Lab 10 Harmonic Motion and the Pendulum L10-1 Name Date Partners Lab 10 - Harmonic Motion and the Pendulum Galileo reportedly began his study of the pendulum in 1581 while watching this chandelier swing in Pisa, Italy OVERVIEW A body is said to be in a position of stable equilibrium if, after displacement in any direction, a force restores it to that position. If a body is displaced from a position of stable equilibrium and then released, it will oscillate about the equilibrium position. We shall show that a restoring force that is proportional to the displacement from the equilibrium position will lead to a sinusoidal oscillation; in that case the system is called a simple harmonic oscillator and its motion is called a harmonic motion. The simple harmonic oscillator is of great importance in physics because many more complicated systems can be treated to a good approximation as harmonic oscillators. Examples are the pendulum, a stretched string (indeed all musical instruments), the molecules in a solid, and the atoms in a molecule. The example that you will be studying in this session is the pendulum: it has a position of stable equilibrium and undergoes a simple harmonic motion for small displacements from the equilibrium position. We will first analyze the motion theoretically before testing the theory experimentally.
L10-2 Lab 10 Harmonic Motion and the Pendulum THEORY A simple pendulum consists of a mass m suspended from a fixed point by a string of length L. If the mass is pulled aside and released, it will move along the arc of a circle, as shown in Figure 10.1. Let s be the distance from the equilibrium position measured along that arc. While the force of gravity, mg, points downward, only its tangential component along the arc, F tan = mgsinθ, acts to accelerate the mass. The minus sign indicates that F tan is a restoring force, i.e. one that points in a direction opposite to that of the displacement s. From Newton s second law, F tan = ma, we then obtain: mgsinθ = ma (10.1) The speed of the mass bob is determined by v = s/ t, and the acceleration by a = v/ t. We can easily express s in terms of θ by noting that s = Lθ. So we have Photogate s = 0 s L Groove marking the center of mass Figure 10.1: (measured from the suspension point to the center of mass) F tan mg Reference bar s = Lθ and θ = s L (10.2) Equation (10.1) is normally studied for small angles, because the solution is particularly simple if we assume small arc length s, and therefore small angles θ 1 (in radians). We expand sinθ in terms of θ to obtain the series expansionsinθ = θ θ3 3! + θ5 5!..., (valid when θ is measured in radians). From this series expansion, we see that we can approximate sinθ by the angle θ itself, as long as we keep θ so small that the higher order terms are much smaller than θ. In other words, when θ 1,sinθ θ. This is called the small angle approximation. If we use the small angle approximation and Equation (10.2), Equation (10.1) becomes ( mg ) F = mg sinθ mgθ = s (10.3) L This is the equation of motion of a simple harmonic oscillator, describing a simple harmonic motion. Note the similarity between this equation for the pendulum and what we found earlier for the mass and spring. In that case the solution for θ is solved for arbitrary values of the amplitude, θ max, and the phase, δ, by: θ = θ max cos(ωt + δ) (10.4) What is the significance of θ max and δ? They are not determined by the previous equations but are fixed by the initial conditions. θ max is the maximum value of θ during the motion. It is determined by how far the pendulum was pulled back initially. The phase δ depends on when the clock was started that is used to measure t.
Lab 10 Harmonic Motion and the Pendulum L10-3 INVESTIGATION 1: MASS INDEPENDENCE OF PERIOD Time: In this activity you will measure the period of a simple pendulum.the materials you will need are: photogate timer step stool large support stands thread (string) pendulum string clamp brass and aluminum pendulum bobs meter stick You are given a task of building a very simple, affordable and robust clock for an outdoor field work. You decided to hang a bob on a string and count number of full swings bob makes during an observation. You are tasked with finding out the time for one full swing (a.k.a. period) and converting the swings into seconds. You need to establish correct relation between the parameters of your pendulum in the lab first, so that you can reliably use your simple clock whenever your research takes you. Naturally, you would like your clock to be as light as possible so the first thing you decided to check is if the time of a swing depends on the mass of the bob. You settled on a light aluminum bob, but you are worried that light material might not work as well as a sturdier but heavier brass bob. Prediction 1-1: List the basic physical parameters of your system relevant to the problem at hand. What parameters of the pendulum besides the material of the bob do you think may affect its period? Explain why how each parameter may affect your results. Prediction 1-2: In order to make sure you are studying the effect of only one parameter at a time (in our case the mass of the pendulum) how would you set up your experiment? What are the things that you think will need to be similar for both set-ups? Are those constraints achievable in the set-up present on your table?
L10-4 Lab 10 Harmonic Motion and the Pendulum Activity 1-1: Period of Aluminum Bob NOTE: To make the pendulums the same length suspend them side by side in the same clamp (held between the clamp and the bar) and adjust their lengths so that the center-of-mass grooves of the two bobs are at exactly the same level. Have your TA check your setup before proceeding! 1. Set the pendulum lengths (measured from the pivot point to the line marking the bob s center of mass) to about 90 cm. There are several ways to measure the length. One way is to elevate the string clamp so you can place the meter stick between it and the table with the 0 of the meter stick flush with the pivot point. This also makes many of the length measurements at eye level rather than down near the table. The string clamp can then be moved down the support rod to where the bob is between the photogates. Record the measured length and estimate the uncertainty: Pendulum length: ± cm 2. Move the brass bob out of the way by draping it over the stand. 3. Open an experimental file L10.A1-1 Mass Independence of Period. 4. Set up the aluminum bob so that it hangs right in the middle of the photogate. The red LED light on the photogate will be lighted up (you can see the reflection of the light on the photogate base). Slowly slide the photogate slightly to a side until you see that the LED light beam is no longer broken by the bob (red light goes off). This is the smallest amplitude we can achieve with our set-up. 5. Now gently push the bob until it crosses the LED light beam again and start collecting data on the computer. Try to minimize the wobbling. Wobbling will result in a poor measurement. Settle for the lightest push of the bob that still breaks the beam. It ll look more like a twitch rather than a swing. 6. Resulting amplitude will be very small but you can verify that the LED light is being broken by the bob by observing the red light reflection going on and off on the bottom of the photogate base. 7. The left most column of the Raw Data table in your software will be displaying the period and resulting statstics in real time. Stop the measurement when you collected about 3 4 values of the period. Copy the live data over to a permanent column of the table labelled accordingly. The statistics tool on the bottom of a column will calculate the mean and the sample standard deviation for you. 8. Record your result and corresponding error below. Aluminum pendulum period (mean): ± s
Lab 10 Harmonic Motion and the Pendulum L10-5 Activity 1-2: Period of Brass Bob 1. Hang the bobs side by side once again and see if they are still the same length. If the length has changed, you may need to measure the length again and repeat your measurements. Move the aluminum bob aside so you can measure the period of the brass bob. 2. Make sure that the aluminum and brass pendulum strings do not cross on top: it will significantly affect your final result. 3. Measure the period of the brass pendulum in the similar manner and enter your result below. Don t forget to copy the live data into a permanent column of the table. Brass pendulum period (mean): ± s Question 1-1: In the measurement that you just conducted for which parameters from Prediction 1 did you not account? As a result, what are your additional experimental errors in the pendulum measurement besides the statistical errors calculated above? Discuss them and make estimates here.
L10-6 Lab 10 Harmonic Motion and the Pendulum Question 1-2: Given that the masses of the bobs differ by about a factor of 3, is there a significant difference between the two periods? How do you know? Would you say that assertion that the pendulum period is independent of the mass is valid? INVESTIGATION 2: DEPENDENCE ON LENGTH Time: Activity 2-1: Dimensional Analysis Hopefully the results of the previous Investigation 1 gave you good news that you can be packing light for your field trip. However to continue the experiment (we can discuss the reasons later) you decided to use the brass bob for your lab studies. You are now ready to do some quantitative predictions. A lot of times simple relations between physical parameters can be deduced by a dimensional analysis technique. The technique is based on the fact that all fundamental measurements are various combinations of the 3 basic measurements: mass, time, and length, or in SI units kilograms, seconds and meters. Let s take a look at already familiar kinematic relation between the displacement and acceleration for a motion with constant acceleration you studied in the early labs. The exact relation is x = at 2 /2. You could arrive to a basic form of that relation by dimensional analysis alone. You know that you have to relate meters on the left side of the equation and seconds and acceleration on the right side. The units of the acceleration are meter/second 2. You can write the equation as follows: meter = (second) n (meter/second 2 ) m where n and m are the unknown exponents to be determined. Since the fundamental units are independent you can equate the exponents on each unit independently. That is meter = (meter) m and 1 = (second) n 2m From the first expression m is 1 and from the second expression n 2m = 0 or n = 2. Since n was the power of time and m of the acceleration we predict that x a t 2. The expression is, unfortunately, only known to a constant that we can not predict from this method. However if the functional dependence is established a constant can be measured experimentally.
Lab 10 Harmonic Motion and the Pendulum L10-7 Another example of the dimensional analysis can be demonstrated for predicting a range of a projectile motion, R, you also studied in the earlier lab. From our everyday experience we know that the range depends on the initial speed, v 0 of the projectile, the launching angle α and we know that the entire phenomenon is related to gravity, thus it s a reasonable assumption that acceleration of the free fall, g, will be involved. Clearly the trigonometric part of the equation can not be predicted from the dimensions since trigonometric functions are dimensionless, but we can still make some valid predictions. Again we have meter for the range, meter/second for the velocity and meter/second 2 for g. or separated by units meter = (meter/second) m (meter/second 2 ) n meter = (meter) n+m and 1 = (second) (m+2n) Simple algebra yields m = 2 and n = 1 or R v 2 0 /g which, up to a trigonometric function of sin(2α), is a correct relation. Moreover, if we experimentally determine that, for example, the range for 45 trajectory is the longest and we know that a simple trigonometric function that can be involved in this relation (it has to be finite for all angles so tan(α) would not work) is less than 1, we can quite confidently say that the dimensional analysis up to a numeric coefficient gave us a correct expression for the range at 45. And furthermore, that numerical coefficient can be determined experimentally as well. Prediction 2-1: What are the physical parameters and global constants you think should be involved in determination of the pendulum period? Remember the results of the previous Investigation 1 and what period does NOT depend on. Derive the expression for the period as function of the physical parameters you chose using dimensional analysis. Write the expression below and use the name ρ for the undetermined proportionality constant. Show your work. T theory = ρ Activity 2-2: Dependence on Pendulum Length The expression you derived above only needs a constant ρ that you can determine experimentally. Prediction 2-2: What kind of measurements can you perform in order to determine the unknown constant ρ? Describe your plan of action.
L10-8 Lab 10 Harmonic Motion and the Pendulum 1. You will now measure T for several lengths, say 30, 60, and 90 cm for same angle. 2. Continue using the same experimental file and the same experimental technique as in the previous Investigation 1. 3. Take 3 4 trials at each length and enter your data into the permanent columns of the Raw Data table. It is not necessary for you to have the lengths set right at 30, 60, and 90 cm. Set it close to these lengths and measure the actual length. In fact you should already have the data for 90 cm from Activity 1-2. Write down the results below: T 90 : ± s L 90 : ± m T 60 : ± s L 60 : ± m T 30 : ± s L 30 : ± m 4. Once you completed the measurement for all three lengths put your data for the average periods and corresponding lengths into Period Dependence on Length table in your software. Prediction 2-3: Look at your expression for the period and think what are the simplest variables/expressions which will give you a linear relation to extract the unknown proportionality constant ρ. Describe your ideas here and explain how you can extract ρ from your measurements. 5. See if the plot in your software matches your prediction. define new variables in the Calculator of your software and redefine the graph to suit your prediction. Use the parameters of straight line fit to the data on the graph to extract ρ. 6. Print a copy of tables and graphs for your report. Include data and graph.
Lab 10 Harmonic Motion and the Pendulum L10-9 Question 2-1: From your fit calculate the missing constant and propagate the error. Show your work. Write down a complete expression for the period of your pendulum using the numerical constant you just derived. ρ: ± T theory = Your quest is nearly complete. You have established a theoretical expression for the period of the pendulum and experimentally found a missing parameter. The only remaining step is to check that your theory actually works!! Remember that in Investigation 1 you measured the period of an aluminum bob as well. You have also measured the length of the aluminum pendulum.
L10-10 Lab 10 Harmonic Motion and the Pendulum Question 2-2: Use your newly minted theory to predict the period of the aluminum pendulum. Propagate the uncertainty from the measurement of the pendulum length AND from your experimental constant ρ. Show your calculations. How well does your theoretical prediction agree with the measurement? Discuss any additional sources of error that may have caused the discrepancy. T aluminum theory : ± s INVESTIGATION 3: CONSERVATION OF ENERGY Time: The conservation of energy is one of the most fundamental laws of physics. We test it more than once in this course. The total mechanical energy, i.e. the sum of potential and kinetic energy, should be a constant. On the other hand, both kinetic and potential energies change with time. Conservation of energy ( KE = PE ) requires that: 1 2 mv2 = mg h = mg(h 2 h 1 ) (10.5) We will use the motion of the pendulum to check Equation (10.5). The change in kinetic energy must be equal to the change in potential energy along the entire path of motion of the pendulum. We measure the change in potential energy by simply measuring the change in height of the pendulum bob. To check the conversion of potential energy into kinetic energy, we need to measure the maximum speed v max of the bob as it passes through the equilibrium position. It will be even easier if we just measure the ratio of the change of the two energies, which must be equal to one: 1 = KE PE = 0.5 mv2 mg h = v 2 2g(h 2 h 1 ) (10.6) The initial speed is zero. The mass has canceled out, and we only need to measure the bob s initial height h 2, the bob s lowest height h 1, and the maximum speed v max of the bob.
Lab 10 Harmonic Motion and the Pendulum L10-11 Activity 3-1: Measurement of Speed To measure the maximum speed of the bob, we will use the motion detector to measure the maximum speed of the bob at the bottom. We could check Equations (10.5) and (10.6) anywhere along the path, but it is easiest to measure the bob s speed at the bottom of the swing, where the speed is a maximum. The motion detector does not easily detect the bob s swing, because it does not move directly towards or away from the detector. We have found the bobs we have been using up to now in this lab to be too small, so we are going to change our experiment somewhat to use a larger bob. You will need the following equipment for this measurement: motion detector large pendulum bob and string short metal rod and clamp hook clamp to suspend pendulum string 1-m stick 1. Place the motion detector on the table. Put the detector on broad beam. The setting may need to be adjusted to improve the data quality. 2. Switch the pendulum string clamp and the clamp with aluminum rod and a hook, so that the aluminum rod and the hook are now at the top of your stand. 3. Hook the pendulum string on the hook clamp. The length L of the pendulum should be about 100 cm: just long enough to have the bob hang over the top of the table surface. Question 3-1: Explain why you do not need to know precisely the mass of the bob for this measurement.
L10-12 Lab 10 Harmonic Motion and the Pendulum 4. Adjust the relative position of the bob and the motion sensor so that the center of the motion sensor is well aligned with the center of the freely hanging bob. Make sure that the ball hangs as far from the stand as possible. 5. The pendulum bob will be pulled back away from the computer and motion detector and let go. Practice pulling back the bob a couple of times and letting it go. Use as large of an initial angle as the set-up allows. Also make sure that the bob does not come within 20 cm of the motion detector or hits it. 6. Because the ball is spherical, you can easily tell where the outside of the sphere touches the ruler. The height of the center of the ball above the table top will be the height h 1, the lowest height of the pendulum bob. Write down the value of h 1 here: Height h 1 of pendulum bob: cm Question 3-2: Explain why you do not need to know the precise value of the angle in this measurement. 7. Open the experiment file L10.A3-1 Conservation of Energy in the computer. The position and velocity of the bob will be visible. 8. Pull back the bob to a desired angle so that the bob moves directly towards the motion detector. Measure the initial height of the bob h 2. Make sure to use the same point on the sphere as you did to measure h 1. Hold your meter stick in position until you let go of the bob. Write down this initial height in Table 10.1 for trial 1. 9. Start taking data and let the pendulum swing through a couple of periods before stopping the computer. 10. You probably will notice the display indicates that the motion detector has several blips where the bob is not seen. This is normal because the motion detector is looking straight ahead, yet the bob swings in an arc. Near large angles the bob can easily go out of view of the detector. However, we are only interested in measuring the bob s velocity at the bottom of its swing where the velocity is a maximum. You should use the initial arc when the bob is swinging towards the motion detector.
Lab 10 Harmonic Motion and the Pendulum L10-13 Parameter Height h 2 Height h 1 Trial 1 2 Height h = h 2 h 1 Speed v KE PE Table 10.1: Conservation of Energy Measurement Question 3-3: Why do we want to use the initial arc of the bob s swing? In this case is the velocity positive or negative? Explain. 11. Find the maximum speed of the bob during this first arc and write the speed v in Table 10.1. Complete Table 10.1 for trial 1. You may use Excel as well. 12. Do not do another trial until you read Question 3-4.
L10-14 Lab 10 Harmonic Motion and the Pendulum Question 3-4: How well do your measurements agree with what you expect? If your results disagree with your expected answer by more than 5%, you should repeat them and be more careful. You do not need to do trial 2 if your measurement is accurate to better than 5%. INVESTIGATION 4: DEVIATION FROM SIMPLE THEORY Time: Activity 4-1: Dependence on Angle There is one remaining issue with your simple clock that we swept under the rug. Similarly to our derivation of the range of the projectile motion using dimensional analysis we should have acknowledged that our simple theoretical calculation can not predict any dependence on the initial angle from which the pendulum is released (which is to say dependence of the period on the amplitude of the oscillations). Intuitively we expect some dependence described, perhaps, by some trigonometric function or a dimensionless quantity such as angle. In this experiment we want to investigate the dependence of the period on the initial angle θ. A simple test would require launching the pendulum bob from various angles and measuring the period. Recall however that to get a reliable measurement with an error consistent with our previous measurements we will need to launch the bob from the same angle several times and measure the angle as well. That might be quite cumbersome to do in the laboratory. However we can use the results of the previous Investigation 3 to measure the initial angle when we know the velocity of the pendulum at its lowest point. Turns out that the photogates you used in the earlier experiments are capable of measuring the speed of and object while inside the photogate. Since the width of the pendulum bob passing through the photogate is known, we can simply divide the diameter of the bob by the time it blocks the photogate s LED to get pretty accurate value of the speed at the bottom of the swing.
Lab 10 Harmonic Motion and the Pendulum L10-15 Prediction 4-1: Using Equation (10.5) and Figure 10.1 derive the expression for the initial angle of oscillation θ. HINT: How are the length of the pendulum, initial angle and the height difference are related? Let s return to our original apparatus and use the brass bob. Re-measure the length of the pendulum if you think it changed since your last measurement in Investigation 2. Length of Pendulum: ± cm 1. Align your photogate so that the bob is in the middle of the gate while it s hanging completely vertically. 2. You want to measure the period T for several initial angles θ. Swing the pendulum several times for various angles to make sure there are no obstacles in its way WARNING: Do not allow the bobs to strike the photogate, ESPECIALLY DURING THE LARGER OSCILLATIONS!! 3. Open experimental file L10.A4-1 Deviation From Simple Theory and enter pendulum length into the software s calculator. The software will use that value, and the measured value of the speed inside the photogate to predict the value of the initial angle using the very same formula you derived in Prediction 4-1. 4. Bring the bob to a relatively small angle, start the software and release the bob. Measure T 3-4 times and make sure the results are consistent and are displayed correctly on your graph of T vs. θ. If not, re-measure until results are consistent. 5. Stop the software, pull the bob to a larger angle, start the software and release the bob again. After several swings a new result will be displayed on the graph again. If it is not make sure that the software allows multiple runs to be displayed on the same graph. 6. Repeat for 2 more angles and make sure that angles are different enough from one trial to another.
L10-16 Lab 10 Harmonic Motion and the Pendulum Question 4-1: Describe what you observe in the data. Is the period constant as the angle increases? Keep in mind a typical error of the period you had in previous Investigations. 7. Print one copy of your graph. Question 4-2: Where on the graph is the value of the period predicted by your simple theory?
Lab 10 Harmonic Motion and the Pendulum L10-17 Question 4-3: What can you say about the consistency of your results with simple theory you developed in this lab? Question 4-4: An intern charged with counting the swings launched your clock from a 30 angle and a particular thesis worthy event lasted 100 cycles (complete periods) by his count. Estimate (to 1 significant figure) by how much your measurement of time will be off if you used your simple theory to convert the swings into seconds. PLEASE CLEAN UP YOUR LAB AREA!