PHYSICS 3 LABORATORY MANUAL BASIC PHYSICS

Transcription

1 PHYSICS 3 LABORATORY MANUAL BASIC PHYSICS updated 9/0

2 TABLE OF CONTENTS ORIENTATION 3 LABORATORY MEASUREMENTS 5 AIR TRACKS 14 ANALYSIS OF EXPERIMENTAL UNCERTANTIES 18 SIGNIFICANT FIGURES 0 FREE FALL VELOCITIES FORCE AND ACCELERATION 8 LINEAR MOMENTUM, AND KINETIC ENERGY 34 ANGULAR MOMENTUM 4 OSCILLATORY MOTION 49

3 ORIENTATION Lab Notebook You will need a notebook suitable for this lab (for example, Engineering and Science Notebook, or Science Notebook which are available in the Bookstore). The notebook must be sewn or spiral bound; loose-leaf is not acceptable. Both you and your lab partner must have your own lab notebook, and must record, with dates, all procedures and data as the experiment is being carried out in the laboratory. Data should never be recorded anywhere except in your notebook. Never leave the lab without your own set of data don t depend on your partner for this! Be sure to identify precisely what is being measured and include units in all of your measurements. Include labeled sketches in your notebook that clarify the experimental procedure and define the data entries. You may be required to produce your lab notebook if questions arise with respect to your lab reports, so don t discard it until you receive your final lab grade. You should have your lab notebook ready for your first lab. General Lab Info You must attend your assigned lab section. If you cannot attend your assigned lab section because of illness or special circumstances, we will ask you to do the lab during the appropriate Makeup lab session. Descriptions and instructions for each experiment are contained in this booklet. You should read these descriptions and instructions before coming to the lab so you will be prepared to start work immediately, after some introductory discussion by the TA. It s important to be prepared so that you can effectively use the limited lab time. If you finish taking data before the lab session is over, we encourage you to stay and use the remaining lab time to begin your analysis and calculations. The lab will be fresh in your mind and the TA is there to help you. Lab Reports You will prepare and turn in for grading a lab report for each lab. Although the work in the lab is usually done with a lab partner, your lab report must be your own work. Your lab report need not be lengthy. We describe below the format for the lab report. The report for each experiment is due at the next meeting of the lab. Your lab report should be handed in to the TA. Remember to keep your data notebook until you receive your final lab grade. Your lab report must be based on the experimental data you have taken. 3

4 The lab grade will be based on the quality of your lab work and report. Late penalties will be applied if your lab report is not turned in on time and will be indicated separately from the lab grade. Late penalties will start with a deduction of 1 point (out of 10) for a report up to one week late, a deduction of points for a report between one and two weeks late, and so forth up to a maximum late penalty of 5 points. In case of illness or other unavoidable difficulty, please see Prof. Pelcovits about special arrangements. Lab Report Format Each of your lab reports should follow this format: 1. Objective: A brief statement of the purpose of the experiment.. Procedure: A brief description of the apparatus used and the steps you followed in carrying out the experiment. Any problems and their resolution should be noted briefly. 3. Data: A tabulation of the data as taken, copied from your original data notebook before any mathematical operations are carried out on it. Organize your data here as efficiently as you can. Be sure to label the quantities you measured and give the units in which they were measured. Be sure to define all symbols used. 4. Calculations: State any equations used in the calculations to achieve the objective of the experiment with your data. Again, be sure to define all symbols used. 5. Results: State the final results of the experiment as obtained from the data and calculations. The results should, of course, relate to the objective. 6. Discussion: Discuss your results, including error analysis. If there is an accepted experimental or theoretical result to which you can compare your result, do so. Discuss how significant your agreement or disagreement with this theoretical number is, in light of the experimental uncertainty of your own result. 4

5 LABORATORY MEASUREMENTS Measuring Length Most of the time, this is a straightforward problem. A straight ruler or meter stick is aligned with the length segment to be measured and only care in matching the segment's boundary points to ruled marks (estimating between marks) is necessary. Greater precision, if the distances are not more than a few centimeters, is afforded by a caliper (see Fig. 1), which can be set to match the distance in question. The distance is in effect memorized by the instrument, which can then be removed to an area where it can be read with ease. The caliper method has a very large application in length measurements. The vernier caliper in Fig. 1 can be used to measure both inside and outside diameters, and even depths, by using the different sets of jaws or the probe at the end. Verniers and Micrometers A vernier scale (see Fig. 1b) provides a better method of reading between the ruled lines of a main scale than simply estimating by eye. To read a vernier scale, first note how many divisions on the vernier are equal to an integral number of main scale divisions. In the ordinary metric vernier, ten vernier divisions are equal to nine main scale divisions. Hence the least count, or smallest value that can be read directly from a vernier scale of this type, is one-tenth of a division. The caliper divisions are millimeters, so that the least count would be one-tenth of a millimeter. The sketch in Fig. 1b shows the reading of a vernier. 5

6 Micrometer calipers (Fig. ) are used to measure diameters quite precisely. Turning a handle moves a rod forward, by a screw thread, until the object to be measured is clamped very gently. In the usual metric instrument, one turn of the handle advances the rod a halfmillimeter. A circular scale around the handle reads fiftieths of this half-millimeter, or thousandths of a centimeter. One more figure is estimated between scale marks. Exactly the same kind of screw, rod, and scale are often incorporated into other measuring instruments. Both vernier and micrometer calipers should always be tested for zero readings when fully closed (so of course should any other instrument). Measuring Mass Several types of balances for weighing objects exist. With any kind, a zero reading (no load and no weights) should be taken first. Care should be taken not to overload any balance nor to spill any corrosive materials on them. The trip balance, or pan balance is used for heavy objects, and for general weighing. The unknown goes in the left pan, the standard masses in the right pan. Beam balances are available in comparable size; there are also more sensitive ones for weighing small objects. In using a beam balance, the unknown is placed in the pan and a weight is arranged to slide along a calibrated beam. 6

7 Ohaus Triple Beam Balance The range of this general laboratory mass balance is increased over the simple single beam form by the use of several beams (see Fig. 3). One beam is for the largest mass increment -- the movable mass on this beam can be placed, in the balances used here, in five positions besides zero. Each corresponds to an increment of 100 gm. Another beam contains a weight that can be placed in any of ten non-zero positions corresponding to increments of 10 gm. In using these two beams, each movable weight must be placed definitely in the appropriate notch. The front beam contains a weight that can slide continuously along a marked scale to a maximum corresponding to 10 gm. The maximum mass that could be measured without additional features would be gm = 610 gm. The least count of the scale is 0.1 gm, and the weight can be estimated to fractions of this smallest division. Before measuring, a zero adjustment check is made with no mass on the pan, and all sliding beam masses at their left-most positions. A thumbscrew under the pan at the left is turned in either direction until the pointer indicates zero. Recheck the zero for each use of the balance. A measurement is made by placing the mass to be measured on the pan, then moving the largest scale-mass (or poise ) to the highest position that does not cause the pointer to change position. Then the second poise is similarly adjusted so the scale reads within ten grams less than the unknown. Finally the front poise is moved until the scale returns to the zero, or balanced, position. The unknown mass will have been determined to within the smallest subdivision of the front scale, a tenth of a gram. 7

8 Other Balances The chemical balance for weighing very small objects has two pans suspended below a light see-saw truss. Again the unknown goes on the left (except in the special technique of double-weighing ). The standard masses for such balances should be handled only with forceps, lest perspiration etch them away. The surrounding glass case is closed while judging a result, so that drafts cannot cause errors. Correct balance is determined by watching the swing of a pointer, not by waiting for it to come to rest (friction may hold it off center). There are also electrically operated balances which give readings of mass directly upon dials, after prescribed settings have been made. For most applications in our lab, the triple beam balance is adequate. Measurement of Elapsed Time: Electronic Time-Measuring Units Note: The following discussion is based on the Thornton Associates modular electronic system. An electronic timer has a manual mode of operating that is identical to that of a stopwatch. Its visual display (or readout) of elapsed time is digital -- the five-digit readout fixes the maximum time interval it can handle depending on the time rate at which it ticks. The Thornton DEC--10 timer used in the physics laboratory accumulates time in milliseconds 3 ( 1ms = 10 seconds) on the display, when using its internal crystal-controlled circuits. 8

9 The DEC--10 timer (also called a counter) has a provision for using an external oscillator instead of the internal circuit for special purposes. Parenthetically, the laboratory timers are not generally built as independent units; you will see that they do not, for example, have a power plug to tap the AC (alternating current) power strips, or even an on/off switch. Instead, they connect to a separate power unit, the APS--101 module, which transforms the AC power into the correct DC (direct current) power for matching modules. By matching we mean that, with more than one supplier's equipment in use in laboratories, one must be certain to use units such as the timing unit and photobridges (soon to be described) from the same manufacturer -- in this case Thornton Associates. Different systems and subsystems may be similar in function but are not compatible electrically; thus, attempts to mix components from different systems will damage the units. Manual Timing When the timer is turned on via the connected power unit ON-OFF switch, a random digital number may (or may not) appear on the display. A RESET button on the timer clears the display to zero. A sliding switch, marked COUNT INPUT, selects whether the timer uses its internal timing circuits (the usual case) in the I KHZ position or an external oscillator in the APS OUT position. The external oscillator connection will not be described here. One other switch labeled HOLD MODE has to be set to either PULSE or GATE MODE. If the GATE mode is used, depression of the START/STOP button will start the timer and release of this button will stop it. If the PULSE mode is used, the first depression of the button will start the timer and the second depression will stop it. Used in this stopwatch mode, a manually operated timer has another feature common to stopwatches -- if it is operated after one timing operation without clearing the timer by RESET, the display simply accumulates more counts -- which may or may not be what you want to do. Manual timing clearly has many applications in the physics laboratory. One of its obvious limitations is the human reaction time (characteristically, hundreds of milliseconds) involved in recognizing the start and stop instants in a process of interest. To avoid this limitation, we consider a sensing device that can make the physical motion itself switch the electronic timer on and off automatically. Photo Transistors The electrical properties of certain materials change drastically when they are exposed to light, compared to their behavior in its absence. Their ability to pass an electrical current is one such property. A phototransistor (used in the application of interest here) constructed of such material can be thought of as a light-actuated switch. It is automatically turned on when 9

10 bright light strikes its sensitive surface and turned off when the bright light is removed. Ordinary room light does not affect it. Phototransistors have, among many other uses, an important application in the laboratory -- because they switch states very rapidly they can be used in measuring precisely even very short time intervals. When the light beam that switches the transistor on and off is mounted properly, across the path of a moving object, the object can turn this kind of switch on and off without disturbing its own motion. The following sections show how mechanical motion can be timed by causing the object itself to control the precision timer by its own motion through light beams. To use the photocell (the mounted, powered phototransistor) effectively, it is made an integral part of a photobridge. The latter consists of both the photocell and the bright light source mounted in a frame that holds the two elements in careful alignment, and separated by a few centimeters. The shape of the bridge allows mechanical objects to pass between the source and photocell, interrupting the beam as they pass through. The standard photobridge unit is a common way to monitor many kinds of motion which involve a well-defined path in space, such as falling objects, or objects constrained to move along a track, or in a circle. When such constraints don't exist, the path an object will take cannot be known well enough beforehand to position a light beam across it, and other methods could be more effective (photographing the object, for example). When a photobridge is used as a monitor, the photocell connects directly to both the power supply and the timer to provide its signals to the HOLD circuits, thus performing the same functions as are performed to operate the timer in the manual mode. Timing with a Single Photobridge Note: Always turn off the model APS-101 power supply when making connections between units. For a single photocell to operate the timer, its white connector (banana plug) must be inserted in the white socket (banana jack) marked --6V on the APS--101 power supply, and its red connector (or plug) must be inserted in the white socket (or jack) marked ELECTRONIC HOLD on the DEC--10 timer. When the photocell is illuminated by the light source, this connection will cause the timer circuitry to be held in the off state so that counting will not take place. A double (red) socket marked 4 VAC at the left of the timer display supplies power to the double plug from the light source of the photobridge; connected here, the lamp should light whenever the APS--101 power unit is turned ON. The photobridge now controls the countercircuits, suppressing timer operation until the light is masked from striking the photocell. The particular way that masking turns the timer on depends on the MODE switch setting: In PULSE MODE, masking the light will start the timer, which will continue running even when the mask is removed. When the mask is restored, the second pulse will turn the timer off. 10

11 In GATE MODE, the timer will start when the light is masked, and will turn back off when the light is restored. Another way to describe the MODE settings is to say that PULSE alternately activates and deactivates the timer on successive transitions from light to dark, at the photocell, while GATE activates the timer only during the dark periods. These operations can be checked simply by turning on the APS--101power supply and using RESET on the DEC--10 timer to clear the display. Use your finger, or a ruler, or a piece of paper to mask the light, and verify the behavior of the counter/timer for each of the two modes. Applications of single photobridge timing GATE MODE timing is useful for measuring the speed of a body passing through the bridge. The length of the body, (or of a mask attached to the body,) divided by the recorded time, gives the speed. If the speed is not constant, the average speed through the bridge is what is measured. When the photobridge is mounted appropriately it can monitor the speed of objects dropping vertically, swinging on a pendulum, moving along a track, and so on. In some arrangements, for example a swinging pendulum, the object will repeat its passage through the bridge over and over, and some thought must be given to the fact that successive passages will each generate a timer reading that adds to the existing reading. PULSE MODE timing with a single photobridge is limited to cases where: (a) (b) (c) Once the body moves through the bridge to start the timer, some other action will generate a pulse to stop it. An example is an arrangement that lets you drop an object through the bridge to hit a pushbutton kind of switch. The bridge would turn the timer on, and the button switch would stop it, giving the travel time between the two devices. An instructive example is to have someone start the timer by tripping the photobridge out of your line of vision. If you watch the display with your finger on a similar pushbutton, and turn the timer off as soon as you see it counting, the timer reading is a good estimate of your reaction time. Some other action starts the timer, and the photobridge pulse turns it off. In analogy with one of the previous examples, you could imagine holding an object against a push button switch, from which it will drop through the photobridge. Here the button turns the timer on, and the bridge pulse turns it off, giving the travel time of an object that starts from rest and falls through a specified distance. Repetitive motion brings the object back through the photobridge, such as the pendulum swinging back and forth. Notice that simply switching the mode of timing from the GATE MODE in which this example was first cited to the PULSE MODE changes completely the physical meaning of the registered time. One GATE MODE reading gives the travel time of the body through the bridge; one PULSE MODE 11

12 reading is the time taken for the body to swing through the bridge to the end of its motion and to swing back through the bridge (which is a time known as the halfperiod of the oscillation). Timing With Two Photobridges and One Counter Possible applications of the Thornton counter timers are considerably expanded when two photobridges are connected to the same timing unit. To make this connection, (again, always make connections with power off) the white plug of one photo-cell is connected to the white --6V jack of the APS--101 power supply, and the red plug of the second photocell is connected to the white electronic hold jack of the DEC--10 counter. The remaining two photocell plugs, the red plug of the first and the white plug of the second, are connected together using a special double jack that accepts a single plug at each end. To check the connection, see that as you trace the path from the --6V jack to the hold jack, you encounter in order the white plug followed by the red plug of the same photocell, then the white plug followed by the red plug of the other photocell. The two lamps (which have double connectors that are both a plug and a jack) are simpler---in either order, the double plug of one lamp enters the 6VAC double jack on the DEC--10 counter as before, and the double plug of the second lamp is connected piggyback into the double jack of the first lamp. Now the lamps of both photobridges should light when the APS--101 power switch is turned on. (REMEMBER, ALL CONNECTIONS ARE MADE WITH THE POWER OFF). Applications of Double Photobridge Counting GATE MODE timing with two photobridges on the same counter is equivalent to having two velocity meters, according to the usage (cited for a single photocell) when a mask of known length is combined with the time to give the velocity. The complication that needs to be handled is that the physical arrangement should allow enough time for the first reading to be noted before the body passes through the second bridge, etc. This is necessary because the time generated at each successive bridge will add to the original time reading, and one has to know how to separate the total time. Notice that it is not necessary to RESET manually between timer activations; that only adds to the separation time required. One area where the reading time requirement can usually be met easily is on an air track, where gliders move slowly compared, for example, to falling body experiments. Air track examples that the double configuration measures well in GATE MODE are recoil collisions and accelerations. The first example can even be done with one photobridge for certain kinds of collisions, but with two photobridges virtually all types of collisions can be monitored. The basic theme is to place the bridges on each side of the area where the bodies will collide. Then, by noting the first readings, which give the velocities before the collisions of two gliders, you can unravel the velocities after collision. The second example that the configuration handles, an accelerated glider, applies because constant acceleration can be deduced from the difference of two velocity measurements. Newton's second law of motion can be checked if the glider is connected via a pulley to hanging masses. If the track is inclined, a component of the gravitational force acts directly 1

13 to accelerate the gliders, and this acceleration is also constant at any particular angle. In general, if the acceleration is not constant, it is the average acceleration that is measured. PULSE MODE timing with two photobridges has the additional sensing element that the single bridge examples lacked. With two bridges, one to start the count in PULSE MODE and the other to stop it, velocities over much longer paths can be measured than those of GATE MODE. Two photobridge units placed side by side give their photocells a separation of 6.3 centimeters. When the two bridges are placed as far apart as possible, with the timer and power supply midway between them, their photocells are separated by approximately 150 centimeters. These two numbers give the minimum and maximum limits of the length over which a motion can be timed to give an average velocity with the two-bridge configuration. While two bridges is the maximum number that will drive a single timer, the number of timers connected to a single experiment obviously can, be increased as necessary. Each timer functions in one of the configurations discussed here. This concludes the introduction to electronic timing with the Thornton Associates modules, either in manual operation or with one or more photobridges. The configurations and techniques used as examples will be used frequently in the elementary laboratories. 13

14 AIR TRACKS The air track is a laboratory device for producing almost frictionless linear motion over a distance of about one and a half meters. The motion of gliders, special light aluminum forms that move on a cushion of air only thousandths of an inch thick is observed. The track over which the glider moves is a perforated tube of triangular cross section; air is pumped into the tube at one end and exits upwards through many tiny holes along the two upper surfaces of the track. Ideally, the air does not impart motion to the gliders, but uniformly supports them just off the track surface; the gliders are therefore free to move on the track, to collide with each other and with spring bumpers at the ends of the track, to be pulled and pushed by string and pulley systems, and so on, with very little friction. Many different experiments are possible, including some in which the entire track is placed at an angle with respect to the horizontal to study forces in two dimensions; the exact experiments are not discussed here. 14

15 The air track is described in this manual because students must think about the equipment components before and while doing experiments that treat the track itself as ideal. Because of the extremely thin air layer that is the basis of the track performance, all these components have to be kept at a high performance level; otherwise friction effects can grow very rapidly within a matter of an hour or two and leave you with data that seems to prove very little about the physics of motion. The Air Supply The supporting air source is a blower that is no more than a canister type vacuum cleaner running backward. One such blower can normally supply air to one, two, or even more air tracks; flexible plastic tubing of four or five centimeter diameter carries the air from the blower to a filter at the air track. Your only concern with this portion of the system is that (1) the air intake of the blower (on its top) is not blocked by debris (paper scraps, books and the like) and that () the plastic fittings that connect the tubing to the blower are firmly in place. Listening for air leaks with the blower running is the surest way to test the delivery system. Very slight hissing is of no consequence -- there is ample excess pressure available in the systems as originally set up. But because the fittings are simply force fit into one another, check for tightness initially and avoid moving the flexible tubing around during the experiment. The most troublesome effects on your data come when the supporting air system changes during your measurements. Air Tube The effectiveness of the track depends in a noticeable way on the temperature of the air that lifts the glider. You will feel the temperature rise to an operating level over the first five or ten minutes of running a track that is initially at room temperature. The heat comes from the blower system and from the slight compression of the air, and spreads along the metal tube. After a few minutes of operation, you should have a stable, warmed track. When the blower is turned off, the system will, of course, slowly cool, but it would return to the operating level within a couple of minutes if turned back on. Most people find the blower systems noisy and distracting, and we heartily agree, but do not simply turn the system on and off during your 15

16 data taking. Run it steadily during actual glider motion observations, and turn it off when your data is complete. Keep warming and cooling times in mind if you want to rerun some parts of your experiment. Leveling the Track YOU WILL BE ADJUSTING ONLY THE ONE SINGLE SCREW AT ONE END OF THE AIR TRACK. The major and most delicate leveling of the air tracks will have been carried out by the Laboratory Technician (Kevin McCabe), leaving only minor adjustments to the single screw located at one end of the air track. This final leveling of the horizontal axis of the track is done through the use of a glider on the operating track. (See section on gliders below.) Briefly, the glider is given an impulse at one end of the track, and its velocity is then measured at a point near each end of the track. The same measurement is done starting the glider at the other end of the track. For each direction, compute the velocity change as a percent of the initial velocity. Evidence of a tilt would be indicated by the effect of gravity. The single screw is then adjusted upwards or downwards, depending on the difference in the two velocity change measurements. This procedure is repeated until no further improvement in the correspondence of the two velocity changes can be made. Sometimes this adjustment must be repeated between data runs. IN ANY CASE, DO NOT TOUCH ANY OF THE OTHER LEVELING SCREWS! IF YOU THINK THERE ARE PROBLEMS OF LEVELING WHICH ARE NOT FIXABLE BY SMALL ADJUSTMENTS TO THE SINGLE SCREW, ASK THE TECHNICIAN (KEVIN MCCABE) FOR HELP. Gliders Proper handling of the gliders (Fig. 8), which are made of the same aluminum as the track itself, is one of your main concerns. Burrs or scoring of the track result from bad handling of the gliders in placing them on and removing them from the air tube. The sharp corners of the gliders can score the track surface, and a scored or burred surface can in turn damage the glider surfaces. The following rules protect the quality of your air track experiments: 16

17 (1) Never place or remove gliders unless the air supply is running, and always gently and vertically. () Never move them on the track unless air supply is running. (3) Always push gliders along an operating track by light pressure at the sides of the glider (not from the top of the glider). (4) If the experiment permits, always send a glider down a track by causing it to recoil a spring bumper at the end of the track. There is a self-aligning character to the floating glider that recoil hardware can maintain better than your direct push. (5) Always add and subtract weights carefully. (6) If you should wish to make some adjustment to the glider on a running track (such as fixing a string to a pulley system) consider slipping a sheet of creased clean paper under the glider. It will isolate the glider temporarily, while the system as a whole will be stable. Of course, major changes should be done by removing the glider from the track. (7) Be especially careful about applying tape to the glider tape is sometimes too useful to avoid, but be sure that traces of its gummy adhesive do not migrate to the lifting surfaces. (8) Throughout, cleanliness may or may not get you closer to godliness, but it does wonders for your air track experiments. 17

18 ANALYSIS OF EXPERIMENTAL UNCERTAINTIES All experimental values are subject to uncertainties which arise from inherent limitations in the instruments used. It is not possible for us to construct a perfect instrument or a perfect observer even in the most carefully designed experiments random effects influence results, so that many repetitions of any measurement will result in a distribution of final results that peaks at the most probable value. We discuss here mainly such random uncertainties. The best way to obtain a good final number is to make a number of independent measurements, as many as feasible. The best number we can then give is the arithmetical average of all the measurements. For N measurements of a quantity L, this arithmetic average, bar L, is obtained from all the individual measurements L i as follows: 1 L = ( L + L + L L N = 1 N L i N i = 1 1 N ) (1) But no experimental measurement is complete without an estimate of the uncertainty, which is reflected in the spread among the individual measurements. This spread is measured quantitatively by the Standard Deviation of the mean, which is calculated (approximately) from the root mean square deviation RMS by the following equation: RMS = 1 N 1 N i= 1 ( L L i ) L = RMS / N. The best complete value is L ± L Sometimes it is not feasible to do multiple measurements of a quantity; in such a case the experimenter has to use his or her judgment about the uncertainty, based on the instruments used. For example, if you are reading a meter stick, you should judge whether the uncertainty in your reading is ± 0. 1 or ± 0. of the smallest division on the meter scale. (Never assume it is better than pm ± 0.1. ) In general, every experimental result must be given in the form X ± X. Frequently, measured values are used to calculate further results. These results will of course be affected by the uncertainties in the measured quantities that enter into the calculation. 18

19 Consider two measured quantities X ± X and Y ± Y. The following rules hold for calculating the uncertainty in X + Y, X Y, XY, and X / Y : For A = X + Y A= ( X ) + ( Y ) For S = X Y S = ( X ) + ( Y ) For M = XY M X = M X Y + Y For D = X/Y D X = D X Y + Y X, Y, A, S, M, D, are called absolute uncertainties. X Y A D M R,,,,,, are called relative uncertainties. X Y A D M R These rules can be extended to any number of factors entering the calculation -- there can be three or more, not just the two used in the examples above. In addition to the random uncertainties or errors discussed above, systematic errors can arise from non-random sources of error. An example of a systematic error is the mis-calibration of your measuring instrument. While random errors can be reduced by making additional measurements, systematic errors are independent of the number of measurements. If you see any evidence of such problems in your data, you should mention it in your lab report and try to suggest specific causes. The uncertainties or experimental errors discussed above do not include mistakes in reading or setting instruments. Also note that the experimental error is derived solely from the precision of the measurements you make and the limitations of the experiment itself. It is not the same as the discrepancies that might exist between your measured value and a known or accepted value. 19

20 SIGNIFICANT FIGURES We understand that all measured quantities have unavoidable uncertainties, because of limitations on the instruments used and our limitations in using them. We have learned to use the STANDARD DEVIATION IN THE MEAN to calculate this uncertainty when we have several independent measurements of the same quantity, and can calculate the mean e.g. g ± g where g equals the Standard Deviation in our determination of g, our average value of g, based on several separate measurements of g. Frequently, we are unable to make multiple measurements, but have to use our judgement to decide on the uncertainty, for instance, when we estimate between the closest marks on any scale, which we do rather than round the reading to the nearest mark. We use a definition of SIGNIFICANT FIGURES which tells us that we don t keep digits in our result beyond the first digit that is subject to our experimental uncertainty. For instance, if we have an experimentally determined number such as S = 16.5 ± 0.3 This (5) is the first digit which is affected by the uncertainty, so we have 3 significant figures. Even if a pocket calculator gave us more figures on its display, it would not make sense to use them i.e. WRONG [ = ± 0.3] S Too many digits, not all significant. Should not be quoted as written, but should be rounded off to PLEASE NOTE: THE POSITION OF THE DECIMAL POINT DOES NOT AFFECT THE NUMBER OF SIGNIFICANT FIGURES. Calculation using significant figures must now be described. Suppose we measure two different lengths, with different instruments, getting 4.5 ± 0. 1 cm ( significant figures) and ± cm (4 significant figures), with every number except the zero before the decimal point being significant (this zero, incidentally, is written just to aid the eye in noticing the decimal point). What is the sum of these two lengths? Simple arithmetic suggests 4.835, but the indicated sum 4.5xxx yyy 0

21 shows that this is not reliable. The three x s after 4.5 represent unknown values and the three y s in the sum are therefore unknown. The rule for addition or subtraction requires us to drop all but the first uncertain figure we must drop the y s and the answer correctly is 4.8 cm. Suppose instead we want to multiply these two lengths, to get an area. Simple arithmetic gives , but it looks suspicious to get six figures with only two in the factor 4.5. By writing 4.5xx and multiplying it by 0.335, you will quickly find the trouble: x s will come into most of the figures in Again the rule is simple: in multiplication or division, find the factor with the smallest number of significant figures (4.5, here), round off the other to the same number of figures (0.34), and carry out the calculation (to get, here, 1.530), which must also be rounded. There is another part of the rule: When the first significant figure is 1 or as in the answer here, keep one more significant figure than when the first figure is 3, 4,. Hence our answer is 1.53, not 1.5. It is understood that 3, the last figure, is an estimate, as is the 5 in 4.5. Taking a square or other root is a form of division, and so similar procedures are followed. Electronic calculators know nothing about significant figures; they will give you results with digits filling out their displays. The user must beware and round off appropriately. Following these rules may seem strange and bothersome at first, but there are good reasons for doing so. If the rules are not followed, the result is distorted and tells a scientific lie the answer to 4.5 x is not when the factors are imperfectly measured numbers. The advanced student has to learn more complicated procedures. For example, in a very long chain of calculations one or two extra, nonsignificant figures are carried to reduce possible cumulative error caused by a great many roundings-off. The important thing to remember is that the final figure should be rounded appropriately. 1

22 FREE FALL VELOCITIES References Laboratory Measurements (Physics 3); and Young and Freedman; University Physics (9th Ed, Extended Version), Chapter. This experiment studies uniform acceleration in one dimension by systematic measurements of a falling body's position and instantaneous velocity. The positions of timing detectors are varied to generate precise values of these quantities from average quantities. The Basis, Plan, and Procedure sections that follow describe the experiment; the function and nomenclature of the timing equipment will be found in the Guide to Laboratory Measurements. Basis of the Experiment It is shown in many texts that if an object is released from rest and allowed to fall, its instantaneous velocity at a distance S is given by (1) v = as where the accepted value of the acceleration due to gravity, a, is about 9.81 m / s. The instantaneous velocity cannot be measured directly, because the body must move over a finite distance S in an interval of time t in order for us to measure a velocity. What we measure is an average velocity v = S / t. There is a way, however, to relate a particular instantaneous velocity to a measured average velocity. The method is based on the fact that the instantaneous velocity v (t) is linearly related to the elapsed time if the acceleration is constant: () v ( t) = at Here the zero of time is defined as the instant of release [ ( 0) = 0] v. Suppose the body falls from an upper point (U) along its trajectory at time t U to a lower point (L) at time t L. The average velocity v ( t, t U L ) in this time interval between t and U t L is defined as the mean of the instantaneous velocities at the instants t U and t L, 1 (3) v ( tu, t L ) = [ v( tu ) + v( t L )].

23 Using Equation () on the right hand side of Eq. (3) ( U, L ) U L U L U L (4) v t t = [ v( t ) + v( t )] = [ at + at ] = a [ t + t ]. The mid-time instant t M in the fall from U to L is, by definition, 1 (5) t M = ( tu + t L ), Substituting this in the right hand side of Eq. (4), we have (6) v t, t ) = at. ( U L M But at M, as shown in Eq. (), is just the instantaneous velocity at time t M, so we have converted our measurement of average velocity over an interval to the instantaneous velocity at t M, the TIME MIDPOINT of that interval: (7) v t ) = v( t, t ). ( M U L We go through all this trouble because the average velocity is easy to measure. The average velocity is s( t = tu ) s( t = t L ) SUL (8) `v = s / t = = ( t t ) T (9) s ( t ) = v t + 1/ ( at ) = 1/ ( gt ) U U 0 U U U (10) s ( t ) = v t + 1/ ( at ) = 1/ ( gt ) L 0 L L L v 0 is 0 because the cylinder is dropped from rest with no initial velocity (free fall ). SUL As a check, let us verify that equation 7 is in fact true. Does v ( t M ) =? TUL (11) v t ) = g(1/ )( t + t ) ( M U L SUL 1/ ( gt L ) 1/ ( gtu ) (1) = = 1/ ( g( t L + tu )) (After a little algebra ) T ( t t ) UL L Equations 11 and 1 are thus equal and hence validate equation 7. U L UL 3

24 Notice that the time midpoint t M is not the space midpoint, as shown in Fig. 1. Because of the acceleration, the body travels farther in the second half of the time interval than in the first half. But if we develop a way of locating M, the space point corresponding to the time midpoint t M, we can take the instantaneous velocity at M to be the measured average velocity over the UL interval, and use Eq. (1) to calculate the acceleration a. Solving Eq. (1) for a, we get for the acceleration (13) a = v / S ZM In our case, S ZM (see figure) is the total distance traveled in the time interval between t = 0 and t = t M. It is not the distance S UL or S UM. So we can calculate g from equation 13 by substituting v = SUL /( tu t L ) and measuring S ZM. But remember this will only be true when the photobridges are set so that TUM = tu t M = 1/ TUL = 1/ ( tu t L ). Electronic timer1 will read T UM and electronic timer will read T UL 4

25 Plan of the Experiment We use photobridges across the path of the falling body to measure the time intervals we need. The apparatus consists of a rigid vertical rod adjacent to the body's trajectory, on which the photobridges, marked U, M, and L in Figure, are mounted. The body, latched magnetically at Z until released, defines an exact zero point in time, distance and velocity. Two electronic timers, marked T UM and T UL in the figure, are set to operate in pulse mode. Not intended to be a wiring diagram (these are present in the laboratory) the figure indicates the logic flow of signals from the photocells to the timers. The pulse from the U cell as the body first cuts its beam is passed to both the UM and UL timers, starting both counters. When the body first cuts the M beam, its photocell sends a second pulse to the T UM timer, which causes the timer to stop, giving the time of fall from U to M. The T UL timer continues until the beam is cut to photocell L, at which time its pulse stops the timer with the time of fall from U to L. All the bridges are movable on the rod. Suppose we start with the U bridge high on the rod and the L bridge mounted about a meter below it. Now let the M bridge be placed midway in space between the other two. When a drop is made, the UL timer will contain the total fall time through the bridges, and the UM timer will show the fall time from U to the space midpoint. The latter, because of acceleration, will be larger than one-half the UL reading. But now, leaving the other bridges locked in their positions, we move the center photobridge upward, searching for the time midpoint. At the next drop, we can verify the constancy of the T UL reading, and check to see if we have reached half that value on the TUM timer. This step is repeated until the position corresponding to the time midpoint is found. Once the time midpoint is found, we can apply Equation (8), using the time readings and distance measurements as described on pages 1 and, to get a precise value of the acceleration. The measured average velocity over the UL interval equals the instantaneous velocity of the body at the instant it cut the M photobeam at the TIME MIDPOINT. The distance S is that from the rest position Z (not merely from the U bridge) to photobeam M. Distance measurements are critical. Note that measurements at the rest position always refer to the lower edge of the body, because that is the edge that activates the photobeam 5

26 switches. The distance from the rest level to the upper photobeam can be made a one-time problem by choosing a good location for the upper bridge (one that allows easy access for placing the mass at the rest position) and locking it there for the entire experiment. All measurements to or between photobeams are best made by using the well-defined metal frame of the photobridge itself. The distance between photobeams, for example, is exactly the distance between corresponding edges on their photobridges. Where the beam itself must be located, as in the case of the upper beam relative to the rest position, use the offset of the beam from the edge of the photobridge that you are using. This can be obtained (again a one-time problem) by measuring the vertical \underline{width} of any bridge with a caliper; the offset is just one-half this width. Procedure and Data 1. KEEP A RECORD OF YOUR PROCEDURE THROUGHOUT THE EXPERIMENT. Align the apparatus so that the beams are cut reliably over the entire drop length. Small shifts of the mounting board on the floor, and small rotations of the bridges, may be needed. Be sure that there is a box at the base to catch the body.. Set the top bridge position high, but allow ready access to the launch position. Make several drops to check for good alignment, for repeatability at fixed bridge positions, and to decide on a good range of positions for the lowest bridge. Note that the highest position of the lowest bridge should not be such as to give small (two-digit or very low three digit) time readings, since any digital reading can inherently be in error by one in the lowest digit. 3. Measure carefully the constant distances discussed above and record them. In your notebook set up a Table in which to enter your data in a clear, understandable way. Always record the numbers as you measure them - leave calculations, even simple ones, for later. Include units for all numbers. 4. Choose a lowest setting of the lower photobridge, and hunt with the middle bridge for the half-time (TIME MIDPOINT) position. Once located and verified, record all distances and the timer readings. These will be used to calculate the acceleration a from equation (8) as described in the Basis section. Record all the UL times for a fixed L position: The variation in this number reflects the reproducibility of the measurements with this apparatus. 5. For at least four more (higher) positions of the lower photobridge, repeat and record the procedure and data as you did in the preceding steps. Calculations For each setting of the lowest bridge, calculate the acceleration by determining v ( U, L) and S from your measurements, and then using Eq. (8). Expect some variation among your values of a. 6

27 Results The best value obtained from a series of N measurements of a quantity is the mean value, simply the arithmetic average of the individual measurements. Using all N independent acceleration determinations a i, (where N is at least 5), calculate a best value for your experiment as the mean, or average of the individual values a i, a = 1 N N i= 1 a i, No experimental result is complete or meaningful without an estimate of the experimental uncertainty. A good measure of the uncertainty in the mean is the standard deviation of the mean, S.D., which is obtained from the mean square deviation (MS) of your measured values from their mean: and MS = 1 N 1 N i= 1 ( a i a), S. D. = MS N, where the a i are your individual determinations of a. A final best value with its uncertainty is then a ± S.D. Discussion and Conclusions Compare your measured value to the accepted value of the acceleration of gravity, and discuss the result, taking into account your experimental uncertainty and the reproducibility of measurements with the apparatus. Try to include a discussion of sources of experimental uncertainties. Note: In your report you are not expected to repeat the plan of the experiment as given in the handout, but to say briefly what you actually did - e.g. how many drops you made to find the time midpoint and what the time was for each drop, any problems you encountered in 7

28 FORCE AND ACCELERATIONATION Study of Newton s Second Law of Motion References: Physics 3 Guide to Laboratory Measurements; A study of Free Fall Velocities; Young and Freedman, University Physics, Ninth Ed., Extended Version, Chapter 4 An object which is free to move horizontally without friction is subjected to a series of known forces and its acceleration is measured for each force. The measured accelerations are to be plotted graphically as a function of the applied force. Basis of the Experiment A linear air track provides a nearly frictionless path for a glider resting on it (Fig. 1). The glider is pulled by a string which passes over a pulley at the end of the track, to a hanger to which known weights can be attached. These weights with the hanger provide the variable applied force. 8

29 Suppose we let: m h = mass hanging on the string (including weight hanger) m t = mass on track (glider plus any weights resting on it) M = m h + mt F = total applied force = m h g T = tension in string (acting horizontally toward pulley, on m t, and acting vertically, upward, on m h ) a = acceleration of m t (horizontal, toward pulley), and also a = acceleration of m (vertical, downward) h Note that m t and m h must move together that is with the same velocity and acceleration as long as the string connecting them remains the same length throughout their motion. Fig. shows the forces acting on m t and m h. The equations governing the motion are: m t a = T, (1) a m h = F T. () Adding these and substituting ( m + m ) a F, gives Ma = F, which can be written t h = a = F / M (3) Equation (3) is what we wish to investigate. It expresses Newton's Second Law of Motion, as applied to the composite system, consisting of glider + weights + hanger + connecting string, while Equations (1) and () are expressions of Newton's Second Law applied to m t and m h separately. 9

30 It is T which actually pulls the glider, and to find T from F we would need to make use of the Second Law as expressed in Eq. (). However, without using the second law we know that M = m h + m t is the mass of the composite system and F = mh g is the only force applied to it; T is an internal force within the two-body system, not a force applied to it. For the twobody system we know the left- and right-hand sides of Eq. (3) independently, once we measure the acceleration $a$ and the masses,and so can test the equation. Plan of the Experiment In the experiment, M remains constant while movable pieces of mass (loose weights) are transferred from glider to hanging support. Each such transfer decreases m t and adds to m h (and therefore increases F without changing M). For each value of F, a is constant during the motion. Since M remains constant, Eq. (3) becomes the expression of a linear relationship between F and a in this experiment. It predicts that if each acceleration a is plotted as a function of the corresponding force F = mh g, the observed values will lie on a straight line passing through the origin, with a slope equal to 1/M. The principle of the acceleration measurement is identical to that used in the free fall experiment (Reference ). Here it is the glider's acceleration that is measured, so to base it on photobridge measurements, the bridges are mounted on a rod suspended horizontally over, and parallel to, the track (Fig. 3). The acceleration is constant (one of our basic assumptions) for a given value of masses because it derives from the gravitational force acting on those masses; it is not of course, equal to the constant gravitational acceleration of free fall at any time. 30

31 The glider rest position ( x = 0, t = 0, v = 0) is at Z. To make the correspondence with Reference clearer, call the photobridge nearestto Z the U (for upstream) photobridge. Call the middle photobridge the C photobridge, (it was the M photobridge in Experiment 1) and the L photobridge is the last photobridge, nearest the pulley. If the glider leaves a fixed point on the track, moving to the right, it will interrupt each of the beams in order. The upstream (U) interruption starts the two timers. The interruptions at C and L turn off the UC and UL timers respectively. The C photobridge position is adjusted until the UC time reading is one-half the UL time reading. The rest of the acceleration measurement involves measuring distances. These measurements are made readily using the metric scale that is permanently mounted on the track. Initial position and photobridge positions require passing a perpendicular line in space down to the track scale. The position of the C photobeam (at the time midpoint between U and L) relative to the leading edge at Z of the resting glider is the distance S in this experiment, at which we find the instantaneous velocity. The value of that instantatenous velocity v(t c ), at the time midpoint between U and L, is equal to the average velocity of the glider in moving from the U to the L photobeam; it is therefore simply the distance UL between these two beams divided by the time registered on the UL timer. With these two numbers, S and v ( t c ), we can then deduce the acceleration of the glider from just as in Experiment 1. Procedure DISTANCE UL v ( tc ) = v( tu, t L ) =, and T a = UL [ v ( t )]/, c S Several such acceleration measurements will be made, for different masses total mass M. m h but constant Three small weights are provided, which can be distributed between glider and hanging support in four different ways (number of weights on hanger support = 0, 1,, 3). For each of the four distributions of weights you should measure the system's acceleration (that is, the glider's acceleration), at least twice. 31

32 First determine the total mass of the system by weighing glider, weight-hanger and the three free weights, all together. This total mass (M) remains constant. You will also need to measure the hanger weight and the individual weights to determine m h for various arrangements. To Guard Against Some Potential Problems in Setting up Your System: Practice starting and stopping the glider to get a feel for the speeds involved and the techniques required. Practice releasing the glider from rest (starting at the backstop position and letting go without imparting any initial velocity). Pick a U photocell position near to the rest position, and an L position near the pulley, note the measured transit time from U to L and repeat several times. If the transit times are not essentially the same, you are giving the glider some initial velocity in the process of releasing it. With all three movable masses loaded on the hanging support (maximum applied force), determine how much room you need to stop the glider near the pulley end, without allowing it to bump into the stop at the end of the track, and without jamming the glider down onto the track. Place the L photobridge to give yourself this room. Record the U and L positions, and leave them there throughout the experiment. With all three movable masses loaded on the hanger, determine the half-time point C of the glider passing through the photobridge array. Do this at least twice before changing the weights, and calculate the corresponding values of acceleration; if there is inconsistency make further adjustments and measurements until the inconsistency is eliminated and you have at least runs in reasonable agreement. (Note the potential problems described in the preceding two paragraphs.) Repeat the last procedure, including at least two runs, with one of the masses transferred to the glider. Repeat again for two masses transferred, then all three. Remember, you need to specify m for each mass arrangement. Analysis of Data h For each mass combination calculate the applied force, F = mh g, and for each run, calculate the value of the measured acceleration. Plot the measured acceleration as a function of the applied force ( F = mh g ). You will have at least two measurements of the acceleration for each of the four values of F. Draw the best-fitting straight line through these points and calculate its slope. Estimate the uncertainty in the slope. Discussion Equation (3) predicts that your graph is expected to be a straight line, with slope ( a / F) = 1/ M passing through the origin. Within the experimental errors: Is your graph linear? Does it pass through the origin? Calculate M from the slope, estimate the uncertainty 3

33 in this value, and compare with the value of M determined by weighing. Are the two consistent? If any of these answers is no, try to suggest what might have caused the discrepancy. Some possible effects which we have ignored, and which might affect the results are: Remaining friction between track and glider Friction in the pulley Track not quite level Mass of string, which we have ignored String pulling glider possibly not exactly parallel to track. 33

34 LINEAR MOMENTUM AND KINETIC ENERGY References: Young and Freedman, University Physics, Ninth Ed., Extended Version, Chapter 8; Physics 3 Laboratory Measurements. The object of this experiment is to investigate the conservation of linear momentum and the conservation of kinetic energy in elastic collisions. We will study collisions between two air track gliders. Velocity measurements are used to study a complicated mechanical event: the collision of two bodies and its effect on their motion. Motions are restricted to a single straight line, and forces other than those of mutual interaction are minimized by the use of the air track. The object is to observe transformations of momentum and energy under various conditions of collision. Basis of Experiment The physics of one-dimensional collisions is presented in the reference text. Referring to the system consisting of the two interacting bodies only, the governing principles are: (1) If no forces act on the system other than the ones that the parts of the system exert on each other, the system's total momentum remains constant and the total energy of the system remains constant (conservation of total momentum and energy for isolated system); this is true for both conservative and non-conservative forces. () If the forces acting between parts of the system are entirely conservative, both the mechanical energy (kinetic plus potential) and the internal energy (heat and chemical forms) of the system remain constant. Principle (1) implies that if the internal forces are non-conservative, the changes in mechanical and internal energy will be equal in magnitude and opposite in sign (i.e. the increase in internal energy will equal the decrease in the mechanical energy). Momentum and total energy are conserved for both conservative and non-conservative forces. In a one-dimensional system, the vector momentum reduces to an algebraic number whose sign denotes its direction according to a consistent designated convention. The energy is a scalar, as always. Since the two bodies of the system do not interact with each other except when in contact, there is no potential energy in the system at times when the objects are separated. At such times, the system's mechanical energy is its kinetic energy only. The different possibilities to which the principles apply are designated by names given to different types of collisions: 34

35 - an elastic collision is one in which the internal forces are conservative, so that both mechanical and internal energy separately stay constant throughout the collision; - an inelastic collision is one in which at least part of the interaction force is nonconservative, so that both mechanical and internal energies change. An endothermic collision is one in which mechanical energy decreases and internal energy increases, that is, mechanical is converted into internal energy (usually heat). A perfectly inelastic collision is an endothermic one in which the maximum possible amount of mechanical energy is converted into internal energy; it is a collision in which the two bodies stick together and move as a unit after the event. Some of these types of collisions will be investigated in this experiment. In all types of collisions, momentum is conserved. Momentum conservation for a two-body system is expressed in Eq.(1), where the m s are masses and υ s are vector velocities, and the subscripts refer to bodies 1 and respectively. The primed velocities are those of the bodies after collision, and the unprimed are velocities before collision. Momentum before collision = Momentum after collision m 1v1 + mv = m1v1 + mv (1) The conservation of kinetic energy (which is conditionally conserved) is expressed in Eq.(): () m v + m v + m ( v ) + m v ( ) In these equations, the left side refers to any time before the collision of bodies 1 and, and the right side refers to any time after the collision. Once the system is defined for 1- dimensional motion, all velocities in one designated direction are positive numbers in the equations, and all in the reverse direction are negative numbers. The signs of the velocities are essential in Eq.(1), a vector equation. In Eq.(), where only the squares of the velocities occur, the vector direction no longer enters -- the energy is a scalar, not a vector quantity. Plan of the Experiment The low friction air track is the same as has previously been used. In order to observe collisions between unequal as well as equal masses, three gliders are needed - two of about the same mass, and one of a different mass. You are to measure glider masses at the start, and use the same gliders throughout the experiment without changing their weights. Elastic collision of the gliders is achieved by setting them on the track with spring-bumpers facing one another. 35

36 Inelastic collisions are arranged by modifying the facing ends of the gliders with wood blocks and velcro tape (sticky bumpers). The photoelectric timer as used here is operated in gate mode; it reads in milliseconds (thousandths of a second) the time during which the light beam is interrupted, from which the glider's speed in passing the beam can be directly calculated. For this experiment, each glider has been fitted with a precise 10 cm aluminum mask, painted black, and it is this mask, rather than the body of the glider, that interrupts the light beam. Thus the velocity of any of the gliders is l = 10 cm divided by the reading of the traversal time. (No correction for acceleration is needed; gliders move with constant velocity both before and after the collision. The Procedure provides for checking and ensuring this.) Two photocell bridges, one on each side of the collision area, are used (Fig. 1). Each is connected to its own timer. Thus each timer, operating in the gate mode, starts when the light of its own photocell is interrupted and stops when the light is resumed. If the timer is not reset between two interruptions, the time interval of the second interruption (after collision) is added on to that of the previous interruption (before collision). The photobridges should be placed so that you can note the first reading before the number is changed by a second passage. You should not reset during the collision -- you could miss counts by trying to reset. Practice a bit to be sure you can do the readings as required. Procedure First weigh the two small and one large gliders and record the weights. Weights should be measured on gliders complete with masks and whatever trimming weights have been added - no changing of masses is called for here. Since we want the collisions to happen in the absence of all external forces, it is important that the track be perfectly horizontal and as frictionless as practical. To check the condition of your track send one glider to your right and note the time it takes to cross each of two photocells placed one meter apart (and operated in gate mode). The difference between the two times divided by the time it took to cross the first photocell will give you the fraction of the energy lost due to friction and gained (or lost) because of gravity. Repeat, the above procedure, this time sending the glider to your left. Check for consistency by again sending it to the right and then left. Before making any adjustments in the air track level, see the Supplementary Procedure Section below. There is always a small amount of friction, which will lead to some time increase in the downstream'' counter. The important points are that the increases should be small, and especially that the percentage increase in a downstream counter be the same, for all practical purposes, whether you are looking at left-to-right, or right-to-left motion. To eliminate the effect of gravity, you may need to adjust the track level in small (e.g. quarter-turn) increments using the single large screw at one end of the track. Do not touch 36

37 any other adjustment screws. Note that there may be washers to fix the position of the track on the table. Stay at that position, since the air track, observed over a long period, has been adjusted to be much more level than the table itself. You should be able to adjust the track so that the loss in velocity is no greater than two-percent, and the same (essentially) in either direction. Readjust as necessary during the course of the experiment. The Various possible collisions, such as a heavy and a light weight glider meeting head on, a heavy hitting a light at rest, a light hitting a light at rest, all with springy'' bumpers (elastic), and a heavy hitting a light at rest with sticky bumpers (inelastic) are detailed in the Data section. More details about the Procedure, especially about achieving elastic and inelastic interactions, are given below: Supplementary Procedure Notes Detailed Cautionary Notes and Explanations A. Do Not Change Transverse Track Level Special levelling methods have been used to correct an assembly error made on several air tracks. Because of the error, the base of the track is no longer necessarily horizontal in the transverse direction when the triangular air tube (the travelling surface for gliders) is horizontal. The tubes have been levelled transversely; do not pay attention to obvious skewness in the cross beam and I-beam of the tracks. Do make adjustments (as you find it necessary as described above) to the single levelling screw that levels the track along the direction of the glider motion. B. Special Collision Hardware on the Gliders READ C. BEFORE HANDLING GLIDERS Each glider in this experiment has been fitted with light aluminum hardware, retaining pins and Velcro in such a way that the glider mass remains balanced and constant whether it is used in elastic or inelastic collisions. Each end of a glider holds a retainer'' behind an elastic (spring) bumper. The retainer keeps the elastic bumper aligned and perpendicular, and also serves as a mounting frame when wanted, for the removable aluminum piece called the inelastic bumper''. Inelastic bumpers (sticky bumpers) have Velcro strips mounted on them; when two gliders collide with these bumpers mounted, they stick together, making a perfectly inelastic collision. The inelastic bumpers of a glider are not simply put aside when the elastic bumpers are in use; they are stored on the two sides of the glider - below the mask, on Velcro strips. By 37

38 doing this (and also keeping their four mounting pins on the retainer) a glider will always have the same mass, whether fitted at the ends for elastic or inelastic collision. The mounting pins are removed from the retainer to mount the inelastic bumper. The bumper is then slipped over the spring, but inside the retainer, until its four holes are aligned top and bottom with those of the retainer. The pins are then returned to lock the inelastic bumper firmly, top and bottom, to the retainer. Note the Velcro material comes in two forms, one of which we can call woolly'' and the other, more plastic in appearance, we can called ribbed''. The high adhesion of Velcro comes in contact between a woolly and ribbed strip; two strips of the same type adhere poorly or not at all. For inelastic collisions, be sure the bumpers of the colliding gliders present the two types of Velcro to each other; similarly store the woolly bumper on the ribbed strip at the side, and vice versa. In any case, always mount and dismount pairs of bumpers on a glider, to maintain the weight distribution symmetrically. C. Additional Handling Care Please do not handle the gliders by their masks; their mounting screws are necessarily small. Do not disturb the trimming weights. Remove a glider completely from the track in order to fit or remove the inelastic bumpers, and note that the bumpers are light and easily deformed by handling until secured by their pins. It is highly recommended that all the elastic collisions called for be done in series, followed by inelastic collisions. This minimizes the time spent in refitting bumpers at the ends of the gliders. D. Glider Masses Weigh each glider with bumpers and pins on it, and the ten centimeter mask mounted. 38

39 Data This is not a data sheet to be filled in - it's meant as a guide. DATA MUST BE WRITTEN DIRECTLY INTO YOUR LAB NOTEBOOK. OBSERVE ALL THE CAUTIONS LISTED IN THE PROCEDURE SECTION AND SUPPLEMENTARY PROCEDURE NOTES ABOVE. I. Check that track is level in the direction parallel to the direction of motion as described in Procedure section, by comparing traversal times through photogates for a glider moving in each direction. When the level adjustment is complete, record the following: Glider Moving to Right t 1 = t =, Glider Moving to Left t 1 = t =, to show that the track is level within acceptable limits as described above. Repeat this to check the level, between collisions, and adjust the level if necessary. 39

40 II. Mass Data Three gliders, two light, one heavy. III. Collision Data The kinds of collisions to be studied are listed below, with the necessary data indicated. The arrows in column show the initial direction of motion. Be sure to include the arrows in your data for the motion both before each collision and after it. Three are elastic collisions, one is inelastic. You are welcome to try other combinations if you like think Mercedes vs. Volkswagen! No. Initial Direction. Means Body At Rest Sticky Bumpers Initial (Before Collision) Traversal Times After Collision Timer Readings* After Collision Traversal Times t 1 t T 1 T t 1 t 1 H L Off (Elastic) DO THREE TIMES H L Off DO THREE TIMES 3 L L Off DO THREE TIMES 4 H L On (Inelastic) DO THREE TIMES *Do not attempt to reset timer immediately after collision; recordt 1 and T, the sums of the two traversals and calculate the after-collision traversal times t 1 = T1 t1 and t = T. 1 t Calculations For each collision, calculate the vector momentum of each body before and after the collision, and the total initial momentum and final momentum of the two-body system. Do this for each of the three sets of data you have taken for each type of collision and for each set calculate the percent difference between initial and final momentum. %DIFFERENCE IN MOMENTUM = Total Initial Mom - Total Final Mom Total Initial Mom Do the same for the kinetic energy, except that you need to do this only for the data in each type of collision which shows the smallest difference between initial and final momentum. 40

41 DIFFERENCE IN K.E. = Total Initial Mom - Total Final K.E. Total Initial K.E. There may be large experimental uncertainties, especially where the initial vector momentum is close to zero. Rather than carry out a detailed uncertainty calculation, you can get an approximate idea of the uncertainties from the variation in difference in momentum among the three values you have obtained for each type of collision. Discussion Discuss your results for momentum and kinetic energy. Keep in mind that we expect momentum to be conserved in every collision, while kinetic energy is conserved for elastic and not for inelastic collisions. Your discussion should include estimates of uncertainties, as indicated by the spread in your repeated (3) measurements of each type of collision, and the effects of residual friction and gravity as determined in DATA Pt.I. Do your results confirm expectations, given the experimental uncertainties? 41

42 ANGULAR MOMENTUM Reference: University Physics, Ninth Edition, Extended Version; H.D. Young and R.A. Freedman, Chapters 9 and 10. In complete analogy with linear momentum in one dimension, we are able to define for rotational motion the quantities L r, the angular momentum, and K, the rotational kinetic energy. For the rotation of a rigid body about a fixed axis, the vector nature of angular momentum is expressed by simply choosing clockwise rotations as positive and counterclockwise rotations as negative (or vice versa, if one prefers). Notice the correspondence: Translation Rotation Velocity v 1 ω Momentum p=mv L=Iω Kinetic Energy T=mv / K=Iω / (1) Here I is the moment of inertia, m the mass, v the linear velocity and ω the angular velocity. [The angular velocity is measured in radians/sec. The rolling condition, when a body (e.g. a disk or ball) of radius r rolls or rotates without slipping, is that v=rω.] In this experiment we use a specially designed air table on which we cause the collisions of a ball and a disk, and so observe the consequence of angular momentum conservation. Notice as you read the description of the collision that it is inherently inelastic, so that the kinetic energy will not be conserved through the collision. The kinetic energy does serve, however, to define the initial state, and in particular the velocity and hence the initial angular momentum. Since no external torques act on the system, the angular momentum is conserved, in analogy with linear momentum conservation when no external forces act on the system. Experimental Apparatus and Basis of the Experiment A steel ball rolls from rest down a ramp to an air table, acquiring both rotational and translational kinetic energy. Because of the curvature of the ramp at the bottom, the ball leaves it horizontally, with a velocity that can be measured. (It can also be calculated, but only approximately, from the change in potential energy.) The ball then makes an inelastic collision with a steel disk free to rotate on the air table (actually, it collides with a low-mass aluminum catcher attached to the disk). The ball is locked to the catcher at some known distance from the axis of rotation of the disk. Disk and ball now rotate together, with an angular velocity that can be measured with reasonable precision. The experiment requires careful alignment (it e.g., the ball must enter the catcher perpendicular to the disk radius) to maintain the simplicity of the data analysis. The experiment will be carried out for four values of initial angular momentum L i. The apparatus is shown in Fig. 1. 4

43 Theory of the Collision If the ball of mass m leaves the ramp with a velocity v h, directed perpendicular to a disk radius, and at a distance r m from the disk axis, its angular momentum will be L I =mv h r m (with respect to the disk axis) before collision. With the disk at rest before the collision, this is the total initial angular momentum of the ball-disk-catcher system. Notice that the angular momentum is a well-defined constant quantity at the time of the collision even though no part of the ball-disk system is in rotation before the collision. If we neglect the very slight influence of gravity on the ball as it moves from the ramp to the catcher, the kinetic energy of the ball remains constant until it strikes the catcher. But this quantity changes considerably as the ball collides and lodges in the catcher. When it lodges in the catcher, the entire system of disk, ball and catcher will rotate with an angular velocity ω f that depends on the moments of inertia of the ball, the catcher and the disk itself, with respect to the axis of rotation. Since angular momentum in any system is conserved in the absence of external torques, the system's final angular momentum is equal to its initial angular momentum. Initial Angular Momentum = Final Angular Momentum L I =L f Mv h, r m = (I b +I c + I o ) ω f. () Here I b is the moment of inertia of the ball treated as a point mass rotating about the disk axis at a distance, r m, I c is that of the catcher (treat as a rigid rod fixed at one end) and I o is that of the disk. (Refer to Chapter 9 of the Reference for details of these moments of inertia.) 43