laboratory exercise simple harmonic motion€¦ · laboratory exercise simple harmonic motion ......

Adapted from: Physics with Vernier/Physics with Video Analysis with permission

PhysicsLabHarmonicMotion2016 of 13

Laboratory Exercise

Simple Harmonic Motion

INTRODUCTION Motion that repeats itself symmetrically around a central point or position is periodic harmonic motion. The motion of a weight on a spring, the pendulum in an old clock, the wheel of your car as it races to get you to school on time, molecules vibrating in a crystal to tell time in your watch or phone, or the oscillations of the 120 electricity that comes out of our wall sockets are all examples of simple harmonic motion. In this exercise we will look at the first two examples (weight on a string, simple pendulum) and finally a comparison of circular, pendulum and spring powered oscillations and their sine wave functions.

Exercise 1 Spring Oscillations

One simple harmonic system that vibrates is a mass hanging from a spring. The force applied by an ideal spring is proportional to how much it is stretched or compressed. Given this force behavior, the up and down motion of the mass is called simple harmonic and the position can be modeled with

ftAy 2sin

In this equation, y is the vertical displacement from the equilibrium position, A is the amplitude of the motion, f is the frequency of the oscillation, t is the time, and is a phase constant. This experiment will clarify each of these terms.

Objectives Measure the position and velocity as a function of time for an oscillating mass and spring

system.

Compare the observed motion of a mass and spring system to a mathematical model of simple harmonic motion.

Determine the amplitude, period, and phase constant of the observed simple harmonic motion.



Materials computer ring stands, long rod, and clamps Vernier computer interface Logger Pro Vernier Motion Detector

Spring twist ties

500 g and 200 g masses

Making predictions - your hypothesis 1. Attach the 500 g mass to the spring and hold the free end of the spring in your hand, so the

mass and spring hang down with the mass at rest. Lift the mass about 10 cm and release. Observe the motion. Sketch a graph of position vs. time for the mass.

2. Just below the graph of position vs. time, and using the same length time scale, sketch a graph of velocity vs. time for the mass.

Procedure Attach the spring to a horizontal rod connected to two ring stands located on two different tables.

Hang the mass from the spring so that it is suspended between the two tables as shown in Figure 1. Securely fasten the 500 g mass to the spring and the spring to the rod, using twist ties so the mass cannot fall.



2. Connect the Motion Detector to the DIG/SONIC 1

channel of the interface. If the Motion Detector has a switch, set it to Normal.

3. Place the Motion Detector at least 75 cm below the mass. Make sure there are no objects near the path between the detector and mass, such as a table edge.

4. Open the file “15 Simple Harmonic Motion” from the Physics with Vernier folder.

5. Make a preliminary run to make sure things are set up correctly. Lift the mass upward a few centimeters and release. The mass should oscillate along a vertical line only. Click to begin data collection.

6. After 10 s, data collection will stop. The position graph should show a clean sinusoidal curve. If it has flat regions or spikes, reposition the Motion Detector and try again.

7. Compare the position graph to your sketched prediction in the Preliminary Questions. How are the graphs similar? How are they different? Also, compare the velocity graph to your prediction.

8. Measure the equilibrium position of the 500 g mass. Do this by allowing the mass to hang free and at rest. Click to begin data collection. After collection stops, click the Statistics button, , to determine the average distance from the detector. Record this distance (y0) in your data table.

9. Now lift the mass upward about 5 cm and release it. The mass should oscillate along a vertical line only. Click to collect data. Examine the graphs. The pattern you are observing is characteristic of simple harmonic motion.

10. Using the position graph, measure the time interval between maximum positions. This is the period, T, of the motion. The frequency, f, is the reciprocal of the period, f = 1/T. Based on your period measurement, calculate the frequency. Record the period and frequency of this motion in your data table.

11. The amplitude, A, of simple harmonic motion is the maximum distance from the equilibrium position. Estimate values for the amplitude from your position graph. Enter the values in your data table. If you drag the mouse from one peak to another you can read the dx time interval.

12. Repeat Steps 8–11 with the same 500 g mass, moving with a larger amplitude than in the first run.

13. Change the mass to 700 g by adding a 200g mass Repeat Steps 8–11. Use an amplitude of about 5 cm. Keep a good run made with this 700 g mass on the screen. You will use it for several of the Analysis questions.

Data

Run Mass

y0

A T

f (g) (cm) (cm) (s) (Hz)

1

2

3



Analysis 1. View the graphs of the last run on the screen. Compare the position vs. time and the velocity

vs. time graphs. How are they the same? How are they different?

2. Turn on the Examine mode by clicking the Examine button, . Move the mouse cursor back and forth across the graph to view the data values for the last run on the screen. Where is the mass when the velocity is zero? Where is the mass when the velocity is greatest?

3. Does the frequency, f, appear to depend on the amplitude of the motion? Do you have enough data to draw a firm conclusion?

4. Does the frequency, f, appear to depend on the mass used? Did it change much in your tests?

5. You can compare your experimental data to the sinusoidal function model using the Manual Curve Fit feature of Logger Pro. Try it with your 700 g data. The model equation in the introduction, which is similar to the one in many textbooks, gives the displacement from equilibrium. However, your Motion Detector reports the distance from the detector. To compare the model to your data, add the equilibrium distance to the model; that is, use

02sin yftAy

where y0 represents the equilibrium distance.

a. Click once on the position graph to select it.

b. Choose Curve Fit from the Analyze menu.

c. Select Manual as the Fit Type.

d. Select the Sine function from the General Equation list.

e. The Sine equation is of the form y=A*sin(Bt +C) + D. Compare this to the form of the equation above to match variables; e.g., corresponds to C, and 2f corresponds to B.

f. Adjust the values for A, B and D to reflect your values for A, f and y0. You can either enter the values directly in the dialog box or you can use the up and down arrows to adjust the values.

g. The phase parameter is called the phase constant and is used to adjust the y value reported by the model at t = 0 so that it matches your data. Since data collection did not necessarily begin when the mass was at the equilibrium position, is needed to achieve a good match.

h. The optimum value for will be between 0 and 2. find a value for that makes the model come as close as possible to the data of your 700 g experiment. You may also want to adjust y0, A, and f to improve the fit. Write down the equation that best matches your data.

6. Does the model fit the data well? How can you tell?

7. Predict what would happen to the plot of the model if you doubled the parameter for A by sketching both the current model and the new model with doubled A. Now double the parameter for A in the manual fit dialog box to compare to your prediction.

8. Similarly, predict how the model plot would change if you doubled f, and then check by modifying the model definition.

9. Click , and optionally print your graph.



Exercise 2 Period of a Pendulum

INTRODUCTION A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for many years the pendulum was the heart of clocks used in astronomical measurements at the Greenwich Observatory.

There are at least three things you could change about a pendulum

that might affect the period (the time for one complete cycle):

* the amplitude of the pendulum swing * the length of the pendulum, measured from the center of the

pendulum bob to the point of support * the mass of the pendulum bob

To investigate the pendulum, you need to do a controlled experiment; that is, you need to make measurements, changing only one variable at a time. Conducting controlled experiments is a basic principle of scientific investigation.

In this experiment, you will use a Photogate capable of microsecond precision to measure the period of one complete swing of a pendulum. By conducting a series of controlled experiments with the pendulum, you can determine how each of these quantities affects the period.

Figure 1

Objectives * Measure the period of a pendulum as a function of amplitude. * Measure the period of a pendulum as a function of length. * Measure the period of a pendulum as a function of bob mass.

Materials

computer string Vernier computer interface 2 ring stands and pendulum clamp Logger Pro masses of 100 g, 200 g, 300 g Vernier Photogate meter stick protractor



Making predictions - your hypothesis 1. Make a pendulum by tying a 1 m string to a mass. Hold the string in your hand and let the

mass swing. Observing only with your eyes, does the period depend on the length of the string? Does the period depend on the amplitude of the swing?

2. Try a different mass on your string. Does the period seem to depend on the mass?

Procedure 1. Use the ring stand to hang the 200 g mass from two strings. Attach the strings to a

horizontal rod about 10 cm apart, as shown in Figure 1. This arrangement will let the mass swing only along a line, and will prevent the mass from striking the Photogate. The length of the pendulum is the distance from the point on the rod halfway between the strings to the center of the mass. The pendulum length should be at least 1 m.

2. Attach the Photogate to the second ring stand. Position it so that the mass blocks the Photogate while hanging straight down. Connect the Photogate to DIG/SONIC 1 on the interface.

3. Open the file “14 Pendulum Periods” in the Physics with Vernier folder. A graph of period vs. time is displayed.

4. Temporarily move the mass out of the center of the Photogate. Notice the reading in the status bar of Logger Pro at the bottom of the screen, which shows when the Photogate is blocked. Block the Photogate with your hand; note that the Photogate is shown as blocked. Remove your hand, and the display should change to unblocked. Click and move your hand through the Photogate repeatedly. After the first blocking, Logger Pro reports the time interval between every other block as the period. Verify that this is so.

5. Now you can perform a trial measurement of the period of your pendulum. Pull the mass to the side about 10º from vertical and release. Click and measure the period for five complete swings. Click . Click the Statistics button, , to calculate the average period. You will use this technique to measure the period under a variety of conditions.

Part I Amplitude

6. Determine how the period depends on amplitude. Measure the period for five different amplitudes. Use a range of amplitudes, from just barely enough to unblock the Photogate, to about 30º. Each time, measure the amplitude using the protractor so that the mass with the string is released at a known angle. Repeat Step 5 for each different amplitude. Record the data in your data table.

Part II Length

7. Use the method you learned above to investigate the effect of changing pendulum length on the period. Use the 200 g mass and a consistent amplitude of 20º for each trial. Vary the pendulum length in steps of 10 cm, from 1.0 m to 0.50 m. If you have room, continue to a longer length (up to 2 m). Repeat Step 5 for each length. Record the data in the second data table below. Measure the pendulum length from the rod to the middle of the mass.



Part III Mass

8. Use the three masses to determine if the period is affected by changing the mass. Measure the period of the pendulum constructed with each mass, taking care to keep the distance from the ring stand rod to the center of the mass the same each time, as well as keeping the amplitude the same. Repeat Step 5 for each mass, using an amplitude of about 20°. Record the data in your data table

Data Part I Amplitude

Amplitude Average period

(°) (s)

Part II Length

Length Average period

(cm) (s)

Part III Mass

Mass Average period

(g) (s)

100

200

300



Analysis 1. Why is Logger Pro set up to report the time between every other blocking of the Photogate?

Why not the time between every block?

2. Use Logger Pro to plot a graph of pendulum period vs. amplitude in degrees. Scale each axis from the origin (0,0). Does the period depend on amplitude? Explain.

3. Plot a graph of pendulum period T vs. length . Scale each axis from the origin (0,0). Does the period appear to depend on length?

4. Plot the pendulum period vs. mass. Scale each axis from the origin (0,0). Does the period appear to depend on mass? Do you have enough data to answer conclusively?

5. To examine more carefully how the period T depends on the pendulum length , create the following two additional graphs of the same data: T 2 vs. and T vs. 2. Of the three period-length graphs, which is closest to a direct proportion; that is, which plot is most nearly a straight line that goes through the origin?

6. Using Newton’s laws, we could show that for some pendulums, the period T is related to the length and free-fall acceleration g by

gT

2 , or

gT

2

2 4

Does one of your graphs support this relationship? Explain. (Hint: Can the term in parentheses be treated as a constant of proportionality?)

7. From your graph of T 2 vs. determine a value for g.



Exercise 3 Oscillations

INTRODUCTION

There are many forms of oscillations. A string on a guitar, a vibrating tuning fork, and a monkey swinging on a vine are examples. Regardless of whether an oscillation is simple or complex, the motion can be described using a sine function or a combination of sine functions. Moreover, since sine functions are related to the components of position of a point moving in a circular path, there are close connections between circular motion and oscillations. In this activity, you will explore these connections using a video of three oscillating systems: a pendulum, a mass on a spring, and a dot on a rotating disk.

The simplest form of oscillatory motion can be described by a sine function, which we can write in the general form

y Asin( Bt C) D. Suppose y is the vertical position of a mass hanging from a spring, measured in meters, and the x axis is t (time), measured in seconds. Parameter ‘A’ of the sin function is the amplitude of the function. The amplitude measures how far the function ranges above (and below) the origin. Parameter ‘D’ is the offset, or vertical shift, of the sin function. The offset is 0 if the function is centered on the axis. The wave will be symmetrical about the axis when the offset is zero. Figure 1 would have an offset of 0. Parameter ‘C’ is the phase angle (Φ) of the sine curve. The angle that is changing is the angle (tilt) of the coordinate axis. Figure 2 shows how a sin function would shift to the right as the coordinate axis was tilted up approximately 120 degrees. Cosine functions describing oscillations are phase shifted 90 degrees (π/2) from sine functions. Finally, parameter ‘B’ is the angular frequency (ω) of the function. The angular frequency is also the time required for one complete cycle. In physical oscillations this is the period of the motion. Figure 3shows all four sine function parameters.

Mass on

Spring

Dot on

Rotating

Disk

Pendulum

Figure 1

Figure 2

Figure 3



Making predictions - your hypothesis

(a) What would the units be for each of the parameters when using a sine function

y Asin( Bt C) Dto describe periodic oscillations?

(b) Sketch graphs of one cycle of y = sin(t) and y = cos(t) on the axis. Don’t worry about units or numbers, just get the shapes right. In particular, think about the relative value of each function at the origin, t = 0. (i.e., are the values at a minimum, zero or maximum for each function?) Use this sketch as a guide later in the activity, when you have to answer questions such as, “Does the graph look like sin(t)?” Label each curve.

(c) When the graph of the sine function sin(t) is shifted to the left by 90 degrees or π/2 radians, it looks exactly like the graph of the cosine function cos(t). Circle the equation that correctly represents this fact:

cos(t) = sin((π /2)t) cos(t) = sin(t – π /2)

cos(t) = sin((–π /2)t) cos(t) = sin(t) + π /2

cos(t) = sin(t + π /2) cos(t) = sin(t) – π /2

Procedure

Part 1. Identifying the components of sinusoidal oscillations

Do the components of circular motion behave as sinusoidal oscillations? Start your investigation by

opening the Logger Pro Experiment file <Oscillations-1.cmbl>. It has the movie <ThreeSystems-1.mov>

inserted in it. This movie shows the dot on the disk moving at a constant speed while the pendulum and

mass on a spring are resting. Ignore the VA_Angle window for the time being. The movie has already

been calibrated in meters.

1. Grab the lower left corner of the movie window and drag it to make the movie larger. Click the Add Point tool ( ) and carefully mark the center of the blue/yellow spot on the disk in every frame. In the Page menu choose Auto Arrange to make the graph visible again. It shows the y component of the dot’s position versus time. Does this graph look like the graph of sin(t)? How is it the same, and how is it different?

2. Select the graph by clicking on it once. In the Analyze menu choose Curve Fit. From the list of possible functions, choose the sine function and then Try Fit and OK. Is it a good fit? Explain why or why not. What are the values of the fit parameters A, B, C and D, with units?

3. Sketch the graph of the fit on the axes shown in Figure 4. Draw it as a solid line.

y

t



4. Click on the Show Origin tool ( ) in the movie window. This reveals the coordinate axes. Note that the origin has been placed at the center of the disk. The parameter VA_Angle shown in a window below the movie is the tilt angle of the x-axis. In the window with the VA_Angle display, click on the Increase Angle tool ( ) about 10 or 15 times to tilt the axes. Describe what the graph appears to do while you change the angle. Which fit parameters stay about the same and which change? Which parameter changes the most, and does it get bigger or smaller?

5. Adjust the angle until the fit line starts at the origin (as a graph of sin(t) would). If you double-click on the window showing the value of VA_Angle, you’ll get a dialog box that allows you to set the angle. Watch what happens to the fit parameter that changes the most. When you tilt the axis a little more than or a little less than the angle where the fit line looks like sin(t), this parameter should switch between zero and 2π = 6.283. Recall that parameter C is the phase angle.

6. (a) Set parameter C as close to zero as you can get it by carefully adjusting VA_Angle. Change the places value from 0 to 1 in order to enter less than a full degree of axis rotation. What is the value of VA_Angle now? When you convert it to radians, how does it compare with the parameters you found in part 2(b)? As previously described, parameter C is called the “phase angle” of the sine curve. It is often represented by the Greek letter

. As you have seen, for the x-coordinate of a point on a circle, the phase angle tells how much you have to rotate the axes to make the graph of x(t) look like a sine function sin(t). Rewind the movie ( ). What is the relationship between the dot on the circle and the rotated x-axis in the initial frame?

Now hold down on the Increase Angle tool ( ) tool and watch the graph cycle through one complete period. Use the down arrow in the VA window to return the value to the value you entered in 5a above and watch as the function cycles through one complete period again.

7. Sketch the graph of the fit on the axes shown in Figure 4 on the previous page. Use a dotted line.

8. Double-click on the graph to get the Graph Options dialog. Choose Axes Options, set it to display both X_circle and Y_circle on the graph and click OK. The Y circle is the dot’s displacement function up and down while the X circle is the displacement right and left in the picture and the repeating cycle of the sine wave through time in the function. Fit X circle to a sine function. Does the graph of the fit look like sin(t)? If not, what function does it look like?

9. You have already set the axes so the phase angle for Y_circle is zero. Convert the phase angle for X_circle from radians to degrees. Is this the value you expect? Why or why not?

Figure 4. Graph of dot on circle.



10. Recall that parameter A is called the “amplitude” of the function. Do X_circle and Y_circle have about the same value of A? Should they be the same? Explain why or why not.

11. Remember now that parameter D is the “offset” of the function. It tells how far the center of the oscillation is from the coordinate origin. If the origin is at the center of the circle, D should be zero. Is it close to zero for X_circle and Y_circle?

12. Parameter B is called the “angular frequency” (ω) of the function. To understand what this means, first determine the time it takes for the dot to go around the circle once. On the graph, this will be the time interval from a point where the graph crosses the time axis to the next time where it crosses with the same slope. Select the graph. With the mouse, point to the third place where the Y_circle graph crosses the time axis. (You adjusted the phase angle so the first place it crosses is the origin of the graph.) The coordinates of that point are shown in the lower left corner of the graph window. What is this value (with units) of the time interval to three significant digits? It is called the “period” of the motion and is denoted T.

13. Calculate 1/T. This is the “frequency” f of the motion. Write its value with units.

14. Multiply the frequency f by 2π radians and write the result to three significant digits with appropriate units. Is the result nearly equal to the value you obtained for the angular frequency? Express the relationship between and f as an equation.

Part 2. Comparing the sine functions of different systems

Different systems of oscillation and sine functions. Start this investigation by opening the

Logger Pro Experiment file <Oscillations-2.cmbl>. It has the movie <ThreeSystems-2.mov>

inserted in it. The pendulum length, the mass and the motor speed were all adjusted so the

systems would have about the same period. The positions of the dot on the circle, the

pendulum bob and the mass on the spring have already been marked for you. To see the

points, step through the movie using the Next frame button ( ) under the movie.

15. Do X_circle and X_pend appear to have about the same phase? Explain how you can tell by looking at the graph or the movie.

16. Fit both X_circle and X_pend to sine functions as you did earlier. What are their phase angles? Are they about the same?

17. From the Options tab remove the graph of the pendulum. Now click on the Show Origin tool ( ) in the movie window. This reveals the coordinate axes. Note that the origin has been placed at the center of the disk. Now select the Set Origin tool (third button from the top on the right side of the picture). If you move the cursor to the center of the coordinate axis and then left click with your mouse you are able to move the axis. Move the axis up and down (you can use the background rod as a reference). Does either the shape of the function or any of its parameters change? Move the axis to the left so that the Y axis lines up with the furthest most reference dot on the disk. Explain what happens to the graph of the sine function. Which function parameter changed?

18. Now that you have some experience in analyzing the sine wave functions of periodic motion you can analyze the motion of the mass on the spring. Set up the axis and obtain the sine function for the motion of the mass. What do the different parameters tell you about its



motion? (There are several ways to do this. You may use either X_mass or Y_mass. You may set the axis at the top, bottom, or in the center of motion. Just so long as you set the coordinate axis appropriately the sine function will provide the same information.

19. Earlier when you compared the fits for X_circle and Y_circle, you found that they had the same amplitude and angular frequency. Do the functions for the circle, pendulum, and mass have the same amplitude and angular frequency? Should they? Explain why or why not.

laboratory exercise simple harmonic motion€¦ · laboratory exercise simple harmonic motion ......

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