplitude of the simple harmonic motion does not change the period. Secondly, changing the mass but not the amplitude or spring constant appeared to change the period. Therefore, the oscillation period is not affected by the amplitude of the oscillation. The graph of the sum of the energies should be a flat line, as energy is preserved throughout the system. The graphs of the kinetic and elastic potential energy should be inverses. After plotting the graphs on LoggerPro, the total energy did not appear to be a straight line. This is because there were other forms of energy not accounted for, such as gravitational potential energy. The individual graphs of kinetic and elastic potential were inverses, as expected. After a while, the amplitude of the oscillation will diminish until the mass reaches equilibrium. This is because of the force of gravity working against the mass whenever it is moving upwards. Appendix Graph 1: 200g 5cm amplitude

Table 1: 200g 5cm amplitude Table 2: 200g 10cm amplitude Table: 300g 10cm amplitude

Energy vs time graph (bottom):