MASSACHUSETTS INSTITUTE OF TECHNOLOGYPhysics Department

Physics 8.01 Fall Term 2010

Experiment 04: Angular Momentum

Purpose of the Experiment:In this experiment you investigate rotational collisions and the conservation of angular mo-mentum in rigid body rotational dynamics. It is the rotary counterpart of linear collisions.

The heart of the experiment is a high quality DC motor to spin a rotor up to several hundredradians per second. When power to the motor is shut off, it serves as a tachometer-generatorwhose output voltage is proportional to the angular velocity of the rotor; thus the angularvelocity of the rotor can be determined by measuring the output voltage. When you holddown the red pushbutton to switch on the apparatus, power is applied to the motor; whenyou release it, the rotor coasts and the output voltage the motor generates can be read bythe computer. This experiment will give you experience in

• measuring and calculating moments of inertia,

• measuring and calculating the effects of external torques, including torque due tofriction, and

• determining the changes in angular momentum during inelastic rotational collisions.

Experimental Materials:

• Logger Lite Software

• Vernier LabPro Interface

• Angular Momentum Apparatus with 3 brass/Velcro disks and power cord

• 2 Vernier Differential Voltage Probes

• 1 m (table height) length of string

• 50 gm weight/hanger

Setting Up the Experiment:Plug the rotary motion apparatus into its power supply; you should see the LED in theplastic pipe elbow come on.

Experiment 04 1 November 9/10, 2010

The experiment is longer than any we have done before and has two parts; extra time hasbeen allotted to do it. First, you will calibrate the equipment and measure the momentof inertia of the rotor. Second, you will use the calibrated apparatus to study rotationalcollisions.

At the start of the experiment, use two voltage probes to connect the output from the appa-ratus to channel 1 and 2 on the Logger Lite interface. Connect the generator output voltage(the jacks farthest from the power input connector) to channel 1 and the phototransistoroutput to channel.

The data acquisition and analysis for both experiments will be done with the Logger Liteprogram called AngularMomentum.gmbl.

It is there because two calibrations must be done in this experiment. Before you can carryout any rotational collision measurements, you must first calibrate the tachometer-generatorand then you must measure the moment of inertia of the rotor. These are the tasks of PartI; in Part II you will study rotational collisions.

The first task is to calibrate the tachometer generator output, i.e., what is ω for 1 V output ?

Part I: Calibration

Calibrating the Generator:Check to make sure there is a black sticker on the white plastic centerpiece of the rotor whereit will be illuminated by the LED; the reflected light will be detected by the phototransistor.(The voltage output will be higher when the light is reflected from the black tape and lowerwhen more light is reflected from the white plastic.)

Navigate to the Experiment menu and select “Data Collection.” Set the voltage sample rateto 500 Hz (the maximum is 588.2353 Hz) and set an appropriate run time (the default valueis 10 sec). Alternatively, collect data manually by pressing the space bar to start and stopcollection. For this calibration you want to measure both the phototransistor output voltageand the voltage from the generator (the two voltage windows that are open).

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Spin the motor up for several seconds, release the red button and allow the motor to coastfor about a second, then click the “Collect” button or press the space bar.

Data is automatically displayed in two voltage vs. time graphs, as in the figure below.The red curve (Plot 1) is the generator output voltage and the blue one (Plot 1) is thephototransistor voltage.

The next step is to determine the voltage output per revolution in order to convert mea-surements of voltage to units of angular velocity (i.e. rad/s). To do this, we will determinethe angular velocity within a given window of time, then divide by the average voltage overthat window to obtain a coefficient that has units of rad/s/V.

First, click on the top graph, navigate to the Options menu and select “Graph Options.”Select a window in time (recommended 0.25 sec). Do this by navigating to the Axes Optionstab, set scaling to manual, and set “Left” and “Right” boxes to any 0.25 sec interval of yourchoosing (e.g. 3.00-3.25 sec). Hit OK.

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If your graphs are not aligned with one another, click outside of the graphs, navigate tothe Page menu, and select “Group Graphs (x-axes).” Your window should now contain acountable number of peaks.

Clicking on the graphs one after the other, navigate to the Analyze menu and select “Ex-amine.” You should see a vertical locus in each window. With your mouse, hover over thefirst peak within your window; the time and voltage should be highlighted in the data box.Record these numbers. Choose the last peak in the window and do the same. Finally, countthe number of full cycles within your window (this should be the number of peaks minus one).

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To determine the conversion factor from Volts to rad/s, we divide the angular velocitywithin this small temporal window by the average voltage over this time frame. The rele-vant calculations include:

ω =2π × # peaks

t1 − t0, and Vavg =

V0 + V12

,

where t0, t1, V0, and V1 are the initial and final times and voltages of the peaks within thewindow. Thus, the conversion factor is simply ω/Vavg, which should be approximately 70rad/s/V.

Moment of Inertia CalibrationThe second calibration you must do is to measure the moment of inertia of the rotatingparts of the apparatus. To measure the rotor moment of inertia, a known torque is appliedby a falling weight and the angular acceleration is measured. Use a 50 gm brass weight witha 5 gm plastic holder. The apparatus should be set up near the edge of the table, as in thephotograph below.

Setting up the string and weight: Take the knot tied in one end of the string and placeit in the kerf cut into the brass washer on the rotor. The string should wrap around therotor several times, pass over the pulley and be fastened to the weight as shown in the photo.The length of string was chosen so that the string will completely unwind and pull out ofthe kerf just before the weight hits the floor.

Measuring the moment of inertia: Navigate to the Experiment menu and select “DataCollection.” Set the voltage sample rate to 500 Hz and the run time to 4.0 s. Wind up the

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string around the rotor so the weight is near the pulley and hold the rotor so it cannot un-wind. Note, wind the string counterclockwise to obtain a positive voltage output, otherwiseyou will see a mirror image of the results displayed in the following figures.

Click the “Collect” button (or press the space bar) and release the rotor a fraction of asecond later. After the 4 seconds have elapsed, you should see a graph like the one at thetop of the next page. As was previously explained, the program displays data in units ofvoltage. Since the voltage is directly proportional to the angular velocity, you are viewingthe same qualitative information in this graph that you would be viewing in a graph ofangular velocity vs. time. To convert to angular velocity, simply multiply your voltage datacolumn by the conversion coefficient that you previously calculated. Export the data as a.csv file, label it AMmassFall, and perform this conversion in Excel.

Theory of the analysis: While the weight is falling, the rotor angular velocity ω increases(α > 0) at a constant rate. After the string pulls out, ω decreases (α < 0) at a constantrate until the rotor stops. The decrease is caused by friction. As the accelerations (slopesof ω vs. t) are constant, the torques must be constant.

Let’s call α1 (a positive number) the angular acceleration while the weight is falling and α2

(a negative number) the angular acceleration while the rotor is coasting to a stop. Let τfbe the magnitude of the torque due to friction. During the coasting period, we must have

−IR α2 = τf

where IR is the moment of inertia of the rotor.

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The equations of motion while the weight is falling are a bit more complicated. Suppose Tis the tension on the string. The white plastic part of the rotor has a diameter of 1.00 in ora radius of r = 12.7 mm. The string will apply a torque to the rotor given by

τs = rT .

A free body diagram of the falling weight will give

T = mg −ma

where m is the mass of the weight and a is the magnitude of the vertical acceleration of theweight. Kinematics tells us that

a = rα1 .

The final equation we need is (the friction opposes τs)

IR α1 = τs − τf .

After eliminating τf , τs, T and a from these equations, the result is (remember, α2 < 0)

IR =mr(g − rα1)

α1 − α2

.

Doing the analysis: Open the file AMmassFall in Excel. The two columns representthe time and the voltage (again, multiply by your conversion coefficient to obtain properangular velocity units). Plot this data in a separate sheet. Use the XY(scatter) formatwithout connecting lines. Minimize the size of the points used to represent the data on thechart. You should see a graph similar to the Logger Lite version on the previous page. Picka range of times that best represents the linear rise of ω with time. Try to avoid the regionsof transitional behavior near the bottom and the top of the curve. Go back to the originalsheet and select the cells containing the times and corresponding angular velocities in thisregion. Plot this region of the data in a new sheet. Use Trendline to fit this region of thedata to a linear curve, thus determining the best fit value for α1. Enter this value on theResults Page. Find and enter α2 in a similar manner. For future reference, estimate theaverage value of ω on this plot and enter it on the Results Page.

The mass of the weight is 55 grams, The radius of the rotor about which the string iswrapped is 1.27 cm. Use the expressions above to find the moment of inertia of the rotorIR and the friction torque τf . Enter these values on the Results Page.

Measuring the Variation in Friction Torque: Actually, the friction torque acting onthe rotor varies slightly with ω. The variation is not enough to matter for the calibrationyou have just completed, but it will be significant when you analyze the rotary collisionsbetween washers, where ω can be much larger than 100 s−1, so it should be measured. Thismeasurement is straightforward. Choose a sample rate of 200 Hz and a run time of 10 s.Run the motor up close to its maximum speed, then start it coasting and click “Collect”(or the space bar) at about the same time. You should get a graph like the one on the next

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page.

You can see this plot is not linear, that is, α = −τf/IR depends slightly on ω since the slopeis higher at higher angular velocities (of course, the plots that are generated display voltagevs. time, but you should make the mental conversion to angular velocity). Export thesedata to a .csv file, label it AMSpinDown, and open it in Excel.

Open the file AMSpinDown. Make a plot of angular velocity verses time on a new sheet. Itshould look like the Logger Lite version above. We can now make a quantitative determi-nation of the angular deceleration as a function of angular velocity.

Use Trendline to make a second order (through X2) fit to the data. You should find thatthe resulting curve describes the data for ω verses t quite well. Take the derivative of theexpression given by the fitting routine to find a model for α as a function of time. However,we want to know α as a function of ω.

Here is a relatively simple way to find an analytic model for α as a function of ω. Go backto the data sheet. Time is in column A and the measured Omega is in column B. Skip acolumn for neatness then in column D make a heading OmegaM and in column E a headingAlphaM. The M stands for model. Using the model expressions for ω(t) and α(t) you gotfrom the fitting routine, and the values of t from column A, fill out the cells in columnsD and E. Now select the contents of columns D and E (starting with the heading row andselecting down to the end of the filled cells) and make a plot of α verses ω in a separatesheet. Note that there is no noise, since you are plotting analytic results. Use Trendline tofit this data to second order. The result will be an analytic expression for α(ω).

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In the next experiments we will be changing the moment of inertia. Thus we will need anexpression for τf . Multiply your expression for α by the moment of inertia you found earlierand enter this expression τf (ω) on the Results Page.

Now do a consistency check. Take the average value of ω you found when determining α2 aspart of the IR measurement. Plug that value into the expression you just found and enterthe result on the Results Page. How does the model τf compare with the measured one?Calculate the ratio of the model value of τf to the measured one. Report this value to theclass TA who will enter it on the board. We would like to see how this fraction varies fromapparatus to apparatus.

Part II: Angular CollisionsFollow the same procedures as in Part I (pages 1 and 2), with the following differences.

• Connect only the generator output (jacks farthest from the power input connector) tochannel 1 of the Vernier LabPro interface via voltage probes.

• Use the conversion coefficient that you determined in Part I of this experiment.

• Find the mass and the inner and outer radii of the washer you will be using. Calculatethe moment of inertia of the washer, IW , and enter it on the Results Page.

• To study the collisions, set the Sample Rate to 200 Hz and the Run Time to 4.00seconds.

Hints for Successful Collisions:You will get better results when you drop the washer onto the spinning rotor in the bestway. Here is what I found helps.

• Hold the washer level with the hole in the washer centered above the rotor axis andjust above the top of the rotor before you drop it. I suggest a separation of about 1/4inch or 1/2 centimeter.

• Release the washer so it stays level as it falls. This is easier to do if someone elseoperates the computer.

Velcro:If you look carefully at the Velcro, you will see it comes in two kinds. One has little hooksand the other is softer and looks more furry. The two kinds stick together when the hookslatch into the “fur.”

The top of the washer that is a permanent part of the rotor has fur. One of the washersyou can drop has hooks on one side and fur on the other. The other washer you can drophas hooks on one side and no Velcro on the other side. This makes three kinds of collisionspossible. If you drop hooks onto fur the washers stick amost instantly and you get a fastcollision. If you drop brass or fur onto fur, the washers slide a while before reaching thesame ω and a slow collision (lasting a few 100 ms) results.

First you should investigate a slow collision, as it is easier to understand the detailed be-havior of it than it is for a fast collision.

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A Slow Inelastic Collision:Try dropping a washer to produce both brass-on-fur and a fur-on-fur collisions. For eachcollision you should get a graph something like this one.

Here are the criteria to use in deciding if you have reasonably good data:

• There is a fairly clear break in slope at the beginning and the end of the collision,making it easier to determine the duration of the collision.

• The slope of ω(t) is fairly constant during the collision and there is little sign ofwobbling of the washer as it falls onto the rotor.

If a brass-on-fur collision seems reasonable, export the data to a .csv file labeled AMslow-CollideBF and open it in Excel.

Repeat for a fur-on-fur collision, and save what seems to be a reasonable collision to anExcel file labeled AMslowCollideFF. Then decide which of the slow collisions you want toanalyze and begin the analysis. The graph above is a brass-on-fur collision.

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Analysis of the Slow Collision:Open the Excel file for the collision you have chosen. The two columns give the time andangular velocity. Plot the data on a separate sheet. Determine the time t1 when the collisionappears to begin and the time t2 when the collision appears to end. Enter these times onthe Results Page.

Go back to the data sheet and select the data up to t1. Plot this range of data on a separatesheet. Use Trendline to do a linear fit to the data. From that fit, determine the value of theangular velocity ω1 at t1. Enter that value on the Results page.

Go back to the data sheet and select the data from t2 to the end. Plot this range of dataon a separate sheet. Use Trendline to do a linear fit to the data. From that fit, determinethe value of the angular velocity ω2 at t2. Enter that value on the Results page. Computethe average of ω1 and ω2 and add this to the Results Page.

The experssion you found in the PreLab for ω2 can be split into a part that would beexpected if there were no friction torque, and a correction due to friction.

ω2 =IR ω1 − τf∆t

IR + IW

=IR

IR + IWω1 −

τf∆t

IR + IW

= ωno friction + correction

Use your expression for τf (ω) together with the average ω during the collision to find thevalue of τf to use in this expression. Enter it on the Results Page. Calculate the twocontributions to the predicted ω2 and enter them on the Results Page. Also enter theresultant sum. If you were careful in carrying out the experiment you should find that thepredicted value differs from the measured value by no more than a few percent.

While admiring your results, keep in mind that the stretching out of the collision time hasnothing to do with the friction in the bearing. It is due to the time necessary for the washerand rotor to reach a common angular velocity. This process is carried out by forces internalto the system under consideration, thus it does not change the angular momentum of thesystem. The external torque, that due to the friction in the bearing, simply has a longertime to act on the system than it does in the fast collision; thus it will produce a largerpercentage correction to the result one would expect if the angular momentum of the systemwere conserved.

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A Fast Inelastic Collision:Keep the same settings for the Logger Lite program and this time drop a washer hook sidedown. You should see a graph like this one.

When you have a collision you like, export it to a .csv file named AMfastCollide, then openthe file in Excel. Proceed to analyze the data exactly the same way you did for the slowcollision. Here, though, not only are t1 and t2 much closer together, but there seem to bestrange things going on at the beginning and the end of the collision. These effects are dueto a slight “springyness” in the interaction between the hooks and fur of the velcro. We willnot pursue these effects here. Simply do your best to find the times and angular velocitieswhere the extrapolation of the slow changes in ω meets the extrapolation of a line throughthe rapid fall of ω. As before, enter the results of the analysis as you go along on the ResultsPage.

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Results Page for Experiment 4

MOMENT OF INERTIA

α1

α2

τf

IR

Average ω while finding α2

Model prediction for τf

VARIATION OF FRICTION TORQUE

Model expression for τf(ω)

FAST COLLISION

t1

t2

ω1

ω2

ω1+ω22

ω2 predictedω2 , no τf τf correction

τf

+ =

IW

SLOW COLLISION

t1

t2

ω1

ω2

ω1+ω22

ω2 predictedω2 , no τf τf correction

τf

+ =

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Physics Department

8.01 Physics I Fall Term 2009

Experiment 4: Material to be Handed In

CLASS

TABLE GROUP

NAME

NAME

NAME

Please fill out the attached Results Page representing your joint efforts in class.

Each person should also attach the required materials from PreLab done out of class.

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