MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.01 Purpose of the Experiment: Experiment 03: Work and Energy In this experiment you allow a art to roll down an inlined ramp and run into a spring that is attahed to a fore sensor. You will measure the position of the art and the fore exerted on it by the spring while they are in ontat. It is a real world experiment, whih means that there are non-onservative fores: frition as the art rolls up and down the trak, and dissipation (internal frition?) in the spring. The goals of the experiment are: To investigate experimentally the work kineti energy theorem, how potential energy in a gravity field onverts to kineti energy whih is then onverted into the potential energy of a ompressed spring. To observe and quantify the effet of non-onservative fores and estimate the work done by these fores at various stages of the art s motion up and down the ramp. Setting Up the Experiment: Refer to the photo to the right and the figure at the top of the next page. A fore sensor should be mounted at the end of the trak that has an adjustable support srew whih should be srewed in enough that the end of the trak an lie flat on the table. Clip the motion sensor to the other end of the trak and raise it by plaing a short piee of 4 under the motion sensor where it lips onto the trak, as you an see in the photo to the right. This should raise the end of the trak about 4. m above the table; as the trak is 1 m long, you an alulate the slope θ = 1.97 and sinθ = 0.0344. The motion sensor works best if it is aimed slightly above the enter of the art rather than pointing diretly at it. (That redues the effet of sound waves that boune off the 1
trak before hitting the art.) The slide swith on top of the motion sensor should be set to the narrow beam position. Usually two springs are available to srew into the fore sensor; if so, use the one that is wound from thinner wire. Plae a art on the trak with the end having the Velro TM pathes faing the motion sensor. Put two 50 gm weights in the art, whih will bring its total mass to 750 gm. (The extra mass redues vibrations and gives less noisy measurements.) Plae the art about 30 m up the trak from the fore sensor and release it. It will roll down the trak, boune most of the way bak up, and repeat that several times. You will notie the trak slides when the art runs into the spring; this is an example of onservation of momentum. To prevent the trak from sliding, plae your thumb on the end of the trak resting on the table and press it firmly against the table. If you don t do this, when the art runs into the spring some of its kineti energy will be dissipated by frition of the trak on the table whih will introdue an unknown error in your analysis. Connet the motion sensor (yellow plug into jak 1, blak plug into jak ) and the fore sensor to the SW750 interfae. The fore sensor should be plugged into hannel A of the SW750. Be sure to tare the fore sensor before eah measurement. The LabVIEW Program WorkEnergy: The LabVIEW program you will use in this experiment is alled WorkEnergy. Like our other LabVIEW programs, it is ontrolled by a pull-down menu above the left side of the graph. There is also a pull-down menu to ontrol plotting of data. The program has two tabs. The Table&Fits tab displays a table of the data plotted on the graph and allows you to ontrol the fits the program an do. It also displays the numerial results of the fits.
The Sample Rate should be set to 100 Hz (this is the rate the position of the art is measured at; the fore will be measured 10 times more often). Set the Run Time to 1 s. In the experiment you will let the art roll into the spring starting from rest about 30 m up the trak from the point where it first touhes the spring. Try this to see how things behave. When you are ready to measure, hold the art in position, and hoose Measure from the pull-down menu. The RUN button will hange to bright green. Be sure to hold the trak so it will not slide and lik the RUN button (or type the Es key) and release the art at about the same time. After the 1 s have elapsed, you should see a graph of raw data something like this one. The top urve is the eho delay of the ultrasoni pulse from the art to the motion sensor and the bottom urve is the voltage output of the fore sensor. These are the raw data, and you may save them to a file in the 8.01 ourse loker for later analysis. You an see from the peaks in the lower urve the times when the art bounes off the spring and you an see from the upper urve that the art bounes bak up to a lower height eah time. You should analyze these raw data in several different ways. The program omputes the position, veloity, and aeleration of the art using the Savitzky-Golay method disussed in the notes for Experiment 1. You may plot any of these three quantities as a funtion of time by hoosing what you want from the Plot Control pull-down menu and then liking the Replot button; you may also plot the fore alone as a funtion of time. 3
You will need to use the ursors and frequently expand the X (time) sale on the graph to arry out the analysis of your results. If you need a reminder of how to do this, the LabView graph ontrols are disussed in an appendix. Eah graph has two ursors, whih you will use to selet whih data points will be fit by the funtions that are available and also to determine the numerial x and y positions of points on the graph. Position the ursors by dragging them when the ursor (left) button is seleted on the graph ontrol palette. Part One: Analyzing Your Measurement: First, make a plot of position vs. time; it should look something like the one below. Carry out your analysis around the seond boune of the art off the spring. The art started a height h 1 above the spring and then bouned bak up to a lower height h. Obviously some energy was lost. Use the ursors and the ursor position readouts 4
above the graph to find h 1 and h ; first measure the positions of the art when it is losest to the motion sensor at the turning points on either side of the seond boune. You may be tempted to determine h 1 and h from the lowest point of the art during the boune. However, that will introdue a signifiant error as that is the point where the spring is maximally ompressed. (The spring ompresses about 15 mm.) You should find the distanes between the high turning points and the point where the art first ontats the spring. To do that, make a plot of the raw data (delay (ms) Fore Sensor (V) vs. time and expand the region in the viinity of the seond boune to fill the plotting area. Use the times when the fore pulse begins and ends to position the ursors and then go bak to the graph of position (m) vs. time (s) and read off the position of the art at the times the art first touhes the spring and then leaves the spring. (In the example, I deided the art first touhes the spring when the position is 0.815 m.) Enter h 1, h into the table below. Calulate the differene in gravitational potential energy between the two high points, and enter it into the table as well. The rest of the analysis you arry out will be to determine as well as you an how this energy was lost. First, assume that there was a onstant frition fore F F ating on the art while it was moving. If you assume that the energy lost was entirely to dissipative work done against F F you an alulate the magnitude of F F ; do this alulation and enter the result into the table in the fourth olumn. h [m] [m] 1 h g 1 Δ U = m g( h h )sin θ [J] F F [N] 5
Next, make a plot of veloity vs. time. Drag both ursors lose to the seond boune of the art from the spring and expand the region from just before the seond boune to just after it so that the region inluding the seond boune fills the entire time axis of the graph (see below). Drag one ursor to the most positive value of the veloity ( v 1 ) and the other to the most negative value ( v ). Calulate the kineti energies orresponding to these two veloities and enter them into the table below. v [m s -1 ] v [m s -1 ] 1 K1 = (1/ ) m v1 [ J ] K = (1/ ) m v [ J ] K K [J] 1 As a final test: alulate the maximum gravitational potential energy, subtrat the nononservative work done against F F and ompare to the kineti energy at the bottom of the roll, both before and after the boune. Enter the results in the table below. ( mg sin θ F F ) h1 [J] K [ J ] 1 ( mg sin θ + F F ) h [J] K [ J ] 6
Part Two: Interation with the Spring When it first hits the spring the art has kineti energy, whih is transformed into potential energy as the spring is ompressed. One the spring is ompressed to its maximum value (and the art is stopped) the fore of the spring will aelerate the art and give it a veloity in the opposite diretion. If the ollision with the spring is elasti, the art will leave with the same kineti energy it arrived with, but moving in the opposite diretion. You investigated this above and summarized your results in the seond table on the previous page. The next step is to see what an be learned from the measurements with the fore sensor. There is some vibration of the spring, espeially after the art has turned around and left the spring. You an see it if you make a plot of fore vs. time. On a fore vs. time plot drag both ursors onto the fore peak for the seond boune of the art and expand the peak to fill the graph. Set one ursor to the start of the fore peak and the other to the end of it. Selet the Table&Fits tab, and hoose Integral from the Fit Funtion? pull-down menu. Then swith bak to the Graph tab. Choose Fit Data from the main pull-down menu. That will alulate the integral Fdt under the peak, whih is the impulse given to the art during the ollision with the spring. You will see a graph like this one: 7
The graph will show the impulse area in green and the numerial value of the integral will be shown on the Table&Fits tab. This integral should be entered in the table below and ompared to the hange in momentum of the art during its ollision with the spring. Impulse [N s] m v ( -1 1 v )[kg m s ] The Shape of F(t): Now onsider a model that desribes the motion of the art while it is in ontat with the spring. Suppose the art is moving in the + x diretion (diagram at the top of page ) and ontats the spring at x = 0 when t = 0. This is shown in more detail in the figure below. 8
To simplify the math, the fore of gravity is ignored while the art and spring are in ontat. The art has mass m and the spring has fore onstant k. Then we an write F = ma for the art (while it is in ontat with the spring) as follows. d x ma = m = kx (1.1) dt You may not have reahed this point yet in a math lass, but this equation is solved by x() t = C osω t+ C sinω t (1.) 1 0 0 The two onstants C 1 and C arise beause you have to integrate twie to get x() t, giving two onstants of integration. You an hek that this is the solution to the differential equation by differentiating twie to get and dx vx () t = = ω0c1osω0t+ ω0csinω0t (1.3) dt dvx ma x() t = m = mω0 C1osω0t mω0 Csin ω0t= mω0 xt () = kxt () (1.4) dt The two onstants C 1 and C are determined by the initial onditions when t = 0, namely xt= ( 0) = 0 and v ( 0) x t = = v1. These are satisfied by C 1 = 0 and C = v 1 / ω 0. Thus x() t = ( v / ω )sinω t (1.5) 1 0 0 The art will remain in ontat with the spring until x() t beomes zero when ω t = π ; at 0 that time v ( x t = π / ω ) 0 = v = v1, the art is rolling bak up the trak and the spring no longer exerts a fore on it. However, what we have measured, and would like the model to explain, is the fore measured by the fore sensor Ft () while the spring and art are in ontat. By Newton s nd law, this is md x/ dt. Thus our model gives 9
F () t = m vω sin ω t,whereω = k/ m (1.6) spring 1 0 0 0 for the fore exerted by the spring while in ontat with the art (0 t π / ω0). You may reognize this as one half yle of a harmoni osillator. The Sine Pulse fit option will fit the spring fore data between the ursors to the funtion Ft () = Asin( π ( t A)/ A) (1.7) 0 1 You an see that the fore amplitude is A0 = mvω 1 0 and the period is A 1 = T = π / ω 0. Thus k = 4 π m / A1. The offset time A is just the start time of the pulse, and the program hooses it initially to be where you set the left ursor. When I adjusted the ursors to the start and end of the fore spike for the seond boune I got a fairly good fit by the Sine Pulse funtion. (My graph is below.) Use your fit results to fill in the table below; Δ X = A / k 0 is the maximum amount the spring is ompressed. A [N] [s] 0 1 A -1 k [N m ] Δ X = A / k [m] 1 0 k X [J] + 1 Δ ( K K )[J] 1 10
Putting it All Together: Fill out your report for this experiment. That will involve filling in opies of the first three tables above and addressing some questions about the meaning of your results. Appendix: the LabView Graph Controls When you lik the enter (zoom) button on the LabView graph ontrol palette a small panel with six hoies will open. Here is what they let you do. Upper Left: After you lik this button, drag a retangle on the visible portion of the graph. When you release the drag, the region inside the retangle will expand to fill the plot. Upper Center: After you lik this button, drag a region in the x diretion on the visible portion of the graph. When you release the drag, the x region you dragged over will expand to fill the plot. The y axis plot will be unhanged. Upper Right: This is like the upper enter hoie, exept the y region is expanded. Lower Left: Clik this button to reset the plotting region to aomodate all the data that were input to the graph (i.e., return to initial state). Lower Center: Cliking this button makes a small rosshair. When you lik it on a point on the graph the plotting range will inrease (your data plotted will shrink) in both diretions while you hold the pointer down. The resaled graph will be entered about the point you liked on. Lower Right: Just like the lower enter hoie exept the plotting range will derease (your data plotted will expand). After using the zoom ontrol, it is good pratie to ativate the ursor (left) button on the ontrol palette. That will prevent surprises the next time you try to use the ursors. The hand (right) button allows you to drag and translate the position of the point you liked on in the plotting region. There is no sale hange. 11