PHY 133 Lab 6 - Conservation of Momentum

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1 Stony Brook Physics Laboratory Manuals PHY 133 Lab 6 - Conservation of Momentum The purpose of this lab is to demonstrate conservation of linear momentum in one-dimensional collisions of objects, and to compare the properties of elastic and inelastic collisions. Equipment air track small glider big glider computer interface box 2 photogates Introduction Conservation laws are very powerful tools in understanding physical phenomena. They allow us to predict the outcome of an event, given information about the input physical quantities. In the case of momentum conservation, which suggests that without any external forces acting on a system, the net momentum vector of that system remains constant, knowing the initial momenta of two colliding objects allows us to predict their final momenta after the collision. As one might expect, this basic principle underlies many important areas of research today, including a range of topics varying from improvement of safety features in automotive vehicles to investigating the properties of high-energy single particle collisions. To develop a fundamental understanding of the principle of linear momentum conservation, this experiment makes use of two gliders colliding along a frictionless air track. If the track is perfectly level, gravity does not affect their collisions, and with the air track's cushion of air beneath the gliders, friction does not affect them either. Hence, the two gliders form an isolated system along one dimension, since no external forces affect their motion. Despite the conservation of linear momentum applying universally to all isolated systems, however, there are two distinct types of collisions that may occur between objects in such a system. In the first, known as an elastic collision, the two objects interact and (sometimes) rebound off one another, conserving not only momentum but also energy. No energy is lost (or converted) into heat or sound or deformations of the objects; instead, all of the input kinetic energy equals all of the output kinetic energy. In the second type of collision, known as an inelastic collision, the two objects interact and stick together, still conserving the linear momentum of the isolated system, but no longer conserving the energy of the system. Instead, some input kinetic energy is converted into other forms like heat, sound, friction, etc., and so, the output kinetic energy of the objects is less than the input kinetic energy. This distinction will become clear when you see the physical outcome of each type of collision! 1/6

2 In order to verify the conservation of linear momentum, you will need to measure the quantities that constitute the initial and final momentum of the system, and compare their total values. As momentum depends on mass and velocity, you will measure the mass of each glider, as well as the initial and final velocities of each glider using the photogate system. However, since there are two gliders interacting in this lab, there must be two photogates present in order to measure their velocities, in case they rebound from one another after a collision. As in previous experiments, the velocity of each glider will be determined by dividing some characteristic distance in this case, the width of a metal tab mounted on top of each glider by the time interval during which that distance blocks and passes through a photogate. Before conducting any trials, you need some preliminary information. First, measure the width w of the metal tab atop the small glider, and obtain the mass m of the small glider using the digital scale in the lab room. Record these values in your notebook. Then, repeat these two measurements for the big glider, recording the width w and mass M. (Note: The width of the metal tab atop the small and big gliders should be identical, so approximate their widths as the same value, which should be approximately 5 cm.) If the gliders are labeled with numbers, you may find their mass values on the list of glider masses posted to the door at the front of the room. Assume that each width has an uncertainty of mm, and that each mass has an uncertainty of g. w σ w = 2 = = 1 Experimental Methods During this experiment, you will conduct three trials, each with a distinct type of collision between the two gliders. In the first trial, you will examine an elastic collision of the small glider colliding into a stationary big glider. In the second trial, you will examine an elastic collision of the large glider colliding into a stationary small glider. And, in the third trial, you will examine an inelastic collision of the big glider colliding into the small glider. To prepare for the measurements, make sure each photogate is connected to the two input ports on the interface box. Turn on the computer, and double-click the Desktop icon labeled Exp5_t1_t2, which is the LoggerPro file for this lab. A Sensor Confirmation window should appear, and you should check that both sensors are listed as Photogate, then click Connect for each of them. A window with a spreadsheet on the left (having columns labeled Time, State 1, State 2 ) should appear. To determine which photogate coincides with State 1 and which coincides with State 2, you should click the green Collect button at the top of the LoggerPro window and pass your finger through each photogate. Whichever State column fills with values corresponds to the photogate you are blocking. Do not confuse the two photogates, so place them, or label them with a numbered sheet of paper aside each, in a manner that will remind you of their State column numbers! Then, enter your value of the metal tab widths w in the same way as entering the d values of previous experiments: under the Data tab, click User Parameters, and enter your width w value (approximately 0.05 m) into each row labeled PhotogateDistance1 and PhotogateDistance2, adjusting the Places and Increment values if necessary. Then, click OK. You are now ready to collect some data! σ M σ m 2/6

3 Collision 1: Elastic (small glider into big glider) For the first collision, you will slide the small glider into a stationary big glider. Once your lab instructor/ta turns on the air track, place a glider on the track to see if it is level. If the glider starts to slide, adjust the screw beneath one end of the air track to level it so that the glider no longer slides on its own. Then, position the two photogates along the track, each one about 1/4 of the length of the track away from each end. The photogates should be tightened to their bases, and positioned so that their beams are perpendicular to the track. Adjust each photogate height so that, when a glider passes through, the metal tab atop the glider blocks the photogate beam. This setup should remain for all three collisions. Again, remember which photogate corresponds to State 1, and which corresponds to State 2. Now, position the big glider in between the two photogates, and carefully hold it steady with one finger if necessary. Place the small glider at one end of the air track, so that the spring bars (non-velcro ends) on the two gliders are facing one another. If either of the spring bars looks weak or disconnected from the glider, get the help of your lab instructor/ta. When you're ready, click the green Collect button on the LoggerPro window to begin a trial. Once the Waiting for data text appears, gently launch the small glider toward the stationary big glider at the center of the track. Carefully observe the direction of travel of each glider through the photogates, both before and after the collision. The small glider should rebound off the big glider, and pass through the same photogate it entered, whereas the big glider should exit through the other photogate. Note which glider passes through which photogate, by watching their entry and exit times on the LoggerPro spreadsheet. However, once each glider has exited through a photogate, you should pick it up and remove it from the track before it collides with the end of the track. If LoggerPro is still running, click the red STOP button at the top of the window. If the data collection ended before each glider exited through a photogate, repeat the trial and either increase the initial speed of the small glider or, under the Experiment tab in LoggerPro, click Data Collection, and increase the duration of the trials as necessary, then repeat the trial. The spreadsheet on the left of the screen should fill with time values and entry/exit values of 1 / 0 in the State columns. Considering which photogate corresponds to each State column, and which glider passed through each photogate, determine the initial and final velocities (and their directions) of each glider for this collision, by dividing the distance w by the difference of each pair of entry/exit times (consecutive State values of 1 then 0 ) on the spreadsheet. Define the positive direction as the direction of the initial velocity of the small glider. For this trial in particular, you should find a large positive initial velocity of the small glider, a smaller negative final velocity of the small glider, and a small positive final velocity of the big glider (with an initial velocity of 0 m/s, since it was stationary). You should begin creating a data table of quantities for this collision, starting with: Glider t i (s) t f (s) v i (m/s) v f (m/s) small big 0 where t i is the time interval during which the glider entered through a photogate (blanked out for the big glider here, since it was w initially at rest), t f is the time interval during which the glider exited through a photogate, = is the initial velocity of the glider before the collision (set to 0 m/s for the big glider here, since it was initially at rest), and after the collision. is the final velocity of the glider Calculate these values and record this table into your lab notebook. As you will see below, there will be many more columns to add to this table during your data analysis! Collision 2: Elastic (big glider into small glider) For the second collision, you will replicate most of what you did for the previous collision, except with a few changes. First, you will instead launch the big glider from outside the photogates into a stationary small glider between the two photogates. Repeat the measurement procedure as above, however, because the big glider will transfer so much of its momentum to the small glider, you must be prepared to catch and remove the small glider once it exits through a photogate after the collision. Also, as the big glider will likely retain its direction of motion, you should not expect it to rebound, and instead exit through the same photogate as the small glider. Again, watch the LoggerPro spreadsheet carefully to see which glider enters and exits each of the photogates. vf v i = w t f t i 3/6

4 As with the first collision, you should create a second data table of quantities for this collision, starting with: Glider t i (s) t f (s) v i (m/s) v f (m/s) small 0 big where all quantities are the same as in the table for the first collision, except the blanked out cell is now the initial time glider, since it was initially at rest this time, and its initial velocity is 0 m/s. for the small Calculate these values and record this table into your lab notebook. As you will see below, there will be many more columns to add to this table during your data analysis! Collision 3: Inelastic (big glider into small glider) For the third collision, you will still replicate most of what you did for the previous collisions, except with a few additional changes. As in the second collision, you will place the big glider outside of the photogates and launch it into a stationary small glider in between the two photogates. However, you must turn the gliders around, so that their ends with velcro attached are facing one another. This velcro will act as an adhesive to physically connect the two gliders after the collision, and simulate a perfectly inelastic collision, in which the two collided objects remain attached after the collision. Lastly, for this trial, you should only record two time intervals: the one during which the big glider enters through a photogate before the collision, and the one during which the small glider (with the big glider connected behind it) exits through a photogate after the collision. You don't need the time interval for the trailing big glider to exit through a photogate because it is already connected to the small glider, and so, its final velocity immediately after the collision should be the same as that of the small glider when it exits through the photogate. Hence, when creating the data table for this collision, it should be modified to look like this: Glider t i (s) t f (s) v i (m/s) v f (m/s) big big+small where all the quantities are the same as in the previous tables, except the row for the small glider is replaced with the final combination of gliders, and there are two blanked out cells: the initial time t i for the big+small glider combination, since it does not exist until after the collision, and the final time t f of the big glider, since only the combination of big+small gliders exists after the collision. In this table, the final time t f for the big+small combination is only the time duration during which the small glider blocks the photogate while exiting (because the big glider is attached and follows behind it with the same velocity). Analysis In order to verify the conservation of linear momentum, and then compare the properties of the two types of collisions (elastic vs. inelastic), some more quantities must be calculated for each of the trials. v i For each of the tables above, you used v = w to find the velocity of each glider; however, to calculate their uncertainties, you must t apply the multiplication/division rule from the uncertainty guide. Also, you may assume that there is negligible uncertainty in the measured times, so that σ t in all cases. Hence, the relative uncertainty of v should equal the relative uncertainty of w for all trials. 0 Next, the linear momentum of each object before or after the collision may be calculated using the general formula p = mv for a mass m moving at velocity v. Since momentum is a vector, and only the vector of momentum should be conserved in collisions, it is important to distinguish the negative momentum values (objects moving in the opposite direction) from the positive momentum values. You can then use the multiplication/division rule to propagate the uncertainties in the mass and the velocity for each case to find. σ p Lastly, the kinetic energy of each object is given by KE = 1 2 mv 2, where m is the object's mass and v is its velocity. Since kinetic energy is not a vector, you don't need to keep track of negative signs as strictly as with the linear momentum calculations. However, for t i 4/6

5 each case, you will need to propagate the uncertainties of the mass and velocity in order to find for each case. You should be able to work out the uncertainty propagation and calculations to complete the table for each collision, filling in each with the following columns (completed below for Collision 1): σ KE Glider t i t f v i σ vi v f σ vf p i σ pi p f σ pf KE i σ KEi KE f σ KEf (s) (s) (m/s) (m/s) (m/s) (m/s) (kg*m/s) (kg*m/s) (kg*m/s) (kg*m/s) (J) (J) (J) (J) small big You should work together with your lab partner through these calculations during lab, in case either of you becomes confused about a particular calculation. The lab instructor/ta will be there for hints and guidance, but not once you leave the lab room! If you're still unsure about a particular calculation, you may try using the number-crunching tool below. Inputting your measured values, it will calculate the remaining values based on the proper formulas and uncertainty propagations. However, you should not trust this calculator, and should attempt all stages of the calculations on your own. Using these values, you should then verify whether linear momentum and/or kinetic energy is conserved in each of your 3 collisions. To do this, you should employ the overlap method, in which you determine whether two estimated quantities may be equal based on the amount of overlap of their estimate ranges with uncertainties. If there is overlap, the two quantities may be equal within experimental error; if the ranges do not overlap, there is no significant evidence that the two quantities are in fact equal. Looking at your data and results, is there a distinction between the two types of collisions? If your results are slightly off from what you expect, what might have influenced your data during the trials? Discuss this all in your lab report! Even if you don't find conservation of quantities like p or KE when you expect them to be conserved, you may still compare how close to being conserved they are in one collision compared to another. Are the results for your elastic collisions closer to overlap than the results of your inelastic collision? Which quantities do you expect to be conserved in each of the two types (elastic and inelastic) of collisions? By making comparisons within your own data set, you can account for any errors that may have affected your entire experiment! Fill in the widths and masses and their uncertainties in the appropriate boxes below: w s = +/- m w b = +/- m m s = +/- kg m b = +/- kg Copy the results from your 3 experiments into the tables below: Table 1. Elastic Collision sliding small glider into big glider: Glider t i [s] t' i [s] Small Big Table 2. Elastic collision sliding big glider into small glider Glider t i [s] t' i [s] Small Big Table 3. Inelastic collision sliding big glider into small glider with velcro t i [s] t' i [s] Pre-collision 5/6

6 Post-collision submit phy133/lab6conservationmomentumlong.txt Last modified: 2016/10/28 13:31 (external edit) 6/6

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