Simple Harmonic Motion - W24

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Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Simple Harmonic Motion Physics Topics If necessary, review the following topics and relevant textbook sections from Serway / Jewett “Physics for Scientists and Engineers”, 10th Ed. • Motion of an Object Attached to a Spring (Serway, Sec. 15.1) • Analysis Model: Particle in Simple Harmonic Motion (Serway, Sec. 15.2) • Energy of the Simple Harmonic Oscillator (Serway, Sec. 15.3) Introduction The simple harmonic oscillator (a mass oscillating on a spring) is the most important system in physics. There are several reasons behind this remarkable claim: • Any system which is in stable equilibrium and disturbed slightly will undergo oscilla- tions. (Think of a ball sitting at the bottom of a valley; moving it slightly will cause it to roll back and forth around the minimum. But this also applies to other systems like buildings disturbed by a gust of wind, or molecules jostled by an electric field). • The mathematics of oscillations - sines and cosines - can be used to describe any system which behaves in a regular, repeating way. For example, most astrophysical systems exhibit periodic motion (rotations of stars, orbits of planets, etc...) and can be described as a 2D oscillation. • The mathematics of oscillations - sines and cosines - can be used to describe waves. Sound, light, (and even, in some sense, subatomic particles) are waves! In this lab, you will use Hooke’s law, and the theory of simple harmonic motion to predict and examine the motion of a mass on a spring. Pre-Lab Questions Please complete the following questions prior to coming to lab. They will help you prepare for both the lab and the pre-lab quiz (Found on D2L). 1.) What is the specific goal of this lab? Exactly what question(s) are you trying to answer? Be as specific as possible. (“To learn about topic X...” is not specific!) What specific measurements or observations will you make in order to answer this question? 2.) Mass and spring at rest Page 1 of 9

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited (a) Draw two diagrams side by side: one with a spring hanging vertically (with no weight attached to it), and one which shows the same spring stretched downward by a distance ∆ y after a mass M is hung on it. (b) Choose a coordinate system. Pick a convenient direction to be the positive y direction, and label this on your sketch. (c) For the part of the diagram where the mass is attached, draw a free body diagram for the hanging mass. (d) According to Newton’s Second Law, what must be true about the forces acting on the mass assuming it is in equilibrium? (e) Write an equation relating the forces acting on the mass. . 3.) Oscillating mass and spring (a) Let’s choose the origin of our coordinate system y = 0 at the new equilibrium position where the mass hangs at rest. Label this position on your sketch. (b) Suppose the system from the previous question has a mass of 100g, is displaced upward from the new equilibrium position by 10cm and then released from this position at time t = 0. Suppose the period of oscillation is 1 second. Draw a careful graph of the mass’ position y ( t ). Be as quantitative as possible by labeling your axes, include units and tick marks. (c) Now suppose the system is identical to the previous question, but is displaced upward from the new equilibrium position by 20cm and then released from this position at time t = 0. Predict the motion of the mass by drawing another careful sketch of the mass’s position y ( t ) on the same set of axes. Be as quantitative as possible by labeling your axes, include units and tick marks. (d) Finally, suppose the hanging mass is increased to 200g and is allowed to come to its new equilibrium position (which we again call y = 0). It is then displaced upward from the new equilibrium position by 10cm and then released from this position at time t = 0. Predict the motion by drawing another careful sketch of the mass’s position y ( t ) on a separate set of axes. Be as quantitative as possible by labeling your axes, include units and tick marks. (e) You should now have three separate graphs. Use your knowledge of simple har- monic motion to write a specific equation for y ( t ) next to each graph. Be as specific as possible, inserting numbers for all known quantities in each case. In- clude proper units in your equation. Page 2 of 9

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Apparatus • Computer • Vernier computer interface (LabPro) • Graphical Analysis software • Vernier motion detector • 50 g hanger, and 50 g masses (5) • Stand, rod and clamp • Spring • Metal cage • Metre Stick Software This lab will be using Microsoft Excel to conduct data analysis. This software is available to all students through Computing and Communications Services (CCS), Microsoft 365 can be downloaded here: https://www.torontomu.ca/ccs/services/software/microsoft/ The software can be run on a PC, Mac or in browser. If you have issues accessing the software please contact CCS. Procedure I - In equilibrium 1.) Download the ‘SHM Worksheet.xlsx’ file from the lab folder on D2L. It will aide you in collecting and organizing your data. 2.) Attach the spring to the horizontal rod connected to the stand. 3.) Remove any weight from the spring, and measure the distance from the table to the bottom of the spring. Record this measurement, including an estimate of uncertainty. Page 3 of 9

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Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited 4.) Hang the 50 g hanger on the end of the spring. Place enough mass on the hanger such that it begins to stretch an observable amount (this may already occur with just the hanger). When the mass hangs freely, measure the distance from the top of the table to the bottom of the spring, be consistent with where you measured the bottom of the spring in step 3.) . 5.) Repeat the measurement above for increasing masses (of 50 g), recording all data and uncertainties. Analysis I - In Equilibrium 1.) Use Excel to plot your data in such a way so that you can determine the spring constant k from the slope of a line. Record your value of k determined using this method. Procedure II - Oscillations 1.) Equipment setup (a) Hang a total of 200g mass on the spring. Ensure the spring, mass, and rod are secured so the mass will not fall and damage the sensor. (b) Position the motion detector below the mass. To protect the motion detector, place a metal cage around the sensor. (c) Connect the motion detector to the DIG/SONIC1 channel of the interface. (d) Take a preliminary run to make sure things are setup correctly. Lift the mass upward a few centimeters and release. Take care that the mass does not fall off the spring as it could damage the sensor below. The mass should oscillate vertically only. (e) Click to begin data collection. After 5 seconds, data collection will stop. (f) The position graph should show a clean sinusoidal curve. If you cannot see any data on the position graph, click the “Zoom All” button below the graph to automatically zoom in on the data. If necessary, you can also zoom into a particular region of the graph by selecting a region and then clicking . The program will zoom into the light blue rectangle you selected. If the position curve has flat regions or spikes, reposition the motion detector and try again, taking care that there are no objects in between the mass and the sensor. You may need to adjust the cage slightly. Once you are satisfied that everything is positioned properly, proceed to the next part. Page 4 of 9

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited 2.) With the mass hanging from the spring at rest, click the position reading in the bottom right corner and zero the sensor. From now on, all distances will be mea- sured relative to this position. When the mass moves closer to the detector than this position, the position will be reported as negative. 3.) Raise the mass about 5 cm above the equilibrium position and release it. The mass should oscillate along a vertical line only. 4.) Click to create a graph of the mass’ position as a function of time. Once the 5s collection time has passed you can begin to analyze your data. Note: By clicking a point on a graph the program will display it’s coordinates. 5.) Determine the average amplitude A avg of the motion by averaging the amplitude of several oscillations together. It is best to examine both the maximum and minimum position to determine the amplitude! If there is a significant difference between the maximum and minimum displacements, you need to re-zero your sensor (see step 2). Record the average amplitude, including uncertainty. 6.) Using the position graph, determine the period of oscillation. Note that if you drag the mouse from one part of the graph to another, the time at both ends of the highlighted area will be displayed, use this to calculate ∆ t . Record the period with an estimate of uncertainty. 7.) Rename your latest run as something significant (ex. Lab1 200g5cm). Click the View Options button in the top right and turn on ‘Data Table’. Click the three dots beside the data set and select ‘Rename Data Set’. 8.) Repeat steps 2 - 10 for the same mass, but with a larger amplitude of oscillation ( ∼ 10 cm). Record your measurements of the average amplitude and period including uncertainty. Note: depending on the spring, you may not be able to reach 10 cm. Do your best to obtain an oscillation amplitude different from your previous measurement. 9.) Hang a total of 300 g of mass from the spring and let the system reach equilibrium. Re-zero the sensor following the procedure from step 2.) and repeat steps 2 - 10, recording all data and uncertainties. Make sure the oscillations are centered around zero, if they are not you should re-zero the sensor and collect another data set. 10.) Rename your latest run as something significant (ex. Lab1 300g5cm). 11.) Click the file menu button and export your data as a .csv file. You should upload your data file to the Lab 1 assignment folder on D2L. Page 5 of 9

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Analysis II - Oscillations 1.) For the last run currently displayed on the screen, determine and record the initial phase ϕ of the measured oscillation. Here are some hints for doing so: • Using Serway’s equations for simple harmonic motion, we have y ( t ) = A cos( ωt + ϕ ) and v ( t ) = − Aω sin( ωt + ϕ ). • Plugging in t = 0 into the simple harmonic motion equations give y 0 = A cos( ϕ ) and − v 0 = Aω sin( ϕ ). • Remember that when you take an inverse trig function there are two solutions, even though you calculator only gives you one. (For example, cos − 1 (0 . 5) = π/ 3 or 2 π/ 3). You will need to decide which solution is the correct one. • Here is another method you can use: determine what fraction F of (a cosine) period had occurred before you started taking data. The phase ϕ = F · 2 π . 2.) Using the ‘Analysis II 2.)’ sheet in the excel file you will apply a sinusoidal fit to your data. (a) In the data table on Graphical Analysis copy a data set, you only need the time and position data. Left click and drag to highlight the data and press Command-C to copy. (b) Paste the data into the Time(s) and Distance (m) columns on the excel sheet ( Command-V ) (c) You should now see two sine waves on the graph. (d) Adjust the fit parameters (A,B,C & D) to align the two sine functions. (e) When satisfied with the fit, record the fit parameters with uncertainty. (f) Save your excel file and upload it to the Lab 2 Report folder on D2L 3.) Given your measured values of amplitude, period and phase, what values would you expect for the parameters A , B , C & D ? Are the actual values found from the fit consistent with your expectations? [Hint: remember the phase ϕ you found above assumed a cosine function whereas here we are using a sine function. How is your ϕ related to C ?] 4.) Using your fit parameters, calculate the value of the spring constant k . Is your result consistent with the value you found in part I (using the mass at rest)? 5.) Save your Excel file and submit it with your report. Page 6 of 9

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Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Wrap Up - Complete In Lab The following questions are designed to make sure that you understand the physics impli- cations of the experiment and also to extend your knowledge of the concepts covered. Your report should seamlessly answer these questions in their noted sections. 1.) [Results] Based on your data, is the oscillation period affected by the amplitude of the oscillation? Explain your reasoning in a sentence or two. 2.) [Discussion] If you combine and plot the sum 1 2 ky 2 + 1 2 mv 2 , what do you expect the graph to look like? Explain in words. Note: For this investigation you will need a data set that is well centered around zero. If you don’t have a trial that meets this criteria, re-zero the motion sensor and collect a new data set. (a) Copy a data set that has oscillations well centered around y = 0 into the third excel sheet ‘Wrap Up 2.)’. You will need the time, position and velocity data. (b) Paste the data into the corresponding columns. (c) Populate the energy columns by inputting the corresponding formulas. For ex- ample to calculate the first entry for Kinetic Energy you could enter =0.5*0.3*C2^2 into cell E9 (assuming a hanging mass of 0.3kg). (d) Once you have entered the formula into one cell you can click and drag it to fill the column. Click the cell → Click and hold the green square in the bottom right corner of the cell → Drag it down to fill the column Alternatively, double click the green square to have excel automatically copy the formula into the cells below. (e) Repeat this process for the other two columns. (f) Once the energy columns are filled the plot should display three data sets: Kinetic Energy, Potential Energy & Total Energy. (g) Include this plot in your report and comment on it’s characteristics. Does the graph agree with your expectations? 3.) [Discussion] If you examine the motion of the mass for many (more than 20) oscilla- tions, what happens to the amplitude of the oscillation? What is the physical reason for this? 4.) Extra challenge (ungraded): Can you guess how to modify the simple cosine/sine equations to account for the effect dicussed in the previous question? Page 7 of 9

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Report Here is a brief guide for writing the report for the lab. The report should include the following sections: • Title Page – Include: Report Title, Your Name, Course, Section Number, Instructor, TA Name, and Date of Submission. • Introduction – What is the experiment’s objective? • Theory – Derivations of the physics being investigated, or reference to a source that provides a description/equation representing the physics being investigated. – Providing graphs that illustrate or predict how the system under study is expected to behave. • Procedure – Briefly explain the systematic steps taken for the experiment. • Results and Calculation – Tabulate the measurements in an organized manner. – Based on the procedure, one should have a sense of how the tables will look like prior to taking measurements. – Graph the main results. – Provide examples of any calculations. • Discussion and Conclusion – Discuss the main observations and outcomes of the experiment. – Summarize any significant conclusions. Page 8 of 9

Toronto Metropolitan University - PCS125 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Appendix A - Using Graphical Analysis Graphical Analysis Interface Details Here is a quick reference to using the Graphical Analysis. For a complete guide, see Venier Graphical Analysis User Manual • View Data Tables by using the View Option button in the top right. Important: Unlike LoggerPro, Graphical Analysis saves every single mea- surement collected (including unwanted sets). This means it is very impor- tant to keep track of your data sets. Be sure to rename data sets (so you can identify them) and delete unwanted data sets immediately to keep your data organized! Renaming or deleting is achieved by clicking · · · next to the data set name. • Hide/unhide data sets by clicking on the y-axis title. • Single clicking anywhere on the plot will reveal the single value of at that position. • For analysis, click and drag a region of interest, and select an analysis option under Graph Options in the lower left. The zoom button is to the right of it. Important: Unlike LoggerPro, Graphical Analysis will apply the curve fit to all data sets shown. It is recommended to curve fit a single data set and record the results before proceeding to the next data set. • Save your experiment using File Menu in the top left. The file format is .gambl which can be opened using Graphical Analysis. You can download a free version of Graphical Analysis to use at home Page 9 of 9

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