Oscillation Lab Report

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Kirsten Voice 10/24/2023 Phys 1051L, Section 003 Oscillating Spring-Mass Lab Report

Names: Advika Sumit, Mihir Barve, Hailey Doerflein, Kirsten Voice Course: PHYS 1051L, Section 003 Date: 10/17/2023 Lab 08: Simple Harmonic Motion Part II Lab 07: Simple Harmonic Motion Part I: Part A: Predicted Graph for Stretch of Spring: The generated graphs of the spring match our prediction. Equations of Trendlines & Uncertainty for each of the Springs Spring ID # Equation for Trendline Uncertainty* 10338 F=5.53x + (-1.18) ± 0.11 11158 F= 7.65x + (-1.88) ± 0.15 13220 F= 16x + (-4.63) ± 0.27 15247 F= 34.1x + (-9.96) ± 0.27 14397 F= 53.6x + (-17) ± 0.51 16159 F= 55.2x + (-16.5) ± 0.78 12122 F= 78.4x + (-24.1) ± 1.0 Questions : 1. What is similar between the mathematical models for each spring? Each mathematical model is linear for each spring. 2. What pattern, if any, do you notice between the numerical value(s) in each mathematical model and the relative stiffness of each spring? The stiffer the spring the steeper the slope of the plot. 3. Use dimensional analysis to determine the units for the numerical value(s) in the equations for the trendlines. Describe what the numerical value(s) physically mean. The units for the numerical values are Newtons/meter. This represents the spring constant of each spring. Factors That May Impact the Period of a Freely Oscillating Spring/Mass System:  Mass hanging from the spring ( hanging mass )  Distance spring is stretched ( amplitude )  Stiffness/stretchiness of the spring ( spring constant ) Experimental Set Up:

Estimation of Uncertainties: Measurement Calculation Uncertainty Value Mass Uncertainty 0.0001/2 +/- 0.00005 kg Length (amplitude) 0.001/2 +/- 0.0005 m Uncertainty of equations for trendlines for each spring excel produced varied based on spring Period 2pi/9.98=0.62957 2pi/ (9.98-0.0031) =62977 2pi/ (9.98+0.0031) =62938 +/- 0.0002s Experimental #1: Distance the Spring is Stretched Research Question What affects the period of an oscillating spring/mass system? Dependent variable (DV): Period of the oscillating spring/mass system (seconds) Independent variable (IV): Amplitude of the spring (cm) = 1 cm, 2 cm, 3 cm Control Variables (CV): Stiffness of the spring = Stiffest Spring (ID: 12122), spring constant = 78.4 Mass hanging from the spring = 0.252 kg Testable Hypothesis: The amplitude of the spring is related to the period of the oscillating spring/mass system. Table 1:

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Table 1: This table shows the compression length (cm) as the independent variable, the period (s) as the dependent variable, and the uncertainty of omega. Graph 1: Figure 1. This graph shows no relationship between Spring Stretch (cm) vs. Period (sec) when the Spring Stretch was the independent variable. Error bars are shown on both the x and y axis but are unable to be seen due to small uncertainty or standard deviation. The equation of the linear slope is acceleration = 0.0001(stretch of spring) + 0.3711 and the R 2 value is 0.999. Conclusion: Based on this experiment, we concluded that the amplitude of the spring has no effect on the period of the oscillating spring/mass system. Based on the graph, as the amplitude of the spring increased (spring was stretched more), the period of the oscillating system stayed the same. We were able to identify this because the error bars of the varying spring amplitudes all overlapped, indicating that there is not a significant difference between the periods of the systems and the slope of the line was almost zero. Additionally, the period values of all the trials were extremely close in values showcasing that they were unaffected by the change in amplitudes. Additionally, the line of best fit indicated a linear relationship with an R 2 value of 0.999.

Experimental #2: Mass Hanging from the Spring Research Question What affects the period of an oscillating spring/mass system? Dependent variable (DV): P eriod of the oscillating spring/mass system (seconds) Independent variable (IV): Hanging Mass = 0.252 kg, 0.302 kg, 0.352 kg, 0.402 kg, 0.452 kg Control Variables (CV): Stiffness of the spring = Stiffest Spring (ID: 12122), spring constant = 78.4 Distance Spring is stretched (amplitude) = 1 cm Testable Hypothesis: The mass hanging from the spring is related to the period of an oscillating spring/mass system. Table 2: Table 2: This table shows the mass (kg) as the independent variable, the period (s) as the dependent variable, and the uncertainty of omega. Graph 2:

Figure 2. This graph shows a positive power relationship between Hanging Mass (kg) vs. Period (sec) when the Spring Stretch was the independent variable. Error bars are shown on both the x and y axis but are unable to be seen due to small uncertainty or standard deviation. The equation of the linear slope is acceleration = 0.7427(mass) 0.5033 and the R 2 value is 0.9997. Conclusion: Based on this experiment, we concluded that the hanging mass of the spring does have an effect on the period of the oscillating spring/mass system. As the graph shows, as the hanging mass of the spring increased, the period of the oscillating system also increased. This represented a positive correlation. We were able to identify this because the error bars of the varying hanging masses did not overlap, indicating that there is a significant difference between the periods of the systems. Additionally, the line of best fit indicated a power relationship with an R 2 value of 0.9997. Experimental #3: Stiffness of the Spring Research Question What affects the period of an oscillating spring/mass system? Dependent variable (DV): P eriod of the oscillating spring/mass system (seconds) Independent variable (IV): Stiffness of the Spring = 5 spring with the highest spring constants Spring IDs = 12122, 16159, 14397, 15247, 13220 Control Variables (CV): Mass hanging from the spring = 0.152 kg Distance Spring is stretched (amplitude) = 1 cm Testable Hypothesis: The stiffness of the spring is related to the period of an oscillating spring/mass system

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Table 3: Table 3: This table shows the spring constant (N/m) as the independent variable, the period (s) as the dependent variable, and the uncertainty of omega. Graph 3: Figure 3. This graph shows a negative power relationship between Spring Constant (n/m) vs. Period (sec) when the Spring Constant was the independent variable. Error bars are shown on both the x and y axis but are unable to be seen due to small uncertainty or standard deviation. The equation of the linear slope is acceleration = 2.4514(spring constant) and the R 2 value is 0.9986. Conclusion : Based on this experiment, we concluded that the stiffness of the spring does have an effect on the period of the oscillating spring/mass system. Based on the graph, as the stiffness (spring constant) of the spring increased, the period of the oscillating system decreased. This represented a negative correlation. We were able to identify this because the error bars of the varying hanging masses did not overlap, indicating that there is a significant difference between the periods of the system and the slope of the best fit line was negative. Additionally, the line of best fit indicated a power relationship with an R 2 value of 0.9309. Lab 08: Simple Harmonic Motion Part II Lab Records

Table 1 . Portions of established scientific model for the behavior of an oscillating spring/mass system. Scientific Models: Description: T=2 mk π  Summarizes relationships in the oscillating spring/mass system  T= period  m= hanging mass  k= spring force constant  derived from Newton’s 2nd Law Fspring=-kx  Hooke’s law  Fspring is the force the spring exerts  K is the spring force constant  x is the distance the spring is compressed or stretched Figure 1: Experimental Outcomes Organizer Comparison Between Experimental Mathematical Model & Established Scientific Model: The results of the experimental mathematical models we discovered during the experiment match with the established scientific model in Table 1 above. Our model, when mass and the spring constant were tested as IV’s separately reveals the same power function as the established model does. We found that distance stretched or compressed does not affect the period as the established model also agrees with. White Board of Model:

Figure 2. This figure shows the whiteboard created with the scientific model compared to the experimental models created in the prior lab. First Challenge Plan: Procedure: We will perform an experiment similar to what we did in the prior lab to find the spring force constants of three springs. This will be done by hanging each of the three springs, one at a time, from the Force Sensor and then gently pulling it down and lifting it while simultaneously running a position vs force graph. The graph will then produce an equation that identifies the spring constant of the spring as the slope. We will then perform the experiment again, but this time attaching the unknown mass to the bottom of the spring. We will determine the period of the spring by running a position vs time graph. We can then plug in the periods and spring constants of each of the three springs into the established scientific model from Table 1 above to solve for the mass. We will do this with all 3 springs to confirm the correct mass and compare it to its true value. Uncertainty Table: Measurement Calculation Uncertainty Value Mass Uncertainty 0.0001/2 +/- 0.00005 kg Uncertainty of equations for trendlines for each spring excel produced varied based on spring Period 2pi/9.98=0.62957 +/- 0.0002s

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2pi/ (9.98-0.0031) =62977 2pi/ (9.98+0.0031) =62938 Data Table: Spring Constant Period (seconds) Spring 1 (16152) 55.9 0.44 ∓ 2pi/17.6 0.0067=0.357s ∓ Spring 2 (12312) 75.0 0.45 ∓ 2pi/20.5 0.00084=0.3064s ∓ Spring 4 (14399) 52.2 0.55 ∓ 2pi/17.3 0.0014 = 0.363s ∓ ….; Calculations: Mass calculation Mass Determined Spring 1 m=kT/2pi= 55.90.357/2pi 0.180 kg Spring 2 m=T2k42 0.178 kg Spring 3 m=T2k42 0.174 kg Average Mass .1773 kg Mass of Unknown Using Balance: Mass of Unknown 0.170 kg Summary: After plugging in the experimentally determined spring constants and period values we determined mass values of 0.180 kg, 0.178 kg, and 0.174 for the unknown which were corresponding to each of the three springs (16512, 12312, 14399) that we used. The true mass of the unknown was measured to be 0.170 kg which was extremely close to the calculated masses we determined with little error. Because of this, we determined the challenges to be successful. Error was expected to be there due to human error, random error, and equipment that may not be that accurate uncertainties throughout the lab are given in a range of 0.44 to 0.55. These values of uncertainty allow us to see how accurate results are compared to the scientific model. The uncertainties can further explain the role of unknown variables including possible air resistance. 1. Adequately explains why each challenge was or was not successful. 2. The explanation includes a discussion about the role of uncertainties. Second Challenge Plan: Procedure :

The purpose of this second challenge activity is to build a spring/mass system that has a period of 0.853 seconds . We will conduct this experiment by first plugging the period we are targeting into the established scientific model from Table 1 above as well as the spring constant of each of the spring being used for this experiment. We can solve for the mass necessary to achieve the targeted period using the model, and then place that amount of mass hanging from the spring to create the oscillating system. We will then use the force sensor to measure the actual period value of the spring with its corresponding mass and compare it to the desired period of 0.853 seconds. Data Table: Period Measured 0.853 seconds Period Measured in the presence of instructor. 0.846 seconds Spring constant= K 6.69 0.072 ∓ Mass= m (calculated) 0.853=2pim/6.69 m= 123.3 g Experimental Period: 0.848 s Summary: After plugging in the experimentally determined spring constants and period values we determined a necessary mass value of 123.3 grams for the spring used after plugging in the corresponding spring constant into our model for the targeted period of 0.852 seconds. The actual period value measured with the added masses was 0.848 seconds, which was extremely close to the targeted period of 0.853 seconds. This showed little error. Because of this, we determined the challenges to be successful. Error was expected to be there due to human error, random error, and equipment that may not be that accurate. These values of uncertainty allow us to see how accurate results are compared to the scientific model. The uncertainties can further explain the role of unknown variables including possible air resistance. The research question at hand is “how does hanging mass, spring constant, and amplitude impact the period of an oscillating spring/mass system?” Throughout the lab there were three independent variables (IVs) tested including change in mass, change in spring constant, and change in the distance the spring is stretched. Each time an IV was tested, the other two IVs were kept constant for accuracy. The first independent variable tested was how far the spring was stretched. A total of 5 data points were taken, each with the spring distance increasing by 1 cm per data point. A

spring constant of 78.4 N/m and the mass was 0.252 kg, each held constant. The amplitude of the spring was graphed along the x axis and the period (s) was graphed along the y axis, as period is the dependent variable. The testable hypothesis was that the amplitude of the spring is related to the period of the oscillating spring/mass system. Through graphing data points and using excel to establish a scientific model, no relationship between amplitude and period was formed (graph 1). A linear scientific model of period = 0.0001 (amplitude) + 0.3711 with an R 2 value of 0.999 was formed in excel. Since the R 2 value was so close to 1, this means that the data and linear model are precise and fit well together. The second independent variable tested was change in mass hanging from the spring. The testable hypothesis was concluded to be, “the mass hanging from the spring is related to the period of an oscillating spring/mass system” (Lab 1). A spring constant of 78.4 N/m and an amplitude of 1 cm were both held constant, while the mass hanging on the spring was changed by 0.05 kg per trial. 5 data points were taken and used to create a graph in excel which then could be used to draw conclusions of possible relationships between the IV and DV. From graph 2, a positive power relationship was found to fit the data. A scientific model of period = 0.7427 (mass) 0.5033 with an R 2 value of 0.9997 was calculated in excel. Since the R 2 value was so close to 1, this means that the data and linear model are precise and fit well together. The third and final IV tested was a change in spring constant. 5 various spring constants were found using the computer and Force sensor. After finding the spring constants of 5 springs that had a wide range of values, a mass of 0.152 kg and an amplitude of 1 cm were held constant through 5 data points. In excel, graph 3 produced a negative power relationship between spring constant and period. The scientific model formed was period = 2.4514 (spring constant) -0.494

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with an R 2 value of 0.9986. Since the R 2 value was so close to 1, this means that the data and linear model are precise and fit well together. All three of these models led to one experimental model based on the equation T=2π  m/k. T represents period, m is hanging mass, and k is spring constant. As the first graph showed no relationship between amplitude and period, there is no variable that represents x in the experimental model. The hanging mass and the spring constant were both found to impact the period, thus they both have a variable (m and k) in the experimental model. As k increases, the period will decrease because k is a denominator. As m increases, the period with also increase because m is a numerator. The equation relationships are shown in figure 2. The relationship between period and hanging mass is correlational because there are other factors other than the hanging mass that impact the period of the system. The relationship would hold true in our situation, but in other conditions these might not hold true. Hypothetically and scientifically, they should hold true, but since we did not test a wider range of variables, we cannot conclude a trustable statement. The relationship between spring constant and period is also correlational, as hanging mass has an impact on the period of the system. The factors hold true in our lab, but taken into an outside setting, the factors could change and not hold true The data collected in each experiment should be trusted because of repeated experimental design and low uncertainty of the data. The force sensor recorded uncertainties of less than 0.01, indicating high levels of accuracy and precision. Each of the IVs were also tested a total of 5 times, with control variables being kept constant to provide accurate and unbiased results. The equations determined by excel should also be trusted as each of the mathematical

models were derived from the scientific model and the fact that the R 2 values were within 0.001 of 1. The equations and R 2 values correlate to each other in the sense that if an equation produces an R 2 value closer to 1, then that equation fits the data points more accurately. Since the values were extremely close to 1, a conclusion that excel should be trusted can be drawn. The two challenges impacted the trust we had in our final model in a positive way. The challenges allowed more strength and support in the final model and showed that mass and spring constant do in fact have opposite effects on the period of an oscillating spring/mass system. Group 1 and 2 had the same scientific model as our group had, which allowed for further support of our own experiments and data. The groups also had similar R 2 values that matched the equations developed in excel. These two groups support our experimental model by indicating mass and spring constant have opposite relationships to period, and that the amplitude of the spring has no correlation to the period of the system. As the scientific model is T=2π  m/k, each of the variables can describe a relationship to the oscillating system according to the rules of physics. The mass (m) has a positive relationship with period, which explains that as the hanging mass increases the period also increases. The mechanism for this variable according to F=ma, indicates that as the mass increases the acceleration decreases. With a decreased acceleration, a slower velocity of the system will also be observed, thus lengthening the period of the system. The spring constant (k) has a negative relationship with period, meaning as the spring constant increased the period decreased. The explained mechanism for this variable is that the force applied to the object has a direct correlation to the acceleration. As the force increases, the acceleration with also increase, and in turn the velocity will increase. An observed increase in velocity will also indicate a shorter

period of the system. Lastly, amplitude was found to have no correlation with period and cannot be included in the experimental model. A few limitations presented in the lab include the fact that only a certain range of mass, spring constant, and amplitude were tested. A larger scale and range of these IVs should be included in a further experiment to obtain for reliable and accurate results. The claim about the effect of amplitude on period holds for a spring with a spring constant of 78.4N/m, a mass of 0.252kg and amplitudes between 1 and 5 cm. The claim about the effect of hanging mass on period holds for a spring with a spring constant of 78.4N/m and an amplitude of 1 cm and masses between 0.252kg and 0.452kg. the claim about the effect of spring constant holds for an amplitude of 1 cm and a mass of 0.152kg and spring constants between 16 and 78.4 N/m. Each of these limitations comes from the range presented in the lab and cannot be applied to wider values and ranges in further experimentation. Some assumptions presented during the lab might have caused biased or inaccurate results. These assumptions include assuming the force of the weight never changes, the spring stretches uniformly, and that the gravity is constant. While the changes in these assumptions might only lead to small and minuet number changes, they can overall lead to mathematical equation changes. We were able to assume these variables did not change, as they had no part in the scientific equation given in the experimental organizer. To improve further experimentation, additional trials should be run, as well as larger variation of IVs. A larger range of IVs would allow for a greater collection of data and increased

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reassurance of experimental models developed in excel. An additional suggestion would be to include air resistance and friction in the models so as to get even more accurate results.

Oscillation Lab Report

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