HM_01_PCS-LabReport-SHM

Department of Physics Course Number PCS 125 Course Title Physics: Waves and Fields Semester/Year Fall 2022 Instructor Yuan Xu TA Name Filip Bodera Lab/Tutorial Report No. 1 Report Title Investigation of Simple Harmonic Motion in a Spring-Mass System Section No. 45 Group No. N/A Submission Date 01/24/2022 Due Date 01/24/2022 Student Name Student ID Signature* Hamza Makrod 93122 HM (Note: remove the first 4 digits from your student ID) *By signing above you attest that you have contributed to this submission and confirm that all work you have contributed to this submission is your own work. Any suspicion of copying or plagiarism in this work will result in an investigation of Academic Misconduct and may result in a “0” on the work, an “F” in the course, or possibly more severe penalties, as well as a Disciplinary Notice on your academic record under the Student Code of Academic Conduct, which can be found online at: http://www.ryerson.ca/content/dam/senate/policies/pol60.pdf

Introduction: When a spring at rest is disturbed by adding a mass placed at different amplitudes and is released, the system will be subject to a restoring force that attempts to return the system to an equilibrium point dependent on the mass added to the system. The motion of the system will be in constant periodic oscillations that can be plotted as sine or cosine functions and utilized to calculate the angular oscillation frequency. Many other relationships can be identified in this motion as well, such as the linear proportionality of a mass’ displacement from the equilibrium position to the size of the acceleration of the system. The purpose of this experiment is to explore and apply these qualitative and quantitative properties of simple harmonic motion in tandem with Hooke’s law to predict the motion of a mass in a spring system. Theory: To theorize and conceptualize important equations of simple harmonic motion we will examine a model of the spring mass system as shown in Figure 1. Figure 1: A diagram of a spring-mass system disturbed by a mass resulting in simple harmonic motion The purpose of this experiment is predicting the motion of a disturbed spring-mass system by utilizing Hooke’s law and the theory of simple harmonic motion. So, the first priority it to ensure proper understanding of these closely related theories. The basic principle behind simple harmonic motion is that it is a unique form of motion that attempts to restore a system back to its equilibrium point when a system is disturbed by applying a force directly proportional to its displacement. A common instance in which this force at work can be observed is in a playground swing. When a swing at rest is disturbed by a swinger applying their force in a direction the swing will continue to travel periodically back and forth like a pendulum until it loses energy and comes to rest at the original position (known as the point of equilibrium). This also holds true for a spring, which when pulled away from its resting position and is released will return

assuming that the spring constant remains the same throughout the experiment than the mass is the important variable. Since the greater the mass the greater the displacement from the resting point, it should be inevitable that the period of oscillation for heavier masses under constant spring stiffness the longer the period of displacement. Conversely by keeping mass constant and manipulating the spring constant, the stiffer the spring the less time it should take to complete a period of oscillation. Stiff springs have their point of equilibrium closer to the position of the spring when undisturbed, therefore the displacement of a mass would decrease as stiffness increases also decreasing period of oscillation. Figure 2: Relationship between increasing mass on a spring and the average time for a period of oscillation Now that reasonable predictions for the motion of a spring-mass system have been established, all that left is to find experimental proof behind these theories. Procedure: The experiment was performed through the use of PhET Interactive Simulations, the simulator had a lab option with a digital spring mass system. The simulator has a scale (varying from zero to three hundred grams) to allow for the user to change the mass and gauge (with ten levels) for the spring constant. It also consists of a ruler to measure height, a stopwatch to time oscillations, and line identifiers to highlight the equilibrium point and displacement from that position. The first part of the experiment kept the mass and spring constant fixed (two hundred grams and the seventh gauge respectively), and solely manipulated the amplitude. The mass was stretched to five different displacements below the equilibrium point (five to forty-five centimeters, with ten-centimeter increments) then released. A stopwatch was utilized to record the amount of time it took to complete ten periodic oscillations for each of the five displacements, then the mean value for each ten cycles was calculated to find the period of

oscillation. Then the values of the periods were compared to identify if changing the amplitude would have an effect of the period of oscillation. The second part of the experiment was completed similarly to the first but just in this case the amplitude and spring constant were kept at a fixed value while the mass of the object was manipulated. The objective was to calculate the change in period of oscillation when the mass is changed, five different masses from one hundred to three hundred grams (with increments of fifty grams) were used and released at 30 cm below their respective equilibrium points. Once again, a stopwatch was utilized to record the amount of time it took to complete ten periodic oscillations for each of the five masses, then the mean value for each ten cycles was calculated to find the period of oscillation. After the periods were calculate the frequency was calculated for each of the five periodic oscillations and graph as function with mass via Microsoft Excel to explore the effect of mass on frequency. The value for frequency was additionally used to calculate the value of the spring constant. The final objective of the experiment was to witness the affect of changing mass on the displacement of the spring. The five masses used in section two were reutilized in section three in tandem with a ruler to measure the distance of the equilibrium position to the hanging point of a spring when unstretched. Afterwards the measurements of distance were plotted into a graph via Microsoft Excel. Results and Calculations: Table 1: Summarizes findings in section 1 when mass and spring constant remain fixed, but amplitude is changed Mass m (g) Height above ground h (cm) Displacement from equilibrium ∆ y ( cm ) Period of oscillation T (s) 200 42 ± 0.5 5.0 ± 0.5 .940 ± 0.005 200 32 ± 0.5 15 ± 0.5 .943 ± 0.005 200 22 ± 0.5 25 ± 0.5 .942 ± 0.005 200 12 ± 0.5 35 ± 0.5 .940 ± 0.005 200 2.0 ± 0.5 45 ± 0.5 .941 ± 0.005 The uncertainty used here for both distance and time were instrument limited, since the instruments used were able to produce clear and precise data. The smallest measurement on the ruler tool was 1cm, so to calculate the uncertainty the smallest measurement was divided by two resulting in an uncertainty of 0.5 cm. The uncertainty for period was dividing the smallest measurement of 0.01 seconds on the stopwatch by two to get 0.005s. The periods above were calculated by dividing the total time it takes for ten oscillations to be completed by ten. For example, the first period was calculated as follows, T = 9.40s/10 = .940s, and the same process it repeated for each section. Table 2: Summarizes findings in section 2 where mass is subject to change while amplitude and spring constant remains constant. m (g) Displacement from equilibrium ∆ y ( cm ) T (s) f (Hz) k (N/m) 100 30 ± 0.5 0.667 ± 0.005 1.499 8.874 ± 0.031 150 30 ± 0.5 0.817 ± 0.005 1.224 8.872 ± 0.031 200 30 ± 0.5 0.943 ± 0.005 1.060 8.879 ± 0.031

∑ ( x i − μ ) 2 =( 6.76 × 10 − 4 )+( 7.84 × 10 − 4 )+( 4.41 × 10 − 4 )+ ¿ ( 1.225 × 10 − 3 ) + ( 1.6 × 10 − 3 ) = 4.726 × 10 − 3 ¿ σ = √ 4.726 × 10 − 3 5 ≅ 0.031 Table 3: Summarizes the findings in the 3 rd and final section of the experiment m (g) Distance of equilibrium to the unstretched spring, ∆ y ( cm ) T (s) k (N/m) 100 19 ± 0.5 0.668 ± 0.005 8.847 ± 0.041 150 26 ± 0.5 0.818 ± 0.005 8.850 ± 0.041 200 33 ± 0.5 0.942 ± 0.005 8.898 ± 0.041 250 39 ± 0.5 1.050 ± 0.005 8.952 ± 0.041 300 46 ± 0.5 1.152 ± 0.005 8.924 ± 0.041 The uncertainty of the displacement and period were calculated the same way as in section 1. The period itself was also calculated in the same manner as in section 1, for example the first period was calculate by taking the total time and dividing it by the number of oscillations (6.68s/10 = 0.668s). Figure 4: Graph of the relationship between increase in mass and distance of the equilibrium position with the spring when unstretched 50 100 150 200 250 300 350 0 5 10 15 20 25 30 35 40 45 50 19 26 33 39 46 f(x) = 0.13 x + 5.8 Mass vs Distance of Equilibrium Point from Unstretched Spring m (g) Δy (cm) The spring constant was once again calculated using m ( T 2 π ) 2 = k . .1 kg ( .668 2 π ) 2 = 8.847 .15 kg ( .818 2 π ) 2 = 8.850 .2 kg ( .942 2 π ) 2 = 8.898 .25 kg ( 1.050 2 π ) 2 = 8.952 .3 kg ( 1.152 2 π ) 2 = 8.924 The uncertainty was calculated using σ = √ ∑ ( x i − μ ) 2 N . μ = 8.8 47 + 8.850 + 8.898 + 8.952 + 8.924 5 = 8.8942 x i ( xi − μ ) 2

8.847 2.22784 × 10 − 3 8.850 1.95364 × 10 − 3 8.898 1.444 × 10 − 5 8.952 3.34084 × 10 − 3 8.924 8.8804 × 10 − 4 ∑ ( x i − μ ) 2 =( 2.22784 × 10 − 3 )+( 1.95364 × 10 − 3 )+( 1.444 × 10 − 5 )+ ¿ ( 3.34084 × 10 − 3 ) + ( 8.8804 × 10 − 4 ) = 8.4248 × σ = √ 8.4248 × 10 − 3 5 ≅ 0.041 The next step is to analyze the experimental data of the sections and consider if the findings are in support of the hypothesized predictions. Discussion and Conclusions: