Purpose The purpose of this lab is to introduce the student to the concepts of simple harmonic motion. Introduction A pendulum consists of a bob at the end of a string which is suspended from a support so that it can swing back and forth. The time required for the bob to make one round trip of its motion (one vibration) is its period. This is a controlled experiment in which each of the elements of the pendulum (the length of the string, the mass of the bob, and the amplitude of the swing) is varied while the other elements remain the same. Any difference in the average time to complete one vibration can be attributed to that element. Differences of less than 10% are assumed to be the result of experimental erron For example, if the string is lengthened but the time to complete a vibration does not vary by more than 10%, the conclusion is that the length of the string has no inuence on the period of the pendulum If the time does vary by more than 10%, the length of the string does inuence it Simple Harmonic Motion Simple harmonic motion (SHM) is a periodic oscillation that traces out a sinusoidal function when the position of the object is graphed against time. Periodic motion is any motion that repeats at regular intervals, like the ticking of a clock or the vibration of a plucked guitar string. A sinusoidal function has the form of a sine or cosine mction, shown in Figure l. Wavelength Wavelength Figure 1 Simple Harmonic Motion Example Oscillations are any repeated motion. Periodic motion and SHM are both specic types of oscillations Oscillations are dened in space and time using the following terms illustrated in Figure l: 0 Period (T) is the time required to repeat one full cycle of motion, returning to the starting position, usually measured in seconds (s). Frequency (f) is the inverse of the period, or the number of full cycles that occur ina given time period, usually measured in Hertz (1 Hz : 1/s). Equilibrium (x0) is the position where the oscillating object experiences no force, usually measured in meters (m). (The green dashed line in Figure 1 above is the equilibrium position) Displacement (x) is the distance from the equilibrium position, usually measured in meters (in). Amplitude (A) is the distance from the equilibrium position to the maximum displacement, usually measured in meters (m), Wavelength (A) is the distance between one point on a wave and the same point on the next wave. (Wavelength is usually measured from crest to crest or trough to trough). Simple Pendulum trictionless pivot amplitude 3 9 ' massless rod bob's'~ ~ . _ trajectory massive bob equilibrium position Figure 2 Simple pendulum Copyright: By Chetvorno Own work, Public Domain, https://commonsiwikimedia.org/w/indexiphp?curid:5276335 The simple pendulum as shown in Figure 2 above is an excellent example of simple harmonic motion a_s__lp_r_1g _a_s the angle is small. When the bob at the end of a massless rod (or string) is pulled a small distance from its equilibrium position, it has stored energy (gravity will pull it back down toward the equilibrium position). At the two extreme positions (far left and far right) the bob has maximum potential energy and no kinetic energy. When the bob is released it starts to move as potential energy is gradually converted to kinetic energy. At the equilibrium position (bottom of the swing) the bob is moving faster than at any other point on it path. Its kinetic energy is maximum and its potential energy is zero Because it is moving fast, it continues to move past the equilibrium position As it swings upward on the opposite side it gains potential energy will it slows down (loses kinetic energy) until it reaches the opposite extreme position, At the extreme position it once again has maximum potential energy and zero kinetic energy (its speed is zero). Gravity still works as a restoring force and pulls it back down toward the opposite side and the motion repeats forever. In the real world, the pendulum will not swing forever because there is some friction, (at the pivot point, air resistance etc.,) causing it to gradually slow down and eventually stop. The Period of a pendulum is the time it takes to complete one full cycle (one complete swing back and forth. If you start with the pendulum at its extreme position on the right, the period is the time it takes to return to the same extreme position on the right. The frequency is the inverse of the period, For a simple pendulum the frequency can be found using the equation: 19 f'i L 7':er 9 Or, since frequency and period are inverse: Since you will be measuring both the Period (T) and the length (L), you can use these equations to solve for the acceleration due to gravity g, T = 211E sguare both sides T2 = 41:2 (En) multiply both sides by \"g\" ng = 47r2L, divide both sides by T2 L 9:4\"277 The Step bx Steg procedures for this lab are on the remaining pages below. Gather the following materials Student Supplied HOL Supplied Digital stopwatch Hanging weight, 50 g Pendulum support Hanging weight, 100 g Tape, clear or masking Protractor String Tape measure Grooved board (optional) 500 g mass (optional) Grooved mler (optional) * Either a physical stopwatch or computer/phone app may be used a_s_lo_ng_a_s_ it records to 001 second ** A war I hook, bracket, towel rack, or similar household item may be used as longas the space below the support is unobstructed Directions to create, a sturdy support are provided starting on page 12.10r using the optional items above. Procedure I Data collection Part 1: Effect of Amplitude on Period Before beginning, nd a solid support from which to hang the pendulum. Ideally, there should be a wall close to the support so that the protractor can be attached for recording the pendulum's movements. A bathroom or kitchen towel bar is ideal for this purpose. A wall hook, hat rack, or over-the-door hanger, shown in Figure 3, are other possible support structures. Alternatively, a support similar to that shown in Figure 3 (or the optional support system described at the end of this lab) can be constructed and placed on a Tall shelf or tabletop. Note you will need at least 1.25 m height to complete all the steps of the lab. A refrigerator is tall enough for this height. Figue 3 Setup support 1. Using the measuring tape, measure a piece of string that is approximately 150 cm in length. (You'll need 125 cm length from pivot to center of the mass, so the extra length is for loops and room at the top). 2. Securely attach the 50 g hanging mass to one end of the string. Using a loop as used in previous labs will make it easy to interchange the masses. Note: the hanging mass will hereafter be referred to as the bob. 3. Suspend the string from the pendulum support so that the distance from the attachment point to the center of the bob is exactly 1.000m (100.0 cm) 4. Use tape to afx the protractor to the wall behind where the string is attached to the support so that the center hole of the protractor is located directly behind the pivot point as shown in Figure 4 Protractor below. Note: the string should hang straight down so that the string aligns with the 90" mark on the protractor. See Figure 4 for correct placement of the protraetor. Figure 4 Protractor 5. Displace the bob by 5s from straight down, and hold it there as shown in Figure 5 (Note the gure is showing 10 to make the angle more visible). Release the hob and observe its swing for one cycle so the bob returns to its starting point (One cycle is from the release point, back to the release point) Note: Keep the string straight as the bob is displaced by holding the boh to create the desired angle. The pendulum must swing without obstruction and should not strike the background as it swings Figure 5 Pendulum Swing Figure 6 Angles 6, Figure 6 above shows the string angle for amplitudes of 5 and 10. Practice holding the bob with the string straight at different angles. 71 Starting with an amplitude of 552. use the digital stopwatch to time the bob as it swings through ten complete cyclesl Remember one complete cycle is from the point of release. .1 swing to the opposite side 11 and return to the starting point. Start the stopwatch when you release the bob being careful the bob swings straight without striking the background. Stop the stopwatch when the bob has completed 10 full cycles. Record the time for 10 full cycles to the nearest 01 seconds in Data Table 1. 8. Repeat step 7 two more times for a total of three trials. 9 Repeat steps 7-8 for the remaining angles in Data Table 1, 101 Calculate the average of the three time trials for each angle and record to the nearest 01 seconds in Data Table 1. 1 1. Find the percent difference between the highest and lowest of the three periods with angle variation using the formula: lThighest Tlowestl C. highest + Tlowest) 2 Percent difference = * 100% record in Data Table 1. 12. Using your percent difference, what can you conclude about the effect of amplitude on the period of a pendulum. Use the m arrows to select the answer that best fits your conclusion. Part 2: Effect ofMass on Period 13. Remove the 50 g hanging mass and replace with the 100 g hanging mass. 14. Adjust the length of the string so that the length from the attachment point to the center of the bob_is exactly 1000 cm (1 000m) 15. Notice the data for the 50 g hanging mass at 100 m and amplitude of 10'; prelled in Data Table 2. Verify the data copied correctly. Edit if needed. 16. Repeat steps 7-8 using the 100 g hanging mass and a 10% Record the time for 10 full cycles in Data Table 2' Figure 7 How to hang 150 g 17. Add the 50 g hanging mass to the 100 g hanging mass so both hooks hang from the loop. Repeat step 12 for this new conguration, Note the center of mass will be between the center of the 100 g hanging mass and the center of the 50 g hanging mass, 18. Repeat steps 1% the combined total of 150 g and an amplitude of 10. Record the time for 10 full cycles in Data Table 2. 19. Calculate the average of the three time trials for each mass and record to the nearest 0.1 second in Data Table 2' 20. Calculate the period for each mass by dividing the average time by 10 and record to the nearest 0.1 seconds in Data Table 2. 21. Repeat steps 11 and 12 except using the effect of mass on the period, Part 3: Effect of length on Period 22 Remove the 100 g hanging mass, leaving the 50 g hanging mass only 23. Adjust the length of the string so that the length from the attachment point to the center of the bob is exactly 25.0 cm, 24. Repeat steps 7-8 for a mass of 50 g, an amplitude of 10\" and the 25.0 cm length and record in Data Table 3, 25. Keeping the 50 g mass, and 10 amplitude, repeat steps 7-8 for the remaining lengths listed in Data Table 3. 26. Calculate the average of the threejime trials for each length and record to the nearest 0.1 second in Data Table 3, 27. Calculate the period for each length by dividing the average time by 10 and record to the nearest 0.1 seconds in Data Table 3, 28. Repeat steps 11 and 12 except using the effect of length on the period. Procedure II Data collection Calculating the Acceleration Due to Gravity 29. Calculate the square of the period for the different pendulum lengths recorded in Data Table 3 and record the value in Data Table 4. 30. Calculate the acceleration due to gravity for each length using the T2 values calculated in step 29 and the equation: 4n2L 9 = T2 31. Record the calculated values for g in Data Table 4. 32. Calculate the percent error between each of your calculated values for the acceleration due to gravity and the accegted value of 981 m/s2 and the equation: gacce ted gcuzmzamd percent error = g x 100% gurcepted 33. Record the calculated values for percent error in Data Table 4. 34 Using Excel, create a graph of Length (L) on the vertical axis versus Period Squared (T2) on the horizontal axis Include a title, axis titles with units, and add a linear trendline with equation that best ts the data on the graph, Copy your graph into Graph 1 in the worksheet. 35 Enter the slope of your trendline in the block below the graph 36. Calculate the acceleration due to gravity using the slope of the trendline from the graph and the equation for the period of a pendulum: Procedure I Data Collection Data Table 1 Angle Varies Length 100.0 cm, bob mass 50.0 g Angle Trial 1 Trial 2 Trial 3 Average Time Time (s) Period (s) (degrees) Time (s) Time (s) S 5 20.43 20.73 20.52 20.56 10 20.15 20.57 20.62 20.45 20 20.52 20.61 20.17 20.43 Find the percent difference between the highest and lowest period from angle variation Based upon this what can you conclude about the dependence of Period on amplitude: Data Table 2 bob mass varies Length 100.0 cm, Angle 10 degrees Trial 1 Trial 2 Trial 3 Average Time Mass (g) Time (s) Time (s) Period (s) Time (s S 50 20.15 20.57 20.62 20.45 0 100 20.64 20.21 20.15 20.33 150 20.52 20.49 20.64 20.55 Find the percent difference between the highest and lowest period from mass variation Based upon this what can you conclude about the dependence of Period on mass:Data Table 3 Length varies bob mass 50.0 g Angle 10 degrees Length Trial 1 Trial 2 Trial 3 Average Time Period (s) (cm) Time (s) Time (s) Time (s) (s) 25 10.47 10.61 10.49 10.52 50 14.57 14.79 14.64 14.67 75 17.71 17.52 17.64 17.62 100 20.15 20.57 20.62 20.45 0 125 22.62 22.79 22.56 22.65 Find the percent difference between the highest and lowest period from length variation Based upon this what can you conclude about the dependence of Period on length:Procedure II Data Analysis Data Analysis Table 4 Acceleration due to gravity, bob mass 50.0 g Angle 10 degrees Length Period Acceleration g from Graph % error graph (cm) Squared (s^2) due to gravity % error (m/s^2) (m/s^2) 25 50 75 100 125 Procedure III Graph Graph #1 paste copy of legth vs period squared graph4TT2 L g = T2 = slope * 4TT2 37. Record the calculated value for g in Data Table 4. 38 . Repeat steps 32-33 for the value of g calculated from the slope of the trendline. 39. Return all HOL supplied materials to the lab kit