Lab 10_ Simple Harmonic Motion

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Virginia Commonwealth University *

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Physics

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Apr 3, 2024

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Name: Partner: Date: 10 Simple Harmonic Motion Introduction Simple harmonic motion is a type of periodic motion where the restoring force is proportional to the displacement. It can serve as a mathematical model for various motions, such as the oscillation of a spring or pendulum. A pendulum can be described as simple harmonic motion provided the displacement angle is small, less than 10 ◦ . The simple pendulum was one of the first timekeeping devices ever made. It was discovered long ago that the time for a pendulum to swing back and forth is always the same, i.e. its period is constant. For a simple pendulum, the period T is given by 𝑇 = 2𝜋√ 𝐿 𝑔 𝐸𝑞?𝑎?𝑖?? 1 where L is the length of the pendulum (from the pivot point to the center of mass of the bob) and g is the acceleration due to gravity. Experiment Part 1: Dependence of the period on mass  Open the pendulum-lab simulator and open the lab tab https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html  Play with the simulator to understand how it functions.  When you are ready to run the experiment, “Pause” the simulator.  Set the pendulum length as 1 m. Note that the pendulum's length is measured from the pivot point to the center of the hanging mass.  Make mass 1 (hanging mass) as 0.50 kg.  As we experiment on Earth, set the gravity as 9.81 ms -2  In a real pendulum, friction will be there from the pivot points and the pendulum chain. L et’s add a little bit of friction. Under the friction tab, set the tick mark at the center.  Click the stopwatch from the bottom left corner on the simulator screen to bring a stopwatch to the screen.  The time a pendulum takes to travel from a starting point back to the starting point is call one oscillation. Bring the mass to one side about 10 ◦ , first run the stop watch and then run the simulator. Measure the time for 10 oscillations.  Repeat the previous step using a 1.0 kg and 1.5 kg mass. Check that the length stays exactly 1 m to the center of the mass. Record the results in Data Table 1.

Data Table 1: Period of a pendulum with various masses [5 pts] m (kg) Time for 10 oscillations (s) T (s) Period 0.50 1.00 1.50 1. Calculate the time period for each mass and complete the Table1. From your data, how does the period of the pendulum depend on the mass? Explain. [5 pts] Note - Only consider the periods in Data Table 1 different, if they differ by more than half a second. 2. Gravity on the Moon is about 1 / 6 th of that on Earth. Keep the pendulum length as 1 m and mass as 1.5 kg. Measure the period on the Moon and record it in table 2. Theoretically, what should the period of this pendulum be on the Moon? Show your work. (Hint: use equation 1) [5 pts] 3. Perform the same experiment on Jupiter and record the period in table 2. [ 5 pts ] Table2: Period comparison in different locations The pendulum length = 1.0 m The pendulum mass = 1.5 kg L0cation Period T (s) Gravity (ms -2 ) Earth Moon Jupiter 4. How does gravity affect the period of a pendulum? [ 5 pts ]

3 Part 2: Dependence of the period on length • Pause the simulator. • Set the pendulum mass as 0.5 kg. • Set the gravity as 9.81 ms -2 . Keep some friction value as well. • Click the period from the left side menu, where you can measure the period directly. • Make a pendulum of length 0.2 m. • Raise the mass by about 10◦, play the simulator, and release the mass. Measure the period. • Open excel and create a table 3 with column headings: Pendulum length, period, and Period square. Create rows for the following pendulum lengths 0.2, 0.4, 0.6, 0.8 and 1.0 m. Your table should include a title, column headings with units. • Repeat the experiment for lengths of 0.4, 0.6, 0.8, and 1.0 m. Record the period in Table 3. 5. Calculate the square of the period for each oscillation and complete table 3. Insert a screenshot of your table below, or attach the excel file with the lab report. [15 points] Insert a clear screenshot of your Table here

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4 6. Plot a scatter graph of Time square vs. length (T 2 vs. L). Make sure to plot T 2 on the y-axis and Length on the x-axis. Add a linear trendline to the scatter graph, and be sure to display the best-fit equation on the graph. Insert a screenshot of your graph below, or attach the excel file. Note that your graph should include title, axis titles(including units), scale, and a trend or fit line [25 points] 7. Square both sides of the equation given in the introduction (equation 1) , w rite the equation of the trend line for your T 2 vs. length graph using meaningful variables and units, define the slope. (Show your work to receive full credit). [10 points ] 𝑇 = 2𝜋√ 𝐿 𝑔 𝐸𝑞?𝑎?𝑖?? 1 Insert a clear screenshot of your graph here

5 8. Determine the experimental acceleration due to gravity using the relationship you obtained in step 7 and the graph's slope. Let’s call th is value the experimental gravitational acceleration (g exp ) and record this value as g expt in Data Table 4. For full credit, show your work [5 pts] 9. Calculate the percent error between the experimental value of g you obtained and the known (theoretical) value of g and record it in Data Table 4. Show your calculation. [5 pts] Data Table 4: Comparison of accelerations [5 pts] g expt m/s 2 g theory 9.8 m/s 2 % Error

6 Results and Conclusions (10 pts) Briefly summarize the objective of today’s lab as well as the results of your experiment. State any applicable errors you calculated and give AT LEAST two possible reasons your results deviated from theoretical values.

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