Pendulum and Springs Lab Report

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University Of Arizona *

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Physics

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Dec 6, 2023

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Pendulum and Springs Lab Report Maria Acosta-Perez Lab Partner: Clarissa Course: PHYS 181-001 Due Date: 10/25/23

Abstract A pendulum undergoes simple harmonic motion as its force and displacement are proportional. A pendulum also has oscillation motion. This motion is where the object in this case the pendulum moves from a fixed point back and forth. In this lab the spring constant in two springs are measured and periods in simple harmonic motion are measured using a pendulum The end result is that Spring 1 the spring constant is 2.97 kg/s^2 and for spring 2 it is 3.17 kg/s^2. The slope of the average periods in correlation to length is 0.1387 cm/s. Introduction Harmonic motion is where an object force and displacement can be considered proportional. Simple harmonic motion can be seen in springs where this mathematically model has been taken and applied to in our everyday lives. A few examples include the regulation of a clock the pendulum in clocks has oscillation that undergoes simple harmonic motions. Guitar strings follow simple harmonic motion when they are plucked as vibrations occur after the sting has been plucked. Theory In the first portion of the lab, we are looking to solve for the tension that is being applied to the pendulum. When the pendulum is in motion, we know that there is an angle that is being created in the vertical axis. Knowing this we can derive an equation to solve for the tension (T) using newtons second law. Newton seconds law is as follows: Σ F = Ma (1) We can create a free body diagram of the pendulum when it is at rest and when it is being released. Released Rest T T Mg Mg

From the Free body Diagrams, we are able to apply Equation (1) and derive the following workup shown in (2) which explains both the forces that are being applied in the X and Y direction in the pendulum to get the total forces being applied to the pendulum as a whole. Σ Fx = m a X = Tsinθ Σ Fy = m a y = Tcosθ -mg (2) ΣT = Iα = m L 2 α = mgsinθ Using equations 1 and 2 we are able to derive equation 3 where T is the period of oscillation. This is measures as the time the pendulum takes to complete a full motion. T = 2 π ω (3) To solve for T we need to first solve for ω which is the angular frequency and the equation for that is shown in equation 4. Where L is the length of the pendulum. ω = √ g L (4) Substituting equation (4) into equation (3) we are able to get the final equation for the period of oscillation that is shown in equation (5) T = 2 π √ g L (5) In the second part of lab, we are measuring the spring constant (x) the equation for this originates from newtons second law as well shown in equation 1. Therefore, we are able to use a free body diagram to solve for the spring constant. Where k is the spring constant and x is the displacement from equilibrium. kx Mg Using equation 1 and the free body diagram we are able to derive equation (6) ma =− kx (6) The angular frequency of the mass and the spring can be expressed with equation (7) k is the spring constant that we are looking to solve, and m is the mass that is being added to the spring.

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ω = √ k m (7) Using equation (3) to plug into equation (7) we are able to create the final expression that was used to solve for the spring constant k that is expression (8) T = 2 π √ m k (8) Procedure In the first portion of the lab, we measured the period of four different pendulums that each had different masses, their masses can be seen in table 1. All the pendulums had the same shape, just a different mass and the pendulum was attached to two strings from the hook that is on each pendulum. To measure the period of each pendulum we used a stopwatch and released the pendulum from the same angular displacement for each run. We were measuring the time that the pendulum took to complete 10 cycles of swinging back and forth. We started the timer as soon as the pendulum was released and stopped it as soon as it completed its 10 th cycle. We then took that time and divided it by 10 to get the measured period for that run. For each pendulum we did this three times and the results for this first part of the lab can be seen in table 1 at the results section. The second portion of the lab was measuring the period of the object when the size of the pendulum changes. The set up was the same as we used in the first part of the lab where we used the two knobs to to hold the strong and we attached the string to the hook on each object. For this part we used wooden balls that varied in diameter, they progressively got bigger. The height of the string was the same as the first lab which measured 53cm in heights, however the hooks on each of the balls varied in length making the overall length of the set up with the string different. The different heights for the hooks can be seen in table 2. The lab was conducted the same as the first part where we used a stopwatch, and measured the time each ball took to complete 10 cycles and divide that time by 10. The data collected for this second part of the lab can be seen in the results section table 2 as well as the diameter for each ball used. The third part of the lab measured the relationship between the period of the pendulum and height. In the previous parts we changed the mass, and the diameter for this one we changed the length of the string used to measure the affect it has on the period of the pendulum. To get more accurate results for this portion of the lab instead of a stopwatch we used a photogate and the PASCO Capstone software to measure the period. As before we used the same angular displacement for each run and used a pendulum for this part of the lab. We measured a total of seven different lengths, and to change the lengths we used the knobs that hold the string to raise and or lower the string to adjust to a new length for each run. Each length was done three different times and the results for this part of the lab can be seen in the results section table 3. For the fourth part of the lab, we began to start working with springs, for the fourth part we measured the extension of two different springs after we applied a known force. We had the spring attached to the force sensor and we added the mass at the bottom of the spring. To measure how much the spring extended we measured what the equilibrium length of the two springs were, to do this we place each spring on the force sensor and measured the spring in length. The equilibrium lengths are found in table 4. A known mass was then attached the bottom

of the spring and we measured the length of the spring after the mass was added. We then subtracted the equilibrium length form the measurement we got after applying the mass to get the extension difference after adding mass. We did this for five different masses, and we did those five different masses on both the springs, the mass added and extension for each spring can be seen in the results section table 5. The fifth and final part of the lab measured the spring constant using the period obtained. For this part of the lab we used the same two springs as used in part four and we used PASCO to measure the force using the force sensor. We placed 50 grams on the end of the spring and released it so that the spring would move vertically in a simple harmonic motion. Using PASCO, we measured 10 oscillations the 50grams we added. PASCO created a graph, and we used that graph to determine the period of oscillation. This was performed on both the springs, and with the period of oscillation we were able to determine the spring constant using equations 7 and 8 for both springs. The period and spring constant for both springs is in table 6 in the results section. Sample Calculation and Results In the first and second part of lab we are measuring the period of the pendulum when changing the mass and diameter of the pendulum. Based off the results in the first lab as shown in table 1 we can see that mass does not affect the period. Table 1 supports this by demonstrating no relationship with mass and period. In the second part of the lab, we were changing the diameter of the pendulum and that does affect the period because the diameter affected the length resulting in a change of period. This can be seen in table 2, where the relationship is that as the length increased the period increases. Part 1: Period vs Mass Material Mass (g) Average Period (sec) Wood 6.6 1.45 Plastic 13.2 1.47 Silver 23.3 1.46 Gold 67.3 1.48 Table 1 : These are the results from the first part of the lab where we were changing the mass of the pendulum each run. The size of the pendulum was the exact same, but they were made of different material making them weigh differently. Part2: Period vs Diameter Length (cm) Diameter (cm) Average Period (s) 52.5 2.5 1.46 54 4 1.48 55.5 5 1.49 57 6 1.51 Table 2: This table shows the results from part two of the lab where we are measuring how diameter affects the period of the pendulum. Based off the results we can see that as the diameter increases, we will see an increase in the average period, meaning that overall changes in the diameter do affect the average period.

Part3: Period vs Length √L (cm) Average Period (sec) 7.1 1.47 7.2 1.5 7.24 1.51 7.3 1.52 7.5 1.55 7.6 1.58 7.8 1.56 Table 3: Is the results from the third part of the lab where we are seeing how and if the length affects the period. As we can see from table three, we can claim that length does affect the average period. As the length increases so does the average period. Equilibrium Length Spring 1 Spring 2 5.5 cm 7.2 cm Table 4: This is the equilibrium length that we used from part 4 in the lab to measure the extension of the springs. Part 4: Measure Spring constant Mass (g) Spring 1 Extension cm Spring 2 Extension cm 10 7.4 11.6 20 9.2 13.2 30 10.5 14.6 40 11.7 16.2 50 13.5 17.5 Table 5: This table shows the results in which were gathered for part 4 of lab where we were seeing the extension between two different springs. Spring 2 was much longer than spring 1 as shown in table 4. This resulted in it having a much larger extension each time, it can also be noted that as the mass increased the extension of the springs increased. This was the same in both springs and that can further be looked at in the table. Part: 5 Spring Constant using period Spring 1 Spring 2 Period 0.65 7.2 cm Spring Constant 2.97 kg/s^2 3.17 kg/s^2 Table 6 : This table is the results for part 5 of the lab where we were looking for the spring constant after we added 50 grams of weight to the spring and released it gently. Using PASCO

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and equations 7 and 8 we were able to calculate the spring constant, and once again spring 2 is the larger spring and it had a larger spring constant compared to spring 1. 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 f(x) = 0.14 x + 0.5 R² = 0.83 Graph #1 Linear () √L (cm) Average Period (sec) Graph #1: This graph shows the relationship between length and the average period. Overall, it can be understood that as the length increases so the does average period. The measured slope of the graph is 0.1387 cm/s. We also calculated the gravitational constant and we got 2. The difference between the measured and calculated slopes can be due to sources of errors, that are further discussed in the discussion and conclusion section. 5 10 15 20 25 30 35 40 45 50 55 0 2 4 6 8 10 12 14 16 18 20 f(x) = 0.15 x + 10.18 R² = 1 f(x) = 0.15 x + 6.05 R² = 0.99 Graph #2 Spring 1 Extension cm Linear (Spring 1 Extension cm) Spring 2 Extension cm Linear (Spring 2 Extension cm) Mass (g) Spring Extension (cm) Graph #2 This graph shows the relationship between mass and spring extension. The overall pattern of the springs is that the greater the mass the greater the extension in the spring. Spring

one had a slope of 0.147 cm/s and the calculated slope for spring 1 was also 0.147 cm/s. For spring 2 the measured slope was 0.148 cm/s and the calculated slope was 0.022cm/s. The slopes for the springs were done by using equation 4. Discussion and Conclusion We can conclude that this experiment as a whole was a good experiment as the results, we obtained were the expected results. Such as in the first part of lab we expected for the mass to not have an impact in the period that is because mass is not in the equation or have correlation to the period as a whole. Based off the results in table 1 we are able to see that mass does not have an effect on the period. However, I did learned that the diameter does change the period and that can be seen in Table 2 whereas the diameter got larger so did the period. A few things to consider in terms of sources of errors for this lab is small angular approximation, and the change in lengths with the change in diameter. When we changed the pendulum to be wooden balls, the hanger on each ball was different in height and then the diameter of each ball was different in diameter. This resulted in having two different changes in one singular experiment which can affect the given results being interpreted.

Pendulum and Springs Lab Report

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