Simple Harmonic Motion II - Lab Experiment 2 | PHYSICS 289, Lab Reports of Physics

Download Simple Harmonic Motion II - Lab Experiment 2 | PHYSICS 289 and more Lab Reports Physics in PDF only on Docsity! PHYSICS 289 Experiment 2 Fall 2006 SIMPLE HARMONIC MOTION − II This is a continuation of the experiment done last week. Only a short report including the worksheets, graphs, and answers to the questions is required. C. Damped Oscillations The purely oscillatory behavior discussed in the previous lab occurs only in the ideal case of a system in which there is no energy loss. When there is energy loss, the amplitude of oscillation decreases in time, and the oscillations are said to be damped. In a simple model of a damped simple harmonic oscillator, the energy loss is due to a frictional force of magnitude f proportional to the velocity. This comes from Stoke’s equation in hydrodynamics, which holds for objects moving slowly through a viscous fluid. The energy is lost because the moving object drags on the surrounding fluid, thus transferring its kinetic energy to the fluid continuously. If the object moves at high speed, like a car driving on a highway, f is actually proportional to v2. For a solid block sliding on a smooth surface, f decreases slightly with increasing velocity. So there is no universal answer to the question of how friction depends on velocity. In this lab, we shall be dealing with a situation where Stoke’s drag is a good approximation, i.e., !"#f . From F = ma, the equation of motion is xbkxxm &&& !!= . The general solution of this equation is of the form )cos()( !"# += $ tAetx t , (1) where the phase φ is a constant determined at t=0, and the constant α = b/2m. There are (at least) four important observations we can make. First, the solution (1) predicts that the motion will be periodic with an exponentially decreasing amplitude. Second, when a time τ=1/α has passed, the amplitude will have shrunk from its initial value A, to Ae-ατ=Ae-1=0.37 A. This time τ for the amplitude to shrink to 37% of its previous value is called the decay time. Also note from the relation α = b/2m that a large amount of friction implies a large b, which gives rise to a rapid decay. Finally, note that the angular frequency ω is not equal to the natural frequency mk / 0 =! of the undamped motion. Instead .)2/( 22220 mb==! "## (2) Measurements: C1) You will use the PC and Data Studio program to record the position of the cart as a function of time for damped oscillations, and then fit the data to the expected functional form. Data Studio measures the position by timing the reflection of ultrasonic waves from the air cart. Log onto the PC using the instructions on the monitor. Double-click the Data Studio (DS) icon on the desktop. Once DS opens, select Open Activity. Navigate up two levels in the directory structure to C:\student temp\p289\, select damped_SHM.ds, and click on OPEN. In the main part of the DS window you should have three windows: Graph, Table, and Experiment Setup. The apparatus will have a motion sensor positioned 20 to 30 cm from the end of the air track. The damping we hope to observe is due to the air resistance of two "outriggers" mounted on the air track cart as an inverted V. The smooth metal surface on top of the outrigger serves as a reflector for the motion sensor. To record data exhibiting damped oscillations, set the cart in motion with an amplitude of 25 to 30 cm, and click the Start button on the DS top menu bar. The data should be plotted as it is collected. Click the Stop button when the amplitude has decreased to about 10 cm, typically 1 to 2 minutes. (This range seems to work best.) The data should be smooth, with no significant jumps. If the data are not smooth, it is likely the motion sensor is detecting something other than the reflector. Adjust the sensor height and angle, and the distance between the sensor and the track until the data are smooth. There are a few tools to allow you to zoom in to see if your data points are okay. If you position the cursor near any axis label in a Graph window, it turns into a spring icon. This allows you to stretch the axis. Near the border it turns into a hand which allows you to drag the graph. Once you have a nice set of data, go to the File menu, select “Save Activity As” and enter a unique file name in the \p289 directory. This saves your data in a format readable by Data Studio. How to Fit Your Data Using the Program “Origin” Now we’ll try to fit the data to equation (1) and determine the parameters A, α, ω, and φ using another program. In the Data Studio top menu bar, click “File” and “Export Data”. Select your data set and enter a filename to save your data in TXT format. Now minimize the Data Studio window then double click the “Origin” icon on the desktop. Select “File”, then “Import”, then “Single Ascii” and browse to find your data file in TXT format. You should see a table of your data. From Origin’s top menu bar, select “Analysis” then “Non-linear Curve Fit”. A new window will pop-up. In this window, select “Function”, then “New”. This will allow us to enter the function in equation (1). Set “Number of Parameters” equal to 5, then enter equation (1) in the “Definition” window by typing “P1*exp(-P2*x)*cos(P3*x+P4) +P5” (without the quotations of course). Note we have added an additional parameter P5 to account for the offset of our position measurement scale from 0. Next we need to tell Origin which column of our data is the x value (time), and which is the y value (position). Select “Action” then “Dataset”. In the top “Variables/Datasets/Fitting Range” window, click on “y”. Go down to the “Available Datasets” window and click on the data set corresponding to position (probably the one ending in “_b”). Click “Assign”. Now go back to the variables window, select “x”, then go to the “Available Datasets” window, select the appropriate data set (probably the one ending in “_a”), and click “Assign”. We’re almost ready to fit. Select “Action”, then “Fit”. Disappointed? Despite pressing “Fit”, the computer doesn’t immediately return a beautiful fit to the data along with a table of parameters, their uncertainties, and completed lab report. The problem is the “parameter space” is enormous. We need to help the computer fit the data by providing initial estimates of the fit parameters. Your skill in doing this will determine how long