Simple Harmonic Motion. Introduction to


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1 Introduction to
2 Contents Part I: Objective Part II: Introduction Interpreting Graphs Part III: Apparatus and Setup Apparatus Force Probe Motion Sensor Part IV: Determining the Period Expectation Collecting Data Analyzing the Motion Part V: Determining the Mass Using LoggerPro Creating Your Graph Analyzing Your Data Part VI: Determining the Spring Constant Creating Your Graph Analyzing Your Data Part VI: Summary
3 Part I: Objective The goal of this experiment is to determine the mass of an aluminium cylinder and the spring constant of the spring. You will be designing your own experiment based on the information given in the following slides.
4 Part II: Introduction In general, any motion that repeats itself at regular intervals is called periodic or harmonic motion. Examples of periodic motion can be found almost anywhere; boats bobbing on the ocean, grandfather clocks, and vibrating violin strings to name just a few. (SHM) satisfies the following properties:! Motion is periodic about an equilibrium position! The restoring force is proportional to and oppositely directed to a displacement from the equilibrium position.
5 Part II The displacement with respect to the equilibrium position x of a system undergoing SHM (as a function of time, t) can be described by x(t) = Acos(ωt + φ) where! A is the maximum displacement or amplitude of the motion,! ω is the angular frequency of the motion, and! φ is the phase constant or phase angle.
6 Part II The angular, or circular, frequency ω is defined in terms of the frequency f: ω = 2π f The frequency is defined to be the number of oscillations that the system completes in one second. The period T is the time taken for one complete oscillation, and can be expressed mathematically as T = 1/f
7 Part II If we consider a mass (m) on a spring as our oscillating system: Hooke s Law states that the there will be a restoring force acting on the mass when it is displaced from its equilibrium position. This restoring force is written as F = k x, where k is the spring constant. The negative sign in the equation indicates that force and displacement are opposite in direction. The motion of any object can be described by Newton s second law. ΣF = ma. k Angular frequency is also given by ω =. m
8 Part III: Apparatus and Setup Apparatus You have been provided with Motion sensor Force probe 200 g mass Aluminium cylinder Metre stick Spring Support rods Stopwatch Clamp
9 Part III: Apparatus and Setup Motion Sensor The motion sensor is a device which measures the distance to the closest object. Connect the motion sensor to DIG/SONIC 1 of the LabPro.
10 Part III: Apparatus and Setup Force Probe The force probe is a device which measures the force acting on it. To obtain accurate results, the force probe must first be calibrated and zeroed: Attach the force probe to a support rod. Set the force probe is to "5 N" or 10 N rather than to "50 N. Plug the force probe into CH1 of the LabPro. Make sure the LabPro is plugged into a power outlet and is connected to the computer.
11 Part III Force Probe To use the force probe and see the measured results, we use a graphing software package: LoggerPro. Click the icon below to launch Logger Pro.
12 Part III Force Probe You will have been provided with a mass of 200 g to use for calibration. Click Experiment then Calibrate then LabPro: 1 CH1: Dual Range Force. Continued
13 Part III Force Probe With nothing attached to the force probe click the Calibrate Now button. The Reading 1 value is 0 N. Click Keep. Now hang the 200 g mass from the force probe and enter the force in the Reading 2 cell. Click Keep. Click OK. Remember that a 1 kg mass weighs 9.81 newtons. Remove the calibration weight, attach the spring to the force probe, and attach the large aluminium weight to the hook at the bottom of the spring.
14 Part III: Apparatus and Setup Motion Sensor Place the motion sensor on the floor directly beneath the aluminium cylinder. Position the motion sensor carefully  the narrow beam of ultrasound it emits can easily miss the hanging mass altogether. Remember, the motion sensor must always be between 15 cm and 100 cm below the mass for it to measure its motion reliably.
15 Part III: Apparatus and Setup Zeroing the Probes Reduce the motion of the aluminium mass as much as possible, and then select Zero from the Experiment pulldown menu. Select both Dual Range Force and Motion Detector and click OK.
16 Lab Report Lab Report 1: Write the objective of your experiment. Lab Report 2: Write the relevant theory of this experiment. Lab Report 3: List your apparatus and sketch your setup.
17 Part IV: Determining the Period Expectation Without collecting any data, Lab Report 4: Sketch the expected form for the graphs of position vs time and force vs time. Explain your reasoning. HINT: Consider the equations for position and force as given in Part II of these instructions.
18 Part IV: Determining the Period Collecting Data Use Logger Pro to plot graphs of the oscillating system: force vs time position vs time and acceleration vs time. Lab Report 5: Do your graphs match the expected form? If they do not match, discuss why. Have an instructor check your graphs and initial your lab report. Turn off the connecting lines on your graph by double clicking in the white space of your graph then deselecting Connect Points.
19 Part IV: Determining the Period Analyzing the motion Logger Pro will display the coordinates of the plots if you click the button. The coordinates will be displayed in a popup box. Use your graph to determine the period of motion. Lab Report 6: Record the period. Include an estimate of the uncertainty. Lab Report 7: Describe how you determined the period using your graph.
20 Part IV: Determining the Period Analyzing the motion Next, use a stopwatch to determine the period of the motion. Hint: Try to be as accurate as possible! Should you measure one oscillation or multiple oscillations? Lab Report 8: Record the period. Include an estimate of the uncertainty. The uncertainty of the stopwatch is reaction time. If you do not know reaction time, you can find it in two ways: (i) do the measurement twice and find the difference or (ii) go online to do a reaction time test. Lab Report 9: Describe how you determined the period using a stopwatch. Lab Report 10: Compare the two values of period: Do they agree? Comment on any differences.
21 Part V: Determining the Mass Using LoggerPro You may change the quantity plotted on the horizontal axis by clicking on the axis label as shown below, and selecting the new quantity from the popup menu that appears. For example, if your plot shows position vs time and you want to plot position vs acceleration, you click on the word time on the x axis and choose acceleration instead. Working with your middle graph: Choose a suitable set of axes that will allow you to determine the mass of the oscillator. Having trouble deciding what to plot? Click here to open the notes from your graphing workshop.
22 Part V: Determining the Mass Analyzing Your Graph To obtain a linear fit to your data, click Analyze then Linear Fit. To find the uncertainties in the slope and intercept, double click on the box that appears and check Show Uncertainty. Use your results to obtain the mass of the cylinder. Lab Report 11: Record the mass of the cylinder. Include uncertainty. Lab Report 12: Describe your method to determine the mass of the cylinder. You may wish to include any equations and discuss the use of a graph. Weigh the cylinder on a triple beam balance. Lab Report 13: Record the mass of the cylinder. Include uncertainty. Lab Report 14: Compare the mass found using the two methods and comment on the agreement.
23 Part V: Determining the Mass Finding the Spring Constant Lab Report 15: Use the mass obtained from the balance and the period you found earlier to determine the spring constant of the spring and its uncertainty.
24 Part VI:Determining the Spring Constant Creating your Graph Working with your lowest graph: Choose suitable axes which will allow you to determine the spring constant k of the spring.
25 Part VI:Determining the Spring Constant Analyzing Your Graph To obtain a linear fit to your data, click Analyze then Linear Fit. To find the uncertainties in the slope and intercept, double click on the box that appears and check Show Uncertainty. For the best printed graph: " Click File then Page Setup and choose the landscape orientation
26 Part VI:Determining the Spring Constant Analyzing Your Graph Click File then Print. To select the only necessary page: Click Pages and choose Single. Format: Include titles and axes labels. Turn off connecting lines. Click Print to print your graph.
27 Part VI:Determining the Spring Constant Analyzing Your Graph Use your results to obtain the spring constant of the spring. Lab Report 16: Lab Report 17: Describe your method to determine the spring constant of the spring. Report the value of the spring constant and it s uncertainty. Does this value agree with the value found previously? Comment.
28 Part VII: Summary Lab Report 18: Outline briefly the steps of your experiment. Lab Report 19: List your experimental results and comment on how they agreed with the expected results. Lab Report 20: List at least three sources of experimental uncertainty and classify them as random or systematic.
29 Wrap it up! Check that you have completed your Lab Report. Your report should include copies of the graphs used to determine mass and spring constant.
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