Download Simple Harmonic Motion

Simple Harmonic Motion Many physical systems exhibit a predictable oscillatory behaviour that can be described mathematically by a sinusoidal wave. A few examples include the motion of a pendulum, the oscillation of a spring-­‐mass system, the oscillation of charge carriers in an inductor-­‐capacitor electrical circuit, or the variation in pressure at the mouth of a bottle when airflow causes an audible resonance. The mathematical physics describing these systems is fundamentally identical, and because of the oscillatory nature of their behavior they are said to exhibit "simple harmonic motion" (SHM). The mathematical similarity between such systems stems from the nature of the forces present. In each case, a displacement from equilibrium generates a restoring force proportional to the displacement. For example, in the case of the spring-­‐mass system we’ll investigate, Hooke’s Law describes the force caused by a displacement from equilibrium according to the formula: F = -­‐kx. These systems exhibiting SHM all possess a physical parameter (analogous to x in Hooke’s Law above) which behaves as a sinusoidal function of time, such that: x(t) = A sin(2πft+φ) where: x(t) is the parameter that exhibits SHM (i.e. position, charge, pressure) t is time A is the amplitude of the parameter f is the frequency of the behavior (related to period, T, by: T=1/f) φ is called a "phase constant” which describes the initial condition of the parameter. In this laboratory we focus on studying the position of a mass in a spring-­‐mass system as a function of time. Figure 1. Example of spring-­‐
mass configuration and detector/cage position. What is interesting from a physics perspective is that once we understand the roles of the different variables describing the SHM of the position of the mass on a spring, we can extrapolate additional characteristics of the system and can study whether these additional characteristics conform to our understanding of the relationships between physical properties. For instance, we'll relate position as a function of time to the behaviour of the velocity and acceleration of the mass on the spring. When we understand the behaviour of the acceleration of the mass as a function of time we can relate that property to the forces acting on the system, via our familiarity with Newton's Second Law (F=ma). And when we understand the forces acting on a system, we can predict the state of the system's kinetic, potential, and total energy, and can make measurements to try to verify that our predictions are sound. So, fundamentally, the study of the SHM exhibited by a mass on a spring enables a much deeper appreciation of many parameters related to the system: position, velocity, acceleration, angular frequency, energy, and energy exchange. And importantly, since the same mathematics applies to many other systems, developing a sound understanding of the SHM of the spring-­‐mass system allows one to infer the properties and behaviour of other SHM systems. In this experiment we'll begin by taking measurements of the position of a spring-­‐mass system under a variety of conditions. We'll spend a portion of the lab analyzing whether our mathematical understanding of SHM can be used to extract those parameters describing the position of the mass versus time (A, f, φ), and we'll ensure that we understand the role of each of these parameters in describing SHM. Effectively, this amounts to "modeling" whether our predicted mathematical description of SHM agrees with observation, and whether we can confidently extract the variables A, f, and φ from our measurements. We'll also touch on the fact that, in reality, simple harmonic motion represents a somewhat simplified mathematical underpinning of more complex physics at play; physics that becomes evident under closer scrutiny. The lab will touch on factors that introduce greater mathematical complexity into our description of the behaviour of our spring-­‐mass system, notably damping. Again, as we understand to greater extent the more detailed behaviour of one physical system, it provides greater insight into those other systems also known to exhibit SHM. This is a common method in the study of physics, where frequently we develop simplified mathematical models to understand certain elements of the behaviour of complicated systems, eventually refining our models to account for more complex aspects of the system. In part two of the lab we'll investigate the state of the potential and kinetic energy associated with the spring-­‐mass system, again identifying whether SHM applies, and investigating whether the principle of conservation of energy applies. Part 1a: The Spring-­‐Mass System in Simple Harmonic Motion Objective • 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 Retort stand, rod, and clamp Vernier computer interface Spring, with a spring constant of Logger Pro approximately 10 N/m Vernier Motion Detector Metal cage 200 g and 300 g masses Procedure 1. Attach the spring to a horizontal rod connected to the vertical stand and hang the mass from the spring as shown in Figure 1. Securely fasten the 200 g mass to the spring and the spring to the rod so the mass cannot fall. 2. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface. 3. Place the Motion Detector approximately 75 cm below the mass. Make sure there are no objects near the path between the detector and mass, such as a table edge. To protect the Motion Detector if the mass should fall, place a metal cage above the sensor. 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. (Be careful not to knock the mass off the spring, as the motion detector below could be damaged.) 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. Measure the equilibrium position of the 200 g mass. To do this, allow 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 as (y0) in the data table provided below. 8. 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 should be characteristic of simple harmonic motion. 9. Using the position graph, determine the average amplitude, Aavg, of the simple harmonic motion and record it in the Data Table below. For each oscillation, A, is the maximum distance from the equilibrium position. Find Aavg by determining A for about 5 or 6 oscillation cycles (the exact number of A values used to calculate Aavg is not critical, though more values allow for a better average). 10. Using the position graph, measure the time interval between maximum positions. If you drag the mouse from one peak to another you can read along the x-­‐axis the time interval. 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 in your data table. 11. Determine the initial phase, φ, of the measured oscillation. To find the initial phase, or phase constant, for the waveform you captured, recall that a full oscillation cycle (or wavelength) corresponds to 2π radians. Take a ratio to determine what fraction of a wavelength cycle had already occurred when your motion sensor began to collect the waveform. This fraction of a full oscillation is the initial phase. 12. Repeat Steps 8–11 with the same 200 g mass, moving with larger amplitude than in the first run. 13. Change the mass to 300 g and repeat Steps 8–11. Use an amplitude of about 5 cm. Store a good run made with this 300 g mass on the screen. You will use it for several of the Analysis questions. Data Table Run
1
2
3
Mass
(g)
y0
(cm)
Aavg
(cm)
T
(s)
f
(Hz)
φ
(rad)
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 maximum? 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 300 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 y = y 0 + Asin(2πft + φ ) where y0 represents the equilibrium distance between the sensor and the mass when at rest. €
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., φ corresponds to C, and 2πf 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 parameter C corresponds the phase constant, φ, 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, φ is needed to achieve a good match. The optimum value for φ will be between 0 and 2π. Find a value for φ that makes the model come as close as possible to the data for your 300 g experiment. 6. Write down the model equation that best matches your data. Does the model fit the data well? How can you tell? 7. Close the Curve Fit window. In your report, show your measured data and your fitted model. Part 1b: Simple Harmonic Motion with Damping In this part of the experiment we study the effect of damping on the behaviour of the spring-­‐mass system. Tape or otherwise attach an index card to the bottom of the mass. Set the mass in motion with an amplitude of approximately 5 or 10 cm and collect additional data. You may want to take data for about 2 minutes. To increase the collection time, click the Data Collection button (next to the green Start Collection button) and increase the length of the measurement. Does the model used in the analysis of Part 1a still fit well in this case? Why or why not? Part 2: Energy in Simple Harmonic Motion In Part 1 we studied the oscillating mass in terms of its position, velocity, and acceleration as a function of time. We can also describe the system from an energy perspective. In this part of the experiment, you will measure the position and velocity as a function of time for an oscillating mass and spring system, and from those data, plot the kinetic and potential energies of the system. Energy is present in three forms for the mass and spring system. The mass m, with velocity v, can have kinetic energy KE KE = 21 mv 2 The spring can hold elastic potential energy, or PEelastic. We calculate PEelastic by using PE elastic =
1
2
ky 2 where k is the spring constant and y is the extension or compression of the spring measured from the equilibrium position. The mass and spring system also has gravitational potential energy (PEgravitational = mgy), but we do not need to include the gravitational potential energy term if we measure the spring length from the hanging equilibrium position. We can then concentrate on the exchange of energy between kinetic energy and elastic potential energy. If there are no other forces experienced by the system, then the principle of conservation of energy tells us that the sum ΔKE + ΔPEelastic = 0, which we can test experimentally. Procedure 1. The Motion Detector should still be connected to the DIG/SONIC 1 channel of the interface. 2. To calculate the spring’s potential energy, it is necessary to measure the spring constant k. Hooke’s law states that the spring force is proportional to its extension from equilibrium, or F = –kx. You can apply a known force to the spring, to be balanced in magnitude by the spring force, by hanging a range of weights from the spring. The Motion Detector can then be used to measure the equilibrium position. Open the experiment file “17b Energy in SHM.” Logger Pro is now set up to plot the applied weight vs. position. 3. Click to begin data collection. Hang a 50 g mass from the spring and allow the mass to hang motionless. Click and enter 0.49, the weight of the mass in newtons (N). Press ENTER to complete the entry. Now hang 100, 150, 200, 250, and 300 g from the spring, recording the position and entering the weights in N. When you are done, click to end data collection. 4. Click on the Linear Fit button, , to fit a straight line to your data. The magnitude of the slope is the spring constant k in N/m. Record the value in the data table below. 5. Open the experiment file “17c Energy in SHM.” You may need to modify the calculations for the energies. Adjust the parameter for mass and spring constant as appropriate. 6. With the mass hanging from the spring and at rest, click to zero the Motion Detector. From now on, all distances will be measured relative to this position. When the mass moves closer to the detector, the position reported will be negative. 7. Start the mass oscillating in a vertical direction only, with an amplitude of about 10 cm. Click to gather position, velocity, and energy data. Data Table Spring constant
N/m
Analysis 1. Click on the y-­‐axis label of the velocity graph to choose another column for plotting. Click on More to see all of the columns. Uncheck the velocity column and select the kinetic energy and potential energy columns. Click OK to draw the new plot. 2. If mechanical energy is conserved in this system, how should the sum of the kinetic and potential energies vary with time? Choose Draw Prediction from the Analyze menu and draw your prediction of this sum as a function of time. 3. Check your prediction. Click on the y-­‐axis label of the energy graph to choose another column for plotting. Click on More and select the total energy column in addition to the other energy columns. Click OK to draw the new plot. 4. From the shape of the total energy vs. time plot, what can you conclude about the conservation of mechanical energy in your mass and spring system?