Phys 275 - June 2021 104

Experiment X Forced Harmonic Motion

I. Purpose The main purpose of this lab is to observe forced harmonic motion and measure the amplitude, resonant frequency, damping time, quality factor, and phase shift of the motion. We also want to show you how Excel Macros can be used to accomplish some complex tasks. II. Preparing for the Lab You really need to prepare before doing this lab. Although the concepts of forced harmonic motion and resonance are important topics that should be covered in introductory physics courses, you may not have had it yet because it requires some mathematical sophistication. Take a quick look now at the Introduction. Not so simple is it? Well before you get to the lab, read through the write-up and consult the references for additional help. Finally, don’t forget to do this week’s Pre-Lab Questions and the Homework from last week’s lab before your section meets. III. References For more discussion of simple harmonic motion, see your Physics 171 textbook, Chapter 14 in Introduction to Physics for Scientists and Engineers by F. Knight, or Appendix B in this lab manual. For more on using the oscilloscope and function generator, see Appendices F and G. IV. Pre-Lab Questions Must be answered on Expert TA before the start of your first lab class #1. If you drive a simple harmonic oscillator at any frequency f, and wait long enough that the

transients have decayed away, the oscillator will move sinusoidally at the same frequency f that it is being driven. True or False?

#2. If you drive the simple harmonic oscillator in this lab at its resonance frequency, what is the difference in phase between the position of the cart and the force exerted by the driving speaker is (a) 0o, (b) 45o, (c) 90o, (e) 180o, (f) 270o?

#3. If you drive the simple harmonic oscillator in this lab at a frequency that is much higher than its resonance frequency, the difference in phase between the position of the cart and the force exerted by the driving speaker is (a) 0o, (b) 45o, (c) 90o, (e) 180o, (f) 270o?

#4. If you change the frequency of the driving speaker, you will typically need to wait much longer than the damping time τd for the system to settle down to its steady state motion. (True or False).

#5. The quality factor Q of the resonator determines how narrow the resonance peak is and how large the motion will be when the system is driven on-resonance (True or False).

V. Equipment

Forced Harmonic Motion setup Mass scale Excel Forced Harmonic Motion template Masking tape & 3x5 index card LoggerPro Forced Harmonic Motion template Oscilloscope

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Figure 1. Photograph of Forced Harmonic Motion apparatus, not including oscilloscope. VI. Introduction In this lab you will study the motion of a driven mass and spring system. The key concept in forced harmonic motion is the phenomena of resonance. Resonance just means that the system exhibits a large motion when it is driven near its natural oscillation frequency or resonant frequency. Harmonic motion underlies or can be used to model many important phenomena in physics and engineering. Examples range from resonant electrical circuits used in tuners, to the oscillations of buildings and bridges during earthquakes, to resonant optical cavities used in lasers. Although these phenomena occur in very different physical systems, they all follow equations that are similar. Figure 1 shows a photograph of the setup you will use. A cart of mass m is connected between two springs and can move left or right along an air track. The left spring goes to a speaker and the right spring goes to a fixed point. The speaker is driven by a sine wave voltage of frequency f that causes the end of the speaker to move back and forth, pushing and pulling on the left end of the left spring. The right end of the left spring then exerts a force on the cart, causing it to move. The spring on the right side of the cart keeps the cart centered in the track and prevents the left spring from collapsing. A sonic ranger is used to measure the position x of the cart as a function of the time t, and we keep track of the motion of the speaker by measuring the voltage from the function generator that drives its motion. When the speaker is being driven at frequency f, the displacement xs of the left end of the spring (which is attached to the speaker) will change with time according to: )2sin( tfAx os π= . (1) This will cause the speaker to push or pull on the cart, driving its motion. The total force acting on the mass is due to both the spring on the left and the spring on the right. This makes it a bit more complicated to analyze. To simplify things we assume the springs have the same spring constant k and define x = 0 as the location of the cart when the speaker is at xs = 0 and the total force of the springs on the cart is zero. With these assumptions, the total force acting on the mass is:

Phys 275 - June 2021 106

2tot sdxF k x kx bdt

= − + − (2)

We can think of the first term as being the force on the cart due to the two springs when the cart is displaced from equilibrium by a distance x, the second term is the effective driving force Fa applied by the speaker on the cart, sin(2 )a s oF k x k A f tπ= = (3) and the last term is the drag force Fd due to air drag on the cart:

dtdxbFd −= , (4)

where b is a positive constant and the minus sign means that the drag is opposite to the direction of the cart’s velocity dx/dt. According to Newton’s second law, the mass times the acceleration of the cart will equal the total force exerted on the object. Using Newton’s second law and Equation (2), the acceleration of the cart will be:

2

2totd xF mdt

= )2sin(22

2

tfm

kAxmk

dtdx

mb

dtxd o π+−−= . (5)

Note that a 2 appears in front of the k in the second term on the right because our setup has two springs connected to the mass. In contrast if you open just about any physics textbook and read through the section on simple harmonic motion, they will be considering a mass connected to one spring and their equation for the acceleration will not have a 2 in front of the k. Equation (5) is a driven linear homogenous second order differential equation. The solutions to this equation are well-known and you will need to understand them in two cases. Un-Driven Motion In the first part of the lab you will measure the free oscillation of the system after the mass has been given an initial displacement. Free oscillation means that no driving force is applied, so Ao=0 in Equation (5). Solving Equation (5) for the position x of the mass yields: dt

osco etfxx τφπ /)2sin( −+= , (6) where the damping time τd is:

bm

d2

=τ . (7)

and the natural oscillation frequency is:

2

22

21

−=

mb

mkfosc π

. (8)

Equation (6) means that the system will oscillate back and forth at its natural oscillation frequency fosc and this motion will decay away exponentially with time constant τd. The phase φ and the amplitude xo are set by the initial conditions, i.e. the position and velocity of the mass at t = 0. We can also rewrite Equation (8) for the oscillation frequency as:

2411Q

ff oosc −= (9)

where the fo is the oscillation frequency when there is no damping and it is given by:

mkfo

221π

= (10)

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and the quality factor Q is: Q = 2π mfo/b = 2π foτd/2, (11) The quality factor Q is a pure number that characterizes the loss in the system. For Q>>1 the system has low loss and will undergo many oscillations (of order Q) before most of its energy is damped away. It is also useful to define the period Tosc of the natural oscillation:

2

22

21

−

==

mb

mkf

Tosc

oscπ . (12)

Equation (12) implies that as the mass m increases or the spring constant k gets weaker, the period increases. Again, the factor of 2 that occurs in front of the k in this expression is because there are two identical springs attached to the mass. Driven Motion The second case we need to consider is when the mass is being driven at frequency f. If the drive starts up at time t=0, in general there will be a transient motion (just like Equation (6)) that will damp out on the time scale τd. These transients are readily visible in the experiment and you should typically wait one or two minutes for them to decay away before measuring the steady state response. For times much longer than τd, the system will settle into a regular oscillatory motion at the same frequency f as the drive and the solution to Equation (5) becomes: sin(2 )ox x f tπ θ= − . (13) At first sight this appears very similar to the driving force sin(2 )a oF k A f tπ= . They are both sine waves with the same frequency. However the amplitude xo of the mass’s motion depends on the drive frequency f and drive amplitude Ao:

2

20

2

20

2

2

00

1

2/

−+

=

ff

fQf

Ax . (14)

and the position x waveform lags the force Fa waveform by a phase angle θ given by:

02 2

0

1Arctan f fQ f f

θ = −

. (15)

Since the Arctan function is multi-valued, it can be confusing to evaluate and different conventions are used. Here we will choose to restrict θ such that it is between 0 and 180o,while Excel reports angles in radians and restricts θ to lie between -π radians and +π radians. Figure 2(a) shows a plot of the amplitude xo of the motion versus the drive frequency f from Equation (14). The amplitude xo of the mass’s motion has a maximum when the drive frequency f is at the resonance frequency:

2211Q

ff ores −= (16).

Notice that this is similar to Equation (9) for the natural oscillation frequency, but it is slightly different, so that fres and fosc are not exactly equal to each other. However, in this experiment you will not be able to measure the difference between fosc, fo and fres because Q>>1. Notice that at

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(a) (b) Figure 2. (a) Amplitude xo of the motion of a mass as a function of frequency f from Equation (14). Here, the resonance frequency is fo = 1.5 Hz and curves are drawn for Q = 50 (highest curve), 20, 10 and 5 (lowest curve). (b) Phase θ as a function of frequency f for a range of Q values. The phase θ is equal to 0 for f =0, and is equal to 180o for f >>fο. At resonance f = fo = 1.5 Hz and the phase is 90o. resonance the amplitude reaches its maximum size of xmax ~ AoQ/2 and this is a factor of Q larger than the amplitude Ao/2 of the mass’s motion at low frequency and Q/2 larger than the amplitude Ao of the motion of the speaker. The factor of 1/2 here occurs here only because the mass sits between two springs. Notice also that the full width of the resonance is fFW = fo/Q; this is the difference in frequency between the two points where xo

reaches 2/maxx . What this means is that the resonance peak gets proportionally higher and narrower as Q increases. Figure 2(b) shows corresponding plots of Equation (15) for the phase versus frequency for different Q values. At zero frequency, the phase shift is zero (in-phase). On resonance, the phase shift is lagging by 90o. When the drive frequency is much greater than the resonance frequency, the phase is 180o (out-of-phase). Note that θ changes more rapidly for larger Q and when the drive frequency f is near resonance (1.5 Hz in this case). To understand what it means for the position to lag the force, consider the following three cases: (1) When the drive frequency f is much less than the resonance frequency fo, Equation (15) gives 0oθ ≈ and we say that the mass is moving "in phase" with the driving force. In this case, as the mass moves back and forth, you will see that when it is furthest to the right (position x is largest) the speaker is also furthest to the right (xs and applied force is largest). (2) When the system is driven on resonance, f = fo and Equation (15) gives θ = 90o. This means that as the mass moves back and forth, you will see that when the mass is furthest to the right (position x is largest) when the speaker is at the center of its range of motion (xs=0) and moving left. You have to wait another 1/4 of a cycle (90o later) for the mass to be at the center of its motion (x=0) and moving left. Since the mass gets to the same location, but 1/4 of a cycle later than the speaker, we say that that the motion of the mass is lagging the driving force by 90o. (3) If you drive the system at frequencies f >>fo, then Equation (15) gives 180oθ ≈ . In this case, when the mass is furthest to the right, the speaker is actually furthest to the left.

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Figure 3. Plot of the position x(t) of the cart versus the applied force Fa(t) with different points on each curve corresponding to different times. For each curve the applied frequency f and the amplitude of the sinusoidal applied force are constant. Black curve is for f much less than the resonance frequency fo. The fact that this curve has a positive slope is because the phase shift is small at low frequency. The green curve is for f a bit smaller than fo, the blue curve is for f = fo, and red curve is for f a bit larger than fo. “Circle and Ellipse Plots” Showing Position Versus Driving Force Notice that the functional form of the driving force Fa from Equation (3): sin(2 )a oF k A f tπ= (17) is very similar that of the mass’s position x from Equation (13): x = xo sin(2 )f tπ θ− . (18) Figure 3 shows that a plot of x versus Fa for different times t produces a closed circle or ellipse. The orientation and eccentricity of the ellipse depends on the phase shift θ, which of course depends on the drive frequency. Figure 3 shows some examples, with the system driven at different frequencies. For example, the blue circle corresponds to the system being driven at resonance and the fact that is not tilted left or right in this figure is because the phase between x and Fa is 90o. As another example, notice that in the black curve, which is a very narrow ellipse, the position tends to increase when the force increases, as expected when the system is driven well below its resonance frequency where the phase difference goes to zero. In the lab, you will be able to directly observe this behavior as the data is collected. VII. Experiment Part A. Getting started (1) Go to the Excel Templates folder and open Forced Harmonic Motion Template. (2) Fill in your name and your lab section number. Note that the date is auto-filled.

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(3) Verify that the sonic ranger is plugged into the USB port and its green light is on. (4) For this part, make sure the function generator is OFF. Also leave the scope off for now. (5) Verify that the cart has a reflector and that the two springs are connected to the cart, speaker

and air-track, as in Figure 1. (6) Check that the sonic ranger is pointing at the reflector on the cart and that the reflector is

perpendicular to the track. Use the web camera to take a photo of your setup and paste it into the spreadsheet.

(7) Turn on the air blower and set the control knob to the 10 o’clock position. Move the cart a few

cm to one side and let go. Verify that the cart oscillates back and forth with little drag. If the oscillations die out in a few seconds,

- check that the spring on the speaker is aligned and not pulling the cart to the side - check that the springs don’t rub against the top of the cart - try increasing the air flow to the 12 o’clock position. If on the other hand the cart keeps oscillating after 100 s, you have a different problem: - make sure the function generator is off! - verify that there are magnets on the cart The magnets produce a small damping force by inducing electrical currents (eddy currents) in

the aluminum track. Ask your instructor if you are still having problems. Part B: Measuring free oscillations of the system In this part you will measure the oscillation of the mass when it is not being driven and then extract the resonance frequency and damping time from this data. (1) Got to the “LoggerPro Templates” folder and open the file “LoggerPro Forced Harmonic

Motion”. The initial display should look like Figure 4. On the left is a data table with columns for the time t, cart position x and voltage V applied to the speaker, which is proportional to the driving force Fa. To the right of the table is a plot of V versus t, a plot of x versus t, and a plot that has both x and V versus t. Finally, underneath the table is a plot of x versus V. This is just like the x versus Fa plot discussed at the end of the Introduction (see Figure 3). Note: no driving voltage is applied for this measurement and you just collect x vs t.

(2) Click on “Experiments”, then “Data Collection”. Verify that the Sampling Rate is 30

samples/second and the Duration is 50 s. (3) Go back to the main panel in Logger Pro and set the y-axis scale on the position versus time

plot so that it shows from 0.03 m to -0.03 m. (4) Let the cart settle to its resting position (don’t hold it), then click on the Zero button (next to the

“Collect” button) to zero the sonic ranger and voltage probe. (5) Pull the cart about 3 cm to the left, let it go, and then hit “Collect”. You should see a nice

decaying oscillation as LoggerPro collects position x as a function of the time t. When the data collection finishes, copy the x and t data and paste into the designated columns in Part B of the spreadsheet.

(6) In Excel, plot x vs t and label the axes. Of course you should use a scatter plot in Excel.

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Figure 4. Initial display panel for the LoggerPro template for measuring forced harmonic motion. (7) From your plot, estimate the initial amplitude xo, the oscillation frequency fosc, and the decay

time τd and offset xoffset. If you zeroed the ranger correctly in step 4, the initial amplitude will be the maximum value of the plot. The oscillation frequency can be estimated by first finding the period (the time interval between peaks) and then taking the inverse. The decay time is the amount of time that it takes for the initial amplitude to decrease to about 1/3 of its original value (it is probably somewhere in the range of 15 s to 20 s). Enter your values into the upper table in part B. Leave the phase φ and χ2 blank.

(8) Calculate the theory for the position x(t) of the mass using /sin(2 ) dt

th o osc offsetx x f t e xτπ ϕ −= + + (19) in the designated column, referring to the estimates for the fitting parameters in the top table.

Add xth vs t to your plot and reformat the points to be a line without markers. Adjust the parameters to get the theory to match the data. Note: This equation is a slightly modified version of Equation (6). We just added xoffset to account for error in zeroing the ranger.

(9) In your spreadsheet, click on the button for the Part B macro. This macro does a χ2 fit to your data and finds best fit values for the decay time τd, natural oscillation frequency fosc, initial amplitude xo, phase φ and offset xoffset. It also makes a list of frequencies to use in the next part. If the fit gets worse after you click the macro button, make sure your data only has a decaying oscillation (if it has a flat part in the beginning you will need to acquire a new data set). If the data is OK, adjust your fitting parameters again to get a better looking fit curve and then click the button again. You need to get reasonably good initial parameter estimates, especially the frequency and phase, before the macro can find the best fit.

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Part C: Measure the motion of the system when it is being driven In this part you measure the motion of the system when it is driven at different frequencies, including in the neighborhood of the resonance frequency.

(1) Verify that the function generator’s 50 Ω output is plugged into the speaker (see Figure 5). Check that black and red leads connect the function generator to the LabPro interface box, and a third set of black and red leads connects the scope’s Ch 1 input to the function generator.

(2) Turn on the function generator (see Appendix G for more on using the function generator)

Make sure the amplitude knob is pushed in and turned to its maximum setting. Do not change the amplitude knob after this step! Push the “1 Hz” frequency range button (see Figure 5) and make sure the Sine waveform button is pushed in. Turn the coarse frequency adjustment knob and the fine frequency knob on the function generator to set the drive frequency close to 500 mHz. There is a bit of a delay in the readout.

(3) Turn on the oscilloscope (see Appendix F) and let it run through its set-up screens. Turn the

horizontal and vertical control knobs until you see a clean sine wave. Check that the wave has a peak-to-peak of amplitude of 2V (use the scope) and frequency of 0.5 Hz. If the amplitude shown on the scope looks much smaller than 2 V, verify that Channel 1 is DC coupled, and that the amplitude knob on the function generator is turned fully clockwise and pushed in.

(4) Examine the apparatus and verify that the speaker is moving back and forth by about 1 mm and

that the cart is also moving as the speaker moves, although only about half as much as the speaker. If the speaker is not moving, check that the frequency display on the function generator is showing about 0.5 Hz (500 mHz), that the scope is showing a 2 V peak-to-peak amplitude, and the low-impedance output of the function generator are plugged into the speaker. If the speaker moving too fast for your eyes to follow, then you are on the wrong frequency scale - switch to the lowest range (1 Hz).

(5) Once everything looks good, hit “Collect” and LoggerPro will acquire 50 s worth of x versus t

and drive voltage V versus t. You should see steady oscillations in V (top plot in the panel), but the oscillations in x (middle plot in the panel) should be small and there should not be a significant phase shift between the x(t) and V(t) waveforms (bottom plot in the panel). After the data collection finishes, copy the t, x and V data and paste it into the designated area in Part C of your spreadsheet.

Figure 5. Function generator with 50 Ω output connected to the speaker and LabPro.

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(6) You can now click the “Fit to data” macro button for the frequency you just collected. This button starts and Excel macro that automatically fits the theory to your data and returns the amplitude and the phase of both the x(t) and V(t) oscillations. It also plots the last few seconds of your data set and the fits. You can expand the plot to see the fit better. The macro is pretty good at fitting the data, but it does fail sometimes and can’t find a good fit. If a fit looks incorrect, just retake the data and rerun the macro.

(7) Turn the frequency adjustment knob to set the drive frequency to the next measurement

frequency indicated in your template. Wait a minute or two for the transient motion to decay and then repeat steps (5)-(6). It typically takes about a minute or two for the oscillation to fully build up and during this time you may see beating and other transient effects.

(8) Repeat the previous step for the rest of the drive frequencies listed in your spreadsheet. You

should have collected a total of nine drive frequencies when you finish. Part D. Analysis of Amplitude and Phase data In this part you will analyze your data and extract the amplitude and phase of the response as a function of the drive frequency. You will then compare your experimental results to theory. (1) Check that you have collected x and V versus t for all of the frequencies listed in part C of your

spreadsheet. (2) Click on the Phase and Amplitude button in the Analysis section of your spreadsheet template.

This will run a macro that automatically extracts the amplitude xo, phase difference Δθ, and drive frequency f for each of the data sets you measured in Part C. The output of this macro is a small table in the Analysis section that summarizes the frequency, amplitude and phase difference found for the data you obtained in Part C.

(3) Make a plot of the measured amplitude xo versus frequency f. Don’t forget to label the axes and

add a title. (4) Fill in the cells for the fitting parameters Ao, fres and Q; we recommend starting with starting

with a trial value for Q of about 70. Then calculate the theory for the amplitude:

2

2

2

22

2

00

1

2/

−+

=

res

th

res

th

theory

ff

fQf

Ax (20)

(5) Add a curve for x0theory vs fth to your plot of your data and reformat the markers to be a line. (6) Adjust the fitting parameters Ao, fres and Q by hand until you get good agreement between data

and theory. This takes a little bit of trial and error. QUESTION D1: How well does the natural oscillation frequency fosc found in part B agree with

your fit value for the resonance frequency fres found here. Should they be the same? Explain. QUESTION D2: Comment on the qualitative agreement between the theory and your measured

results for the amplitude xo versus the drive frequency f.

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(7) Make a plot of your data for the phase difference θ versus frequency f. Be sure to label the axes, and add a title.

(8) Calculate the theory for the phase difference θ :

02 2

0

1Arctanof f

Q f fθ θ

= + − . (21)

This is the same as Equation (15) except we added an offset θo that you will need to adjust. Add θth vs fth to your θ vs f plot and reformat the markers to be a line. For this theory, use the values for fres, Q and Ao that you found from your previous plot of the amplitude. You should get a good fit with θo=0, but you may find you need to adjust θ0 to get good agreement between data and theory. Hint: if something looks wrong with your theory for θ(f), recall that Arctan is multi-valued and also Arctan in Excel gives an output in radians, not degrees.

QUESTION D3: Discuss how well the theory for the phase shift θ versus the drive frequency f

agrees with your data. QUESTION D4: From Equation (11) you can show that τd = Q/(π fo ). Using the Q and fres that you

just got in part D, use this equation to find the damping time τd. Compare this value for τd with the value for τd that you obtained in Part B from fitting to the free oscillation decay curve.

Part E. What Else is Going On? - While working on parts C and D, you may have noticed some other interesting things going on in

this system. To explore some of the other interesting effects that can occur in this system, pick just one of the options below. The point is to choose the option that looks most interesting to you. If you can’t choose based on that, figure out which one looks easiest. If that is not the way you roll, pick the hardest one. After you have picked one of the options, briefly discuss with your instructor why you chose that one and then go at it.

- Use the worksheet labelled “Part E” for completing this part. Unlike the cells in sheet 1, you can

access all the cells in the Part E worksheet. You will still need to use the Save button on sheet 1 to save your file.

Option #1: Transient Response of the System This option involves observing transient beating phenomena. One thing you already should have noticed is that after you changed the drive frequency in Part C, that it took time for x(t) to settle down to a steady oscillation, and during this time the envelope or amplitude of the motion got bigger and smaller. This is called beating and happens because the system is moving simultaneously with two different frequencies. One part of x(t) is a transient (decaying) sine wave with the frequency fosc and the other part is a growing sine wave at the drive frequency f. - Carefully adjust the frequency of the drive so that it is about 0.1 or 0.06 Hz above the resonance

frequency. You can use almost any drive frequency, but either of these choices for the drive frequency should yield some nice looking beats.

Phys 275 - Jan 2020 Update 115

- Next, turn off the power to the function generator and wait for the cart to stop oscillating and settle to its resting position. Of course the air blower should still be on.

- Tell LoggerPro to start Collecting data, wait about 1 or 2 seconds, and then turn on the power to the function generator. After the drive starts up you should see a nice beating transient response, which settles towards the steady oscillation you observed in Part C. - Copy and paste the t, x and V data into the part E worksheet. Make a plot showing x versus t and make sure everything (plot, axes and cells) is properly labelled. - Identify two successive peaks in the envelope of the x(t) curve and find the time tbeat between them. From this get the beat frequency 1/tbeat. QUESTION E1 - What is the frequency of the beats you observe? QUESTION E2 - From your measurements (clearly state numerical values you obtained), how is

the beat frequency related to the drive frequency and the natural oscillation frequency of the system?

- Figure out how to get Excel to extract the peak-to-peak position xpp for each cycle of the x(t) data,

i.e. the maximum and minimum envelope for the position waveform. Then make a plot of this xpp versus t.

Option #2: Position versus Force Plots This option involves generating an experimental version of the circle and ellipse plot of x versus Fa shown in Figure 3 in the Introduction. Although you can’t measure the driving force directly, you can measure the voltage V driving the speaker, which is directly proportional to the driving force Fa exerted by the speaker. In fact as you collect data LoggerPro already plots x versus V in a small graph on the lower left side of the display, right under the data table. - Run LoggerPro and watch the x vs Fa plot as you carefully adjust the frequency of the drive; this

plot is under the data table. Find four frequencies that qualitatively reproduce the circle (resonance) and ellipses (above and below resonance) shown in Figure 3. You don’t need 50 seconds of data, just grab a few seconds.

- Record the four drive frequencies on your Part E worksheet as well as x(t) and V(t) for these

frequencies. - Make one plot with the four x versus V curves on it. Choose the colors as in Figure 3 and be sure

to label the axes, add units, the legend and a title. QUESTION E3 - Does your plot look qualitatively like the one shown in Figure 3?

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Option #3: Resonance Frequency versus Mass of the Cart In this option you observe how the resonance frequency varies with the mass of the cart. - You will first need to find the mass of the cart and reflector. Carefully detach the springs from the

cart by pulling up on the reflector and sliding the springs off the post - hang onto the springs while you are doing this so they don’t go shooting off. Replace the reflector on the cart and measure the total mass of the cart and reflector.

- Record the mass value in the Part E worksheet. - Put the cart and reflector back on the track. Reattach the springs and make sure they don’t rub

against the cart. - Turn on the blower and function generator. Adjust the drive frequency to resonance and let the oscillation settle to its steady state behavior. Click on Collect on the LoggerPro template and watch the x versus V plot at the bottom left side of the screen (See Figure 3). If the plot circular (or at least not tilted to the left or right of the y-axis) then you have done a good job of driving the system at the resonance frequency. If it is an ellipse tilted to the right, then you will need to slightly increase the drive frequency, while if it is tilted to the left you will need to slightly decrease the drive frequency. Carefully do a fine adjust on the frequency until you have a nice circle or an ellipse that is not tilted left or right. Record the frequency fo where his happens - this is the resonance frequency. QUESTION E4 - What is the smallest change in the drive frequency that still produces a noticeable change in the x versus V plot? - Add a 10 g mass to the cart and repeat the previous step to find the resonance frequency. Repeat for a 50 g mass and then a 100 g mass added to the cart. Make sure that the springs don’t drag against the mass. Record in your spreadsheet each resonance frequency and the corresponding total mass of the cart, reflector and weight. - Make a plot showing the resonance frequency of the cart versus the total mass of the cart. Don’t forget to include a point for the data you collected in Part C, for which the total mass will equal just the mass of the cart and reflector. - Fit your fo vs Mtot data to Equation 10, using the spring constant k as a fitting parameter. QUESTION E5 - What is your best fit value for the spring constant k? Option #4: Changing the Drag Force on the Cart This part involves measuring the effect of air drag on the cart. The main drag force acting on this system is due eddy current damping from two small magnets that you can see attached to the side of the cart by screws. Air drag from the cart and the reflector also contributes significant damping to the system. In this part you will measure the change in the drag parameter b when you increase the area of the reflector.

- From Equation (7) you can see that the damping parameter b is related to the relaxation time τd by:

2 / db m τ= . (22)

Phys 275 - Jan 2020 Update 117

- To get a numerical value for b, you will need to find the mass of the cart and reflector. Carefully detach the springs from the cart by pulling up on the reflector and sliding the springs off the post. Hang onto the springs while you are doing this so they don’t go shooting off. Replace the reflector on the car and measure the mass of the cart.

- Record the mass value in the Part E worksheet. - Put the cart and reflector back on the track and reattach the springs. QUESTION E4 - Use Equation (22) and your data from Part B to get b for your cart with the

unmodified reflector. What do you find for b? - Get a 3x5 index card and masking tape from your instructor (there should be some at the

instructor’s desk). Securely attach the card to the top of the reflector. Take a photograph using the web camera and paste in your spreadsheet.

QUESTION E5 - By what percentage did the 3x5 card increase the area of the reflector? - Repeat the procedure in Part B to obtain a free-oscillation decay curve. Fit this curve to Equation

(6) and find the new oscillation frequency and decay time τd of the resonator when the card is in place. DO NOT OVERWRITE YOUR DATA OR ANALYSIS FROM PART B - do a new fit to your new data in the Part E worksheet.

- Use Equation (22) to get the damping parameter b when the card is in place. QUESTION E6 - Did you see a significant increase in τd? What is the new value of the damping

parameter b? By what percentage did b increase when the reflector was added? - When you are finished, be sure to remove the 3x5 index card and any tape you used from the

reflector. VIII. Finishing Up Check over your spreadsheet to make sure that you have not missed any steps, have filled

everything in, and have no feedback messages displayed. The automatic feedback system on the template has limited ability to detect problems, so check carefully.

Save your spreadsheet and submit it to ELMS before you leave the Lab. Turn in your checksheet to your instructor before you leave. If you did not complete the entire lab in class, finish everything at home and submit a revised spreadsheet to ELMS before the start of your next lab. If you finished early, use the remaining class time to work on the homework.

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IX. Homework (Submit your answers to the homework to Expert TA before the deadline) Questions and multiple choice answers on Expert TA may vary from those given below. Be sure to

read questions and choices carefully before submitting your answers on Expert TA. #1. In this lab a periodic force is applied to the cart using: (a) the sonic ranger, (b) the air track, (c)

friction, (d) drag, (e) a spring that is attached to a speaker, (f) the function generator. #2. A mass and spring system has a resonance at fo =1.40 Hz with a full-width of fFW=0.021 Hz.

What is the quality factor Q?

#3. If the drive frequency f equals the resonance frequency fo, the phase difference θ between the position of the drive and the mass is (a) 0o, (b) 45o, (c) 60o, (d) 90o, (e) 180o.

#4. Using Equation (14), find xo when Ao= 1.00 mm, f = 1.40 Hz, fo = 1.45 Hz, and Q = 80.0 #5. Using Equation (15), find θ when f = 1.47 Hz, fo = 1.44 Hz, and Q = 90.0 where Q is the

quality factor.