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1 (ta initials) first name (print) last name (print) brock id (ab17cd) (lab date) Experiment 5 Harmonic motion In this Experiment you will learn that Hooke s Law F = kx can be used to model the interaction of countless physical systems the basics of the Harmonic Oscillator, a cornerstone model in the analysis of physical problems that harmonic motion is esentially the movement of an object around the circumference of a circle this model cam easily be described by the rules of trigonometry how to set and check the calibration of a data gathering device; to extend your data analysis capabilities with a computer-based fitting program; to apply different methods of error analysis to experimental results. Prelab preparation Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report and as reference material during your final exam. Compile these printouts to create a lab book for the course. To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the content of this document and that of following FLAP modules ( Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the module in depth, then try the exit test. Check off the box when a module is completed. FLAP PHYS 1-2: Errors and uncertainty FLAP MATH 1-5: Exponential and logarithmic functions FLAP MATH 1-6: Trigonometric functions Webwork: the Prelab Harmonic motion Test must be completed before the lab session! Important! Bring a printout of your Webwork test results and your lab schedule for review by the TAs before the lab session begins. You will not be allowed to perform this Experiment unless the required Webwork module has been completed and you are scheduled to perform the lab on that day.! Important! Be sure to have every page of this printout signed by a TA before you leave at the end of the lab session. All your work needs to be kept for review by the instructor, if so requested. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 34

2 35 The simple harmonic oscillator A simple harmonic oscillator (SHO) is a model system that is widely used, and not just in the elementary physics exercises.the reason for this is profound: the same fundamental equations describe the motion of a mass-on-a-spring, and also of the interatomic forces that hold all matter together. Literally, everything we touch, at the atomic level, is held together by springs connecting pairs of atoms. The details of these interactions are studied in Quantum Mechanics and Solid-State Physics, but two masses connected by a coiled elastic wire represent an excellent model system, from which much can be learned. With no external forces applied to the material, the interatomic springs are at their equilibrium length, neither stretched nor compressed. The application of an external stretching force to the material will cause these springs to extend, thereby increasing the bulk length of the material. When the applied external force is removed, the springs return to their equilibrium lengths, restoring the material to its original dimensions. This restoring force may be overcome by a large enough applied external force that will cause the object to deform permanently or to break. The maximum force applicable without permanent distortion is called the elastic limit of the material. Hooke s Law states that the amount of stretch or compression x exhibited by a material is directly proportional to the applied force F. The proportionality constant, called the spring constant k, has the units of Newtons per metre (N/m) and is a measure of the stiffness of the material being considered. This proportionality between force and elongation has been found to hold true for any body as long as the elastic limit of the material is not exceeded. Perhaps the most convenient way of expressing a relationship of the Hooke s Law is to write; F = kx, (5.1) The minus sign expresses the fact that the force exerted by the spring, F s, is always opposing the deformation. It is often described as a restoring force If F s is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency, which does not depend on the amplitude. Harmonic motion is equivalent to an object moving around the circumference of a circle at constant speed. In the absence of friction, the oscillations will conrinue forever. Friction robs the oscillator of it s energy, the oscillations decay, and eventually stop altogether. Frequently, this friction term is proportional to the velocity of the moving mass (think of the air resistance), F d = Rv. If a frictional force F d = Rv, proportional and opposed to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the strength of the friction coefficient R, the system can either oscillate with a frequency slightly smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator); or decay exponentially to the equilibrium position, without oscillations (overdamped oscillator). Mass on a vertical spring When a mass m is attached to a hanging spring, the spring is subjected to a force F g = mg and will be stretched to a new equilibrium position y 0 where F g = F s mg = ky 0 m = k g y 0. (5.2) If the mass is displaced from y 0 and released, it will begin to oscillate about y 0 according to y = A 0 cos(ω 0 t+φ), ω 0 = k/m = 2πf 0 = 2π T 0 (5.3)

3 36 EXPERIMENT 5. HARMONIC MOTION where A 0 and φ are the initial amplitude and phase angle of the oscillation, T 0 is the period in seconds, and ω 0 is the angular speed in radians/second. If a damping force F d is present, the oscillation decays exponentially at a rate determined by the damping coefficient γ: y = A 0 cos(ω d t+φ)e ( γt), γ = R 2m, ω d = ω0 2 γ2 (5.4) Note two interesting details; first, the damped frequency of oscillations, ω d, is made smaller than ω 0 by the subtraction of γ 2 under the square root. However, if R is not very large, this reduction is not all that noticeable, even though the decrease in the amplitude due to the e γt term may be readily observed. Second, in the case of the air resistance, F d = Rv is an approximate relation, valid only for slow speeds. As the speed increases, the turbulent air flow offers resistance that is proportional to the square of the velocity, F d = Cv 2. Here C is the so-called drag coefficient. Unfortunately, this expression cannot be solved analytically and would have to use numerical methods (computer) to evaluate. Note also that the hanging spring is stretched by its own weight and may exhibit twisting as well as lateral oscillations when stretching. One other simplifying assumption: this experiment assumes an ideal massless spring connected to a point mass m. Even with all these approximations, Equation 5.4 lends itself very nicely to an experimental investigation. Procedure and analysis The experimental setup consists of a vertical stand from which hangs a spring. A platform attached to the bottom of the spring accepts various mass loads. The spring-mass system is free to move in the vertical direction. When the platform is static, it s height above the table can be measured with a ruler. When the system is in motion, Physicalab can be used to record the time-dependent change in distance from a rangefinder to the bottom of the platform. The rangefinder measures distance by emitting an ultrasonic pulse and timing the delay for the echo to return. Part 1: Static properties of springs In the following exercises you will determine the force constants of two different springs, distinguished as #1 and #2. You can then verify that their combined force constant satisfies the equation derived in the review section of the experiment. Note: Use positive k values in all the following steps. Equation 5.2 represents a straight line with slope k/g = m/ y so that k = m/ y g. By varying m, measuring y 0 and fitting the resulting data to the equation of a straight line, the force constant k for the spring can be determined. Note that the absolute distance y 0 does not matter, however the change in distance y with mass m is critical. Suspend spring #1 from the holder and load it with the 50 g platform. Use a ruler to carefully measure the distance y from the top of the table to the bottom of the platform to a precision of 1 mm. Record your result in Table 5.1. Determine the mass required to bring the platform near the top of the table and record this new distance, then select several intermediate masses and fill in Table 5.1.? Why do you want to determine a mass that brings the platform near to the table? Enter the data pairs (y,m) in the Physicalab data window. Select scatter plot and click Draw to generate a graph of your data.

4 37 mass (kg) y (m) Table 5.1: Data for determining the spring force constant k 1 of spring #1? Didyouchoosetoincludethemassoftheplatform(50g) aspartofthestretchingmassmintable5.1? How would the graph and the resulting value for the spring constant k be affected by your choice? Select fit to: y= and enter A+B*x in the fitting equation box. Click Draw to perform a linear fit on your data, then evaluate k 1 and the associated error σ(k 1 ). Click Send to: to yourself a copy of the graph for later inclusion in your lab report. B =... ±... A =... ±... k 1 =... =... =... k 1 =... =... =... k 1 =... ±... Repeat the above procedure using spring #2, recording your results in Table 5.2, then determine the spring constant k 2 for the spring. mass (kg) y (m) Table 5.2: Data for determining the spring force constant k 2 of spring #2 B =... ±... A =... ±... k 2 =... =... =... k 2 =... =... =... k 2 =... ±... Connect together springs #1 and #2, and repeat the procedure, entering your data in Table 5.3, to determine the experimental effective spring constant k e of the two springs: B =... ±... A =... ±...

5 38 EXPERIMENT 5. HARMONIC MOTION mass (kg) y (m) Table 5.3: Data for determining the effective spring force constant k e of springs #1 and #2 k e =... =... =... k e =... =... =... k e =... ±... Calculate the theoretical effective spring constant k e and k e for the two springs #1 and #2: k 1 =... ±... k 2 =... ±... k e = k 1 k 2 =... =... k 1 +k 2 ( k1 ) 2 ( ) 2 k2 + =... =... k e = k 2 e k 2 1 k 2 2 k e =... ±... Part 2: Damped harmonic oscillator You will now explore the behaviour in time of an oscillating mass with the aid of a computer-controlled rangefinder. This device sends out an ultrasonic pulse that reflects from an object in the path of the conical beam and returns to the rangefinder as an echo. The rangefinder measures the elapsed time to determine the distance to the object, using the speed of sound at sea level as reference. By accumulating a series of coordinate points (ω,m) and fitting your data to Equation 5.4, you can use several methods to determine the spring constant for the oscillating mass system. Begin by selecting one of the two springs to use: With each of the masses used previously with this spring, raise the platform a small distance, then release it to start the mass swinging vertically. Wait until the spring/mass system no longer exhibits any erratic oscillations. Login to the Physicalab software. Check the Dig1 box and choose to collect 200 points at 0.05 s/point. Click Get data to acquire a data set. Select scatter plot, then Click Draw. Your points should display a smooth slowly decaying sine wave, without peaks, stray points, or flat spots. If any of these are noted, adjust the position of the rangefinder and acquire a new data set. Select fit to: y= and enter A*cos(B*x+C)*exp(-D*x)+E in the fitting equation box. Click Draw. If you get an error message the initial guesses for the fitting parameters may be too distant

6 39 from the required values for the fitting program to properly converge. Look at your graph and enter some reasonable initial values for the amplitude A of the wave and the average (equilibrium) distance E of the wave from the detector. C corresponds to the initial phase angle of the sine wave at x=0. The angular speed B (in radians/s) is given by B= 2π/T. Estimate the time x= T between two adjacent minima of the sine wave then estimate and enter an initial guess for B. The damping coefficient D determines the exponential decay rate of the wave amplitude to the equilibrium distance E. When D = 1/x, the envelope will have decreased from A 0 to A 0 e 1 = A 0 /e = A 0 /2.718 A 0 /3. Make an initial guess for D by estimating the time t=x required for the envelope to decrease by 2/3. Check that the fitted waveform overlaps well your data points, then label the axes and include as part of the title the value of mass m used. yourself a copy of all graphs as part of your lab data set. Record the results of the fit in Table 5.4, then complete the table as before. mass (kg) y 0 (m) ω 0 (rad/s) γ (s 1 ) Table 5.4: Experimental results for damped harmonic oscillator The spring constant k can be determined as before from the slope of a line fitted through the (y 0,m) data in Table 5.4. Generate and save the graph, then record the result below: k(y 0 ) =... ±... The spring constant k can also be determined fron the period of oscillation of the mass. Rearranging the terms in Equation 5.3 yields m = k/ω0 2. Enter the coordinate pairs from Table 5.4 in the form (ω 0,m) and view the scatter plot of your data. Select fit to: y= and enter A+B/x**2 in the fitting equation box. This is more convenient than squaring all the ω 0 values and fitting to A+B/x. Click Draw to perform a quadratic fit on your data. Print the graph, then enter below the value for the spring constant: k(ω 0 ) =... ±... The damped harmonic oscillator equation 5.4 predicts that the frequency of oscillation depends on the damping coefficient γ so that your experiment actually yields values for ω d rather than ω 0. Calculate ω 0 from ω d using an average value for γ.? Which ω d value should be used? Why? ω 0 =... =... =...? Is the difference between ω d and ω 0 significant? Explain.

7 40 EXPERIMENT 5. HARMONIC MOTION Calibration of rangefinder It was assumed that the rangefinder is properly calibrated and measuring the correct distance. It is always a good practice to check the calibration of instrument scales against a known good reference, in this case the ruler scale. Calibration errors are systematic errors and can be corrected. Once the amount of mis-calibration is determined, the mis-calibrated data can be adjusted to represent correct values. Check the calibration of your rangefinder against the ruler. Compare the distance results for two points from the (y 0,m) data in Part 1 (ruler) and Part 2 (rangefinder). The mass difference between the points must be the same for both parts. If the rangefinder is calibrated, the distance differences should also be the same (within measurement error).? How do you choose the two calibration points that give the least relative error? Part 1: y 1 =... y 2 =... y 2 y 1 =... ±... Part 2: y 1 =... y 2 =... y 2 y 1 =... ±...? Why are two calibration points sufficient? IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Lab report Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero. Notes:...

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