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1st Year Physics Laboratory, Department of Physics 2009-10 PX110 ES2 Experimental Skills and Data Analysis Experiment: Simple Harmonic Motion Aims To use simple pendulums and mass-spring systems to explore simple harmonic motion, and determine the acceleration due to gravity and an unknown spring constant. To understand why we take multiple readings to reduce random errors as well as how to process them. To use the data collected to practice using Origin and learn how to plot and fit straight line graphs. Investigate the limitations of the small angle approximation. Safety Provided the experiment is performed in the manner described it should present no danger to the student or others. The risk assessment form is available on request. Preparatory Tasks 1) Read the entire script; familiarise yourself with the experiment and what you will be doing. Write a brief plan in your lab-book of what you will do. 2) The theory for a pendulum undergoing simple harmonic motion makes use of the ‘small angle approximation’. Use a suitable spreadsheet package such as Origin or Excel and create a table of angles and their sine to determine roughly when this approximation breaks down – i.e. the difference between the two is greater than 10%. (HINT: Make sure your angles are in radians!) 3) Make notes in your lab-book clearly identifying sources of random and systematic errors in the experiments you will be doing. In the two experiments to be conducted, identify the dominant error. In your lab-book discuss your experimental strategy to mitigate the effects of the small angle approximation and reduce errors to obtain accurate and precise results. [1 Mark] 1st Year Physics Laboratory, Department of Physics 2009-10 1. Background - What it’s about On being displaced from an equilibrium position, many objects will oscillate about that position. Examples include pendulums, masses attached to a spring, electrons in circuits containing an inductor (L) and a capacitor (C) as well as the motion of charged particles trapped in a potential. We call this Simple Harmonic Motion and it arises whenever the acceleration of the object is proportional to the displacement of that object. Mathematically we can describe simple harmonic motion by the following equation of motion: d2 x 2 2 x . (1) dt The general solution to this differential equation is a sinusoidal function: x t A sin t B cos t . (2) A sin t The motion, x t is periodic and is characterised by its amplitude, A and frequency, . The period of the oscillation is given by T 2 . The initial conditions define the second constant which is related to the phase of the wave, (either B or ). In most practical applications, the amplitude of the oscillations about the equilibrium position is seen to decrease with time and the oscillations are said to be damped. Resistance to the motion through effects such as air resistance cause energy to be removed from the system and equation (1) to be modified. You will explore damping in simple harmonic motion during the foundations course. In this practical session we will be primarily concerned with the period of oscillation which is much less affected by these damping terms. This experiment is about understanding simple harmonic motion. It introduces you to the measurement of oscillating systems and makes use of repeated readings. You will learn how to process this data and undertake simple error propagation within Origin. You will find the acceleration due to gravity using a simple pendulum and investigate the effect of the angle of oscillation on the period. By measuring the period of a mass-spring system you will determine the unknown spring constant. The role of random and systematic errors will be introduced and you will conduct statistical operations on your data as well as plot and fit your processed data using Origin 8.0. 1st Year Physics Laboratory, Department of Physics 2009-10 2. Preparation To find the equation of motion for a simple pendulum we need to derive the forces that are acting on the system (figure 1). At equilibrium the mass of the bob, m, on the end of the length of string, l, is balanced by the tension in the string, T. When the bob is displaced from equilibrium we need to resolve the weight (mg) into tangential and radial components: FRadial mg cos . (3) FTangential mg sin As the bob is forced to move in a circle, the Fig. 1: The forces on a pendulum linear acceleration of the bob is only consisting of a mass m suspended determined by the tangential restoring force. on a string of length l displaced through an angle . Thus we can apply Newton’s second law and obtain: d2s ma mg sin m 2 . (4) dt The minus sign comes about because the tangential force is always trying to move the bob back towards its equilibrium position. The arc equation relates the angle, to the displacement, s: s l , (5) giving for the tangential velocity and acceleration: ds d d 2 s d 2 vTangential l and aTangential l 2 . (6) dt dt dt 2 dt Thus we can re-write equation (4) in terms of the angle, : d 2 g sin 0 . (7) dt 2 l If we further make the assumption that the displacement angle is small, i.e. that sin we arrive at an equation that has the general form for simple harmonic oscillation: 1st Year Physics Laboratory, Department of Physics 2009-10 d2 g 2 . (8) dt l Now the angular acceleration, d 2 is proportional to the initial displacement dt 2 angle, (c.f. eqn. (1)) with 2 g ). Thus, within the small angle approximation we l expect a pendulum to oscillate with a period: 2 l T 2 . (9) g 3. Ready to Start Under the assumption that the initial angle of displacement is small, equation (9) allows one to use a pendulum to measure the acceleration due to gravity, g. To make a measurement of g one needs to measure the length and period of a pendulum. Before beginning the experiment make a note in your lab-book of the precision of the measuring devices and consider: How can any random and systematic errors be minimised? How will you measure T? How many repeat measurements will you make? How many swings will you use? When during a swing will you start/stop the stop-watch? Task 1: Measurement of the Period: Construct a pendulum using some string and a bob. Record in your lab-book the length of the pendulum and the mass of the bob along with their errors. Take several measurements of the period T. What is your best estimate of the period and is uncertainty? In eqn. (9) the mass of the bob does not appear. Show that the period of the pendulum is independent of mass by taking a series of measurements with bobs of different masses. For each mass, take several repeat measurements of the period and note your observations and conclusions in your lab-book. Use equation (9) to determine the acceleration due to gravity, g. What is your error in g? (HINTS – the error on g, g , can be calculated either using the tables or the functional approach. The error on the mean is the standard error, where is the standard N deviation of the measurements and N the number of measurements taken.) [1 Mark] 1st Year Physics Laboratory, Department of Physics 2009-10 4. Experiment In calculating the acceleration due to gravity in task 1 you assumed that there were no systematic errors and that equation (9) was valid. We will now verify that this is the case experimentally. Before conducting the experiment discuss with your partner making some clear notes in your lab-book the following: How many points do you need to get a precise measurement of g? What graph will you plot? How will you linearise your data? How will you determine g from your graph? Task 2: Calculating the acceleration due to gravity using a Pendulum: Verify equation (9) by varying the length of the pendulum and taking repeated measurements of the period for each pendulum length. Enter your data into Origin and use the statistical package to calculate the mean, standard deviation and standard error in your measurements of T. Linearise this processed data and calculate the appropriate error bars. Plot your processed and linearised data using Origin. Ensure that you know how to: Incorporate the error bars on your graphs How to rescale the axis and adjust the font size of the titles How to produce a line of best fit. Use the layout feature to resize the plot to print Use the linear fitting function and calculate the weighted least-squares best fit line to your data. Adjust your graph to enhance clarity and print it out and stick it in your lab-book. If the data is consistent with eqn. (9) what is your best estimate of g and its error? (HINT: Operations on columns in Origin can quickly be implemented using CTRL+Q) [2 Marks] With your partner consider the following making clear notes in your lab-book: How could you use your graph to identify systematic errors? Is your best-fit line a good fit?, Is so justify your conclusions Is your experimentally determined value of g accurate? If you can clearly identify any systematic errors, correct your data and re-plot your graph to obtain a more accurate determination of g and its error. You should have reached this point within 2½ hours 1st Year Physics Laboratory, Department of Physics 2009-10 5. Exploration Mass on a Spring Another example of a system that undergoes simple harmonic motion is the motion of a mass on a spring (shown left). Here the restoring force after the system is moved by a small displacement, y, from the equilibrium position is provided by Hooke’s Law: F ky , (10) where k is the spring constant. As in the pendulum example the equation of motion for the system can be found by balancing the forces: d2y m ky mg . (11) dt 2 As before the solution to this equation is of the form: y Asinwt B , (12) substituting (12) into equation (11) gives: m 2 Asint mg k Asint B . (13) d2 y This can be re-written to look like equation (1), namely 2 y if: dt 2 k B mg k and 2 . (14) m The value, B is simply the amount the spring stretches when the weight is initially added. This shifts the equilibrium position of the spring down. After an initial displacement from this new equilibrium position, the spring will oscillate under SHM with a period given by: 2 m T 2 . (15) k 1st Year Physics Laboratory, Department of Physics 2009-10 Task 3: The Mass-Spring System: Replace the pendulum with the mass-spring system. Note that the spring must be securely attached to the holder and avoid excessive displacements of the masses. For each of the five different masses record the period. Ensure that you repeat your measurements several times to reduce the effects of random error. Use ORIGIN to perform a similar analysis as you undertook in task 2. Plot a suitable graph and use origin to find the best-fit weighted least-squares fit to your data. Stick the graph in your lab-book along with the best-fit values and their uncertainties of the gradient and intercept. What can you conclude from the intercept and its error? What is your value of k and its error? [2 Marks] 6. Further Work – The pendulum explored The derivation of the equation of motion for a pendulum assumed the small angle approximation. If we no-longer restrict ourselves to small angles, it can be shown that the period of a pendulum depends on the maximum angle of the swing, max. If only the first order terms in the expansion are considered, the equation for the period becomes: l 1 2 T 2 1 sin2 max ... g 2 2 . (16) l 1 1 16 max ... 2 2 g This theory still assumes that the pendulum moves in a fixed circle with the string remaining the same length at all times. How will you measure the period accurately? Consider: How many swings you will measure How many repeats you will measure 1st Year Physics Laboratory, Department of Physics 2009-10 Task 4: The ‘Correct’ Pendulum – Testing equation (16): Use the protractor to find max and using the solid bar pendulum record the period as a function of the maximum angle of swing. Clearly note the length and its uncertainty of the inflexible ‘string’ in your lab-book. Does your data show a dependence of period with angle? If so, plot a graph of period against max . Fit your data in turn to a constant (eqn. (9)) and then to a second order polynomial (eqn. (16)). Print out your graph with both best fit lines shown. Which model results in a better fit? Justify your answer in your lab-book. Are you fitted parameters in agreement with eqn. (16)? 2 Marks are available for the quality of your lab-book and a further 2 Marks will be awarded for progress. 7. To conclude You will cover simple harmonic motion in the foundations course. More details can be found by reading chapter 13 in Young and Freeman 12th Edition. Simple Harmonic Motion permeates many areas of physics and you will come across it repeatedly during your physics career. The use of repeated measurements is common in experimental physics and you will need to be confident in their use. Review of Learning Outcomes After this experiment you should feel confident in: Understanding the physics behind simple pendulums and the small angle approximation Understanding how and when to take multiple readings. Using the statistical operations within Origin to process multiple readings. Performing simple error analysis Plotting and fitting a straight line graph with correct error bars in Origin The correct use of significant figures in reporting data Estimating Systematic and Random errors
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