Physics 1202 - SHM- worksheets - Winter 2023 - Rishan

Simple Harmonic Motion - 5 — Data and Work Sheets Simple Harmonic Motion - Physics 1202B 2022-2023 Please circle the appropriate values Course 1102B 1202B 1402B 1502B Lab Section 002 003 004 005 006 007 008 009 010 013 014 Lab Subsection A B C D Name First: Rishan Last: Rajakulathilakan Student # 2 5 1 3 1 4 5 5 5 Lab Partner First:Yanishka Last:Gahlot Lab Station # 1 Date 2023 – 01 - 31 Demonstrator Rina and Baria Disclaimer: Please note that some but not all questions in this lab writeup will be graded. EXPERIMENT 1: MEASURING THE SPRING CONSTANT OF THE SPRING APPARATUS: steel helical spring, masses, meter ruler and mirror (to eliminate parallax) METHOD You should have a setup as shown in Figure 1. The spring is hung on a force sensor. The force sensor and the ultrasonic device are not used in Experiment 1 when the spring constant is measured, but will be used in Experiment 2. Record in Table 1 the position of the lower end of the helical spring with no masses attached. This is the equilibrium position, x 0 , of the spring. Now, add a mass to the spring and record the mass ( m ) attached to the spring, and the position ( x ) of the lower end of the spring, in Table 1. (Note that you should always start with the lowest mass so as to not elongate the spring past its breaking point. The displacement ( x x 0 ) should be at least 3 cm and no more than 20 cm . Continue to increase the load on the spring by small increments, by either adding masses or by substituting a heavier mass, and record the position of the lower end of the spring for each mass. Add masses gently so that the spring extends monotonically, i.e., without bouncing the load on the spring. Repeat this process for at least 4 different masses.

Simple Harmonic Motion - 6 − − Figure 1: Setup of vertical mass-spring system. The spring is suspended vertically and extended due to the weight of a mass attached to the end. The mirror is used to reduce measurement errors due to parallax. The force and motion sensors are not shown in this schematic. Calculate the corresponding displacement ( x x 0 ) of the spring for each mass, and record ( x x 0 ) in Table 1. The displacement is the increase in length from the equilibrium position of the spring. Include an estimate of uncertainty in the column heading for each variable. Table 1: Static (non-oscillatory) force-displacement data for determining the spring constant of the mass-spring system Mass m (g) ±0.1g Position x (cm) ±0.5cm Weight (N) (mass x gravity) ±0.1N Displacement ( x - x 0 ) (m) ±0.5 0.0 42.0 cm 0.0 0.0m 50g 42.0cm 490.0N 0.0m 100.0g 44.9cm 980.0N 2.9m 200.0g 55.5cm 1960.0N 13.5m

Simple Harmonic Motion - 8 We want to choose a mass such that oscillations will be about an equilibrium position that stretches the spring by approximately 8-12 cm. 1. Start up the Logger Pro software, and select the use of both the force and the position sensors. 2. Hang your chosen mass, and let the spring-mass system come to an equilibrium. You may have to wait a minute or so for it to come to rest. 3. Under the menu “Experiment”, choose “zero” to zero both the force and the position sensors (see Figure 3). 4. To start collecting data, click on the green arrow button labelled “Collect”. 5. The Logger Pro is setup to collect 25 measurements per second over a period of 10 seconds. 6. Without any motion, collect a baseline set of data (see Figure 4). 7. The vertical axis range of each graph can be changed. You may click on any number on the vertical axis to change the range. EXPERIMENT 2(b): WHAT TO MEASURE 1. Each station will accommodate two students who work in pairs using the same spring. You will use two different masses, allowing the spring to stretch by approximately 8-12 cm from the equilibrium each time. 2. You will then setup and record two sets of oscillations using two different amplitudes (e.g., amplitudes between 1-3 cm for the first set and 3-6 cm for the other set). Label and save your data. You will need to use the data for the two sets of amplitudes later in the analysis section. 3. While working together, lab partners will need to take turns using the equipment. Once you finished acquiring data for one mass and one amplitude, save your work on the Desktop, and open it with Excel to do the analysis (see the Analysis sections). Be sure to give your file a meaningful name when saving your work so you know which mass and amplitude you used. Then acquire data for the same mass using a different amplitude. Analyze the data and save your work under a different name. Repeat the above procedure for a different mass and the above two amplitudes. You should acquire 4 sets of data (i.e., two different masses, two different amplitudes).

Simple Harmonic Motion - 9 Figure 2: Setup for Experiment 2. The spring is suspended vertically and extended due to the weight of a mass attached to the end. This resting position is the equilibrium position, and the mass will oscillate about this point during simple harmonic motion.

Simple Harmonic Motion - 11 Figure 4: While the system is still at equilibrium, the baseline data can be acquired to ensure that all values at equilibrium are indeed zeroed. EXPERIMENT 2(c): MEASUREMENTS Once you are satisfied with having the sensors zeroed with the mass hung on the spring (at the equilibrium position), you may start the oscillation by stretching the spring to an amplitude you have chosen. 1. Choose an amplitude and record it below (do not allow the spring to stretch beyond its breaking point of 20 cm). 2. Set the mass to oscillate vertically. Wait and observe that it is oscillating vertically, then click on the green button to collect data. If you wished to start over, let the system finish collecting the data before attempting to restart. If you are happy with the oscillation, for example as shown in Figure 5, then save the data. Go into the File menu, and save the data as *.cmbl files. 3. Choose a meaningful file name, and record that into your lab manual. 4. Repeat the above with a different amplitude and save your results. Amplitude 1 = 1cm ±0.5cm Amplitude 2 = 5cm ±0.5cm

Simple Harmonic Motion - 12 Figure 5: An example of force, position, velocity and acceleration data recorded by the force and motion sensors. Change the vertical axes to a range suitable for the oscillations in your experiment. OBSERVATIONS 1. Starting with the Position vs. Time graph, identify times when the velocity should be a) zero and b) a maximum, and confirm that on the Velocity vs. Time graph. The velocity is 0 when the mass is at the amplitude displacements. Velocity is at its max when the mass in SHM is at the equilibrium position. 2. Starting with the Velocity vs Time graph, identify times when the acceleration should be a) zero and b) a maximum, and confirm that on the Acceleration vs. Time graph. Acceleration should be zero at the equilibrium constant. Acceleration should be at its maximum when the mass is at the amplitude (furthest distance from the origin) 3. Are the oscillations in the Force vs. Time graph consistent with the Position, Velocity and Acceleration vs. Time graphs? Explain why. Yes, the force vs time graph is consistent with the position, velocity and even the acceleration time graphs. This is because force is proportional to displacement – (F=-kx), and the force is also proportional to acceleration. (F=ma). Force max is when x = 0 and acceleration max is when x=amplitude(A) DATA ANALYSES

Simple Harmonic Motion - 14 — ANALYSIS 1: Determine parameters from Excel plots Analysis 1(a): Plot Force ( F ) versus Position ( x ) Hooke’s law, F = kx , tells us that the restoring force exerted by the spring to bring the mass back to the equilibrium position is proportional to how far the spring has been stretched from its equilibrium position. 1. Using the data acquired during Experiment 2, plot Force versus Position in Excel . Done on excel 2. Determine the spring constant (k) from the slope. Slope (spring constant) – 27.160 N/m 3. Compare with the spring constant obtained in Experiment 1, and compute the percent differ- ence. Are the two values within 10%? Percent diff = [(|K2 – K1|)/K1] x100% = [(|27.160N – 30.127N/m|)/30.127 N/m] x 100% [( 2.967N/m / 30.127N/m ] x 100% = 9.84% The two values are within 10%

Simple Harmonic Motion - 15 Analysis 1(b): Plot Force ( F ) versus Acceleration ( a ) Newton’s second law, F = ma , tells us that the force exerted on a mass is proportional to its acceleration. 1. Using the data acquired during Experiment 2, plot Force versus Acceleration in Excel . 2. Determine the effective mass of the system, m e f f , from the slope. In addition to the mass that you hung on the spring, this effective mass also accounts for the mass of the spring itself. m e f f = 519.3g = F(a) 1 (the slope of the function) _ Analysis 1(c): Plot Acceleration ( a ) versus Position ( x ) 1. Using the data acquired in Experiment 2, plot Acceleration (a) versus Position (x) . 2. What physical quantity does the slope represent? F = -kx = ma Isolate for a (divide both sides of the formula -kx = ma by “ m ” a = -kx/m - Slope represents spring constant over mass 3. Determine the slope. -k/m = -79.112 N/m*kg 4. Determine the measured angular frequency, ω . ω = sqrt(k/m) = sqrt(79.112 N/m*kg) = 8.894 rad/s

Simple Harmonic Motion - 17 e f f ANALYSIS 3: Relationships involving the angular frequency ω in simple har- monic motion For these steps, use percent difference tests to compare quantities. a) Compare the angular frequency ( ω ) that you determined in Analysis 1 with q m k , where k is the spring constant and m e f f is the effective mass as measured and determined in Analysis 1(a) and 1(b), respectively. Sqrt(k/ m e f f ) = sqrt(27.160N/m / 0.5193kg) = 7.232 rad/s % diff = (|7.232 – 8.894|/8.894) x 100% = 18.76% > 10% (not within 10 percent) b) Plot Acceleration (a) versus Position (x) in Excel for the second data set acquired with a different amplitude and determine the angular frequency. Compare the angular frequencies of the oscillations from two different amplitudes. Slope of a(x) = -52.224 (amplitude of 2cm) ω =sqrt(52.224) = 7.2266rad/s % diff = |7.2266 – 8.894|/8.894 x 100% = 18.75% > 10% (since the % difference is not within the acceptable 10%, the angular frequencies were not close enough in the experiment) c) From the acquired Velocity vs. Time data, determine an averaged value of the maximum velocity, v max , for the oscillation. Compare this with the theoretical value, as expressed in Equation (2), using the values of A and ω determined in Analysis 1 and 2. Theoretical: Vmax = A ω = (0.03172)(8.894) = 0.28211768 m/s Experimental: Vmax = 0.280997725m/s %diff = |((0.28211768m/s – 0.28097725m/s)/0.28097725m/s)| x 100% = 0.40% <10% - Since the values are within 10%, therefore the values are very close

Simple Harmonic Motion - 18 DISCUSSION AND CONCLUSIONS 1. You are given a mass and a spring, setup either in a vertical configuration, as in this lab, or horizontally. When we set the mass in motion about its equilibrium position, can it oscillate with more than one frequency? Explain. root(K/m) can be used to calculate the angular frequency. (K: spring constant, and m: mass). Both K and m do not change in their values – constants--. After this analysis, we can see and conclude that a certain frequency will remain constant to its an assigned constant mass since the K value is always constant. This demonstrates that a mass in motion (which is not altered (g)) will not oscillate with more than one frequency in SHM. 2. The frequency of a simple harmonic motion does not depend on its oscillation amplitude. What must then be changing to allow for the larger amplitude of oscillation within the same period? As specified above, the frequency stays constant when the mass is constant. In order to allow for a larger amplitude within the same period would be to increase the initial amplitude by pulling the mass attached to the spring down further, then letting it go to oscillate. The formula T = 2(pi)root(m/K) --where T = period — demonstrates how changing the initial amplitude in order to increase it, will not affect the period of the oscillation since they are not interrelated in this formula – amplitude is not present--. Marks Table