Introduction to Simple Harmonic Motion

Transcription

1 Introduction to

2 Prelab Prelab 1: Write the objective of your experiment. Prelab 2: Write the relevant theory of this experiment. Prelab 3: List your apparatus and sketch your setup.! Have these ready to be checked by lab staff at the door on the day of your lab.

3 Introduction 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 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 Introduction 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 = A cos(ωt + φ) where A is the maximum displacement or amplitude of the motion, w is the angular frequency of the motion, and ω is the phase constant or phase angle.

6 Introduction The angular, or circular, frequency ω is defined in terms of the frequency f: ω = 2pf 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 Introduction 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 3 = 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. SF = ma. Angular frequency is also given by ω = 9 :.

8 Lab Report! Before collecting any data: Lab Report 1: 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 are given in the instructions.

9 Apparatus and Setup You have been provided with Motion sensor Force probe 200 g mass Aluminium cylinder Metre stick Spring Support rods Stopwatch Clamp

10 Apparatus and Setup 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.

11 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.

12 Force Probe To use the force probe and see the measured results, we use a graphing software package: LoggerPro. Click the icon to launch Logger Pro.

13 Force Probe Callibration 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

14 Force Probe Callibration 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.! Remember that a 1 kg mass weighs 9.81 newtons. Click OK. 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.

15 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.

16 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.

17 Data Collection Use Logger Pro to plot graphs of the oscillating system: l force vs time l position vs time and l acceleration vs time. Lab Report 2: Do your graphs match the expected form? If they do not match, discuss why. CP 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.

18 Data Analysis Logger Pro will display the coordinates of the plots if you click Analyze, and Examine, or click the button. The coordinates will be displayed in a pop-up box. Use your graph to determine the period of motion. Lab Report 3: Record the period. Include an estimate of the uncertainty. Lab Report 4: Describe how you determined the period using your graph.

19 Data Analysis 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 5: Record the period. Include an estimate of the uncertainty. The uncertainty in time using the stopwatch can be based on taking multiple measurements of the period and estimating based on the variance in your values. Lab Report 6: Describe how you determined the period using a stopwatch. Lab Report 7: Compare the two values of period: Do they agree? Comment on any differences.

20 Data Analysis 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? Consider the physics equation that relates the data you have.

21 Data Analysis 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 8: Record the mass of the cylinder. Include uncertainty. Lab Report 9: 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 10: Record the mass of the cylinder. Include uncertainty.

22 Data Analysis Lab Report 11: Compare the mass found using the two methods and comment on the agreement. Lab Report 12: Use the mass obtained from the balance and the period you found earlier to determine the spring constant of the spring and its uncertainty.

23 Data Analysis! Working with your bottom graph, choose suitable axes which will allow you to determine the spring constant k of the spring. 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

24 Data Analysis Click File then Print. To select the only necessary page: Click Pages and choose Single.! P Include titles and axes labels. Turn off connecting lines. Click Print to print your graph.

25 Data Analysis Use your results to obtain the spring constant of the spring. Lab Report 13:Describe your method to determine the spring constant of the spring. Report the value of the spring constant and it s uncertainty. Lab Report 14:Does this value agree with the value found previously? Comment.

26 Summary and Conclusions Lab Report 15: Outline briefly the steps of your experiment. Lab Report 16: List your experimental results and comment on how they agreed with the expected results. Lab Report 17: List at least three sources of experimental uncertainty and classify them as random or systematic.! Include printed copies of your graph and all data analysis with your report.