PHY201 Lab8 Pendulum

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Feb 20, 2024

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Physics 201L Lab #8: Pendulum Motion Introduction: This week our experiment will investigate the periodic motion of a simple pendulum. The periodic motion of a simple pendulum is another discovery of Galileo (which originated from observations of swinging lamps in chapel) and resulted in the pendulum clock being used for centuries as the first precise timepiece. A requirement for periodic motion is a restoring force that is opposite the direction of motion and pulls the object back to an equilibrium position. The word periodic tells us that the motion repeats itself – it runs on a cycle. The period T is the time in seconds it takes to complete one cycle of its motion. Some common examples: a child swinging back and forth on a swing-set, a floating object that bobs up and down as waves travel through the water, or an object bouncing on a spring. One example of periodic motion is a simple pendulum that bobs back and forth (like the child on a swing). A pendulum consists of a bob of mass m suspended by a light string of length L fixed at its upper end, as shown in the figure below. Since the string is light, we ignore its mass. When the bob is released it swings back and forth over its path. Note that mgsin  is the tangential force that acts as the restoring force that pulls the bob back to its equilibrium position, causing the pendulum to have periodic motion. For small angles this force can be simplified using the small angle approximation, which says that sin    when the angle is in radians (not degrees). For small angles the restoring force of a simple pendulum is approximated as F r  -mg  . This restoring force is directly proportional to the displacement and in the opposite direction of motion. When the restoring force is directly proportional to the displacement, the object exhibits simple harmonic motion . (The restoring force of Hooke’s l aw is also directly proportional to the displacement, so masses bouncing on springs will also experience simple harmonic motion).

2 In this experiment you will examine what variables influence the period T of a pendulum (and which do not influence the period of a pendulum). You will consider the pendulum variables: pendulum mass (m), the string length (L), and release angle (  ). Each measurement run you will measure the period T as you change one of the variables (and keep the others constant). Prelab Reading Assignment: Read sections 15.1, 15.2, and 15.5 of the textbook for background information. Equipment: Vertical stand with suspension arm, meter stick, string, pendulum bobs, PASCO 850 Universal Interface and Capstone program Procedure: Part 1: Equipment Setup The experiment will be conducted using a simple pendulum that will swing between a photogate at the bottom of its motion. We will vary the pendulum length, mass, and angle amplitude and measure the period T of the pendulums motion. 1. Setup your simple pendulum by suspending the string from two support points on the horizontal suspension arm. 2. Check that the photogate is plugged into the PASCO 850 Universal Interface. Turn on the PASCO 850 Universal Interface, and open the PASCO Capstone program from the lab computer’s desktop. 3. Click on the Classic Templates to begin. Now select Hardware Setup on the left- hand side. Click on the Channel 1 port input and select the photogate option. 4. Now select Timer setup on the left-hand side of the screen. Click next for pre- configured timer. The photogate should appear and already be selected. If not ask for help from your lab instructor. Click Next. 5. Select the Pendulum Timer option. 6. For the measurement data select the period option. Deselect all the other data options. Click Next. 7. Check the diameter (width) of your pendulum bob and input it into the requested box. The standard pendulum bob width is 1.60 cm. Click next. 8. Now rename your timer or leave as is. Click Finish. Click again on Timer setup to minimize the window. 9. Add a data table to your experiment. Display the period data in the data column.

3 Part 2: Pendulum Period vs. Angle. 1. Select the brass bob and measure its mass. 2. Place the brass bob on your pendulum string and measure the pendulum length L from the center of the bob to the lower edge of the support arm at the midpoint between the two string attachment points (directly above the bob). 3. Using a protractor, swing the bob up by about  = 5 degrees from vertical. 4. Release the bob from the 5-degree amplitude. Let it swing a couple times so it stabilizes, then start the timer to record data. Let it record for about 30 seconds or at least 15 cycles. 5. Press stop to record the data. Add significant figures on your data table to have at least 3 significant figures . Using the statistic button in Capstone, record the average value of the period and its standard deviation (period uncertainty). 6. Repeat the period measurements by increasing the release angle to  = 10, 15, and 20 degrees. 7. Create a small data table for part 2 with data columns of angle  , period T, and period uncertainty  (standard deviation). Question #1: In part 2, do you observe any noticeable trend when you varied the angle? Does the pendulum period depend on the release angle? (Does the period change in relation to the change in the angle?) Explain . Question #2: Why do we measure the pendulum length to the center of the bob, and not to the furthest end of the pendulum bob mass? Explain . Part 3: Pendulum Period vs. Mass Repeat the period measurements by now varying the pendulum mass. Record the mass and period for the 3 other pendulum bobs. Use the same length pendulum as part 2. Also use the release angle of  = 10 degrees each time. Create a small data table for part 3 with data columns of pendulum bob mass m, period T, and period uncertainty  . Question #3: In part 3, do you observe any noticeable trend when you varied the pendulum mass? Does the pendulum period depend on mass of the pendulum bob? Explain .

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4 Part 4: Pendulum period vs. Length. Repeat the period measurements by now varying the pendulum length. Choose the brass bob again and release angle of  = 10 degrees each time. Vary the length 4 more times (for 5 measurement lengths in total). Vary the lengths in consistent large increments, from a minimum length of about 20 cm up to a maximum length of 100 cm (or if you have a short vertical stand the maximum length possible). Create a small data table for part 4 with data columns of pendulum length L, period T, period uncertainty  , and period squared T 2 . Question #4: In part 4, do you observe a noticeable trend when you varied the length of the pendulum? Does the pendulum period depend on the pendulum length? Explain . Question #5: Using your data from part 4, make a graph of period squared T 2 vs. length L. Period squared should be on the y-axis and length of the x-axis. Your data should look like a straight line if done correctly. Make a linear trendline fit and display the equation on the graph . Record the slope of the line. Now take the inverse of the slope (1/slope). Multiply the inverse slope by 4  2 . What value do you get? What are the units of this number? Does this number look familiar? (It should if you did the experiment accurately). What physical quantity did you just measure? Show your calculations. Question #6: Based on your results from question #5, determine the equation for the period T of a pendulum. Show your work. (Hint, think about the algebra in question 4 and what physical quantity it resulted in, then work backwards to find the equation of the period T).

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