EXPERIMENT 1: ONE-DIMENSIONAL KINEMATICS

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1 TA name Lab section Date TA Initials (on completion) Name UW Student ID # Lab Partner(s) EXPERIMENT 1: ONE-DIMENSIONAL KINEMATICS MOTIONS WITH CONSTANT ACCELERATION 117 Textbook Reference: Walker, Chapter LABS - GENERAL INFORMATION The lab exercises in this manual are all concerned with measurements of the motion and interaction of material bodies. The frame of reference for measurements of motion is the laboratory. Within the precision of the measurements possible with our equipment, the laboratory is an inertial reference frame,thatis,oneinwhichnewton slawsofmotionaply.notethat in order to make this statement it is necessary to know the precision of the measurements. If you are attempting to compare your measurementstothepredictionsofaphysicalmodelsuchasnewton slaws,astatementofthe precision of measurement should always accompany the measurement. This is usually done by writing the uncertainty of a measurement as well as the value of the measurement. For example, a velocity could be expressed as 1.25 ± 0.05 m/s. The uncertainty is 0.05 m/s and is 4% of the value of the velocity, so the measurement can be described as a measurement with a 4% uncertainty. This is a shorter way of saying that the precision of the measurement is 4%. An uncertainty expressed in this way is usually a statistical uncertainty. In the case of a velocity measurement that involves measuring a distance interval and a time interval, small differences are likely to occur in repeated measurements ofthetimeandistance.perhapsyoudidn tclickthestopwatchwhentheobjectwasexactlyathe first or last point of the interval, or your estimate of the distance interval varied because you had to estimate distance to 1/10 mm on a meter stick with marks every millimeter. The differences in these measurements tend to be random and center on some average value. For most measurements, the averagevalueofthemeasurementsisthebestestimateofthe true value(the true valueisthe average value as the number of measurements approaches infinity). A detailed mathematical treatment of such random uncertainties exists and should be part of the background of all scientists. [Mathematicaltreatmentsofuncertaintiesareoftencaled eroranalysis.wewilnotusetheterm eror sincerorstendtobeconfusedwithmistakes,i.e.gofs.theseare not subject to statistical treatment, but among beginners one often hears the nonsensical statement that one of the sources of uncertaintyinameasurementis humaneror. ].Somelabswill use simpler estimates of uncertainties that will not differ greatly from the more formal ones. A good part of the reason for this is that repeated measurements must be made to determine formal uncertainties and there is simply not enough time for much repetition in our 3-hour labs. A detailed description of uncertainties as we will use them is contained in the eight-page document that is part of the introduction to the lab entitled Uncertainties. You are encouraged to read this at some time early in the quarter and to bring it to the lab as a reference. For this first lab, the relevant descriptions of uncertainties will be introduced as required (this writeup is longer than usual!). SYNOPSIS - LAB 1 In this lab, you will study the motion of an object with nearly zero, positive and negative acceleration. You will measure the position of a cart on a track and of a ball moving under the influence of the gravitational force (and therefore a constant acceleration). You will approximate the instantaneous velocity, v(t), from the position versus time, x(t), graph and the instantaneous Physics 117 Copyright 2007, Dept. of Physics, U. of Washington Winter 2007

2 acceleration, a(t), from the velocity versus time, v(t), graphs. For the case of uniformly accelerated motion, a(t) = constant, you should learn: (1) how the shape of the position versus time, x(t), graph depends on the acceleration for three cases, a > 0, a = 0 and a < 0 and (2) how the slope of the velocity versus time, v(t), graph depends on the acceleration for three cases, a > 0, a = 0 and a < 0. APPARATUS You will use a computerized motion detection system. The motion sensor is an ultrasonic detector and is used to measure the position of a cart or ball. The detector (see figures below) sends out busts of 49kHz ultrasound waves and detects the echo reflected back by any objects in the path of the ultrasound beam. The pulse rate for this lab is set at 50 pulses per second. The detector measures the time difference between the time it sends out a pulse and the arrival time of the echo. Since the speed of sound in air is known (~344 m/s), the distance between detector and the object can be determined. The minimum of the range of our motion detector is about 15 cm. It does not give the right value of distance if the object is closer to the detector. Figure 1-1 Motion Sensor Note there is a switch on top that allows you to choose narrow beam or wide beam. Always set this switch to narrow beam when you measure the distance of a cart. The velocity of a moving target is obtained from the difference in distance between pulses (the so-called first difference of the distance data) and the acceleration is obtained from the difference in velocity between pulses (the second-difference of the distance data). The DataStudio program measures positions directly, but it computes velocities by taking the difference of positions at small time intervals and computes the accelerations from the differences in velocities. Because such small differences have relatively large uncertainties, the velocityandacelerationgraphswilbeabit raged. Thisistobexpected. PROCEDURE Horizontal (or nearly so) track Add two 500g bars to the cart. This will reduce the importance of the frictional forces acting between the cart and the track. 1. Level the track as well as you can with the small pocket level. [The tabletops are not completely level, so you should not change the position of the track after you have leveled it.] When the track is level, use the cart to check the leveling. Describe how you used the cart to check the leveling. Physics Autumn 2006

3 Attach the motion detector to one end of the track so that the transducer disk is aimed down the track. Give the cart a push so it moves away from the detector to the opposite end of the track. Do not allow the cart to collide with the motion detector or with the floor. Click the Open Activity option of DataStudio and the Lab 1-One Dimensional Kinematics folder (Located in My Documents folder incaseitdoesn tcomeupimediately). Open the file Lab 1- Cart Practice taking data with the motion detector. Click Start to begin data taking, push the cart and then click Stop at the end of the cart motion. Note that the motion detector reading becomes more positive as the object moves away from it, i.e., the positive x direction is away from the detector. Remember that the data are inaccurate if the cart is closer than 15cm to the detector. Adjust your push so that the velocity is in the range 0.30 m/s to 0.50m/s. 2. When you have a good run, maximize the graph of position vs time, and then use the zoom select feature to fill the screen with a portion of the graph that is nearly a straight line. The zoom select icon is the magnifying glass fourth from the left. Click on it, then click-drag the mouse icon (a magnifying glass) to create a rectangle about the region of interest. The selected area will automatically expand. Print a copy for both partners. Write your name on the graph and a description of the data. The description can be a reference to the lab instructions. For example, thisgraphcouldbelabeled question2 or leveltrack. Thishould be done for all graphs in Examine the x(t) graph you have just printed. Does the velocity of the cart increase with time, decrease or remain the same? Explain how you know. 4. Determine the velocity and the uncertainty in the velocity from the graph of velocity vs time for a time t 0 that you choose near the center of the graph. Use seven data points centered on t 0. Find the velocity at t 0 by averaging the velocity for times symmetric about t 0. For example, if t 3 immediately precedes t 0 and t 4 immediately follows t 0, the velocity at t 0 is approximated by v(t 0 ) = (1/2)[v(t 3 ) + v(t 4 )], the average of velocities just before t 0 and just after. Calculate v(t 0 ) by means of the three pairs of velocities symmetric about t 0 and the velocity at t 0. Label the values 1, 2 and 3 in the order of increasing time difference between pairs. You can read the values of the velocity and time directly from the graph, but a simpler method is to use a feature of Data Studio called Smart Tool. Click on the Smart Tool icon in the toolbar at the top of thegraph(it stheone6 th from the left that looks like an x,y coordinate). An x,y coordinate system will appear on the graph. Drag it until it locks onto the data point at t 0. The screen will show values for v(t 0 ) and t 0. They will probably not be as precise as you need. Double-click on the Run#1 icon under Velocity, Ch 1&2 (m/s) in the Data column to the left of the screen. In the Data Properties window that appears, click the Numeric tab. In the new window in the Style box select Fixed Decimals and set Digits to the right of the decimal to 4. Click OK. The time and velocity values in the Smart Tool display should now have 4 decimal places. Lock onto and record the values for seven data points you have chosen. These data should be taken in pairs, as described above, with t 1 some interval below t 0 and t 2 the same interval above t 0. Then t 3 is a larger interval below t 0 and t 4 is that same interval above t 0, and so on. Thus each pair gives a value for v(t 0 ) and you will compare these values. Physics Autumn 2006

4 t 1 = s v(t 1 ) = m/s t 2 = s v(t 2 ) = m/s t 3 = s v(t 3 ) = m/s t 4 = s v(t 4 ) = m/s t 5 = s v(t 5 ) = m/s t 6 = s v(t 6 ) = m/s t 0 = s v(t 0 ) = m/s 5. From your measurements in 4, find four values for the velocity at t 0. (0) v(t 0 ) = m/s (1) v(t 0 ) = m/s (2) v(t 0 ) = m/s (3) v(t 0 ) = m/s Find the average of the four measurements and from the spread in them find the uncertainty v i ( t 0 ) in an individual measurement, Equations U1 and U2 as described in Appendix B: Uncertainties. v ( t 0 ) v(t 0 ) ave = m/s = m/s v i ( t 0 ) The best estimate of the value of a quantity for which there is a set of measurements is the average value of the quantity from the set. The uncertainty of an individual measurement, such as v i ( t 0 ) above, answers the question, if I were to make another measurement of v(t 0 ), how closely would it agree with the average? The question you are more likely to ask is, if I make another set of measurements, how closely do I expect the average of the second set to agree with the average of the first set? You likely feel instinctively that the averages would agree with each other more closely than an individual measurement would agree with the average. This is indeed the case. Part 9 in Appendix B: Experimental Uncertainties provide the following result for cases in which the uncertainty, x, is i the same for all measurements. n where n is the number of measurements in the set and i x i is the uncertainty in an individual measurement. 6. Copy the best estimate of the value of v(t 0 ) that you calculated above, and Calculate the uncertainty in the best value. v(t 0 ) best = ± m/s = v ( t 0 ) v(t 0 ) best and its uncertainty must bewritenin proper form.(see Part 5 in Appendix B: Experimental Uncertainties.) Warning: In this lab and future labs, any result and/or uncertainty not written according to the two rules above will be marked WRONG, independent of their values. Physics Autumn 2006

5 7. With this warning in mind, rewrite your values of t ) v(t 0 ) best, and v t ), below. v i ( 0 ( 0 = v i ( t 0 ) v(t 0 ) best = ± m/s Inclined track Having completed measurements with the level track, a 0,youwilnowmakemeasurementsfor a > 0 and a < 0 by tilting the track. In each of the cases below, release the cart so that it will roll downhill. DO NOT LET THE CART RUN INTO THE MOTION DETECTOR! 8. For a positive acceleration, a > 0, is the end of the track closest the motion detector higher or lower than the other end? Explain. 9. Place one block under the leveling screws at one end of the track so that a positive acceleration is produced. Start the data collection, and release the cart. Sketch below the relevant portions of the resulting x(t), v(t), and a(t) graphs below. Provide appropriate scales. Be sure to label the graphs. [Heretheterm relevantportions includesthesmothlinerepresentingthemotion,butdoesnot include jagged parts of the trace that may arise from spurious reflections. In a physics lab, it is oftenecesarytodeterminewhena spike inagraphasrealmeaningandwhenisitjustbad data. If you have questions about this, discuss it with your TA.] x(t) v(t) a(t) Physics Autumn 2006

6 10. Maximize the x(t) graph and select a time, t 0, near the middle of the graph. Zoom-select the data so that you can obtain a precise reading of the time and distance near t 0. Print a copy for both partners. Label your graph with your name and a title that describes the subject of the graph. For example, this graph could be labeled, Question10,or Velocityvs.timeforoneblock. Provide labels for all graphs you print in the 117 lab. Fortunately, the printer provides time and date as well as the lab name. From the graph, determine the slope x t of the curve at t 0. To approximate the slope at t 0, requires four data values, two times and two distances. Write below the data values from which you determine the slope, their uncertainties and their units. Data Studio states that the accuracy of a distance measurement is 1.0 mm. Using a sampling rate of 50/s, the time between samples is 0. 02s, so using a value of s for the time uncertainty is reasonable. Use Smart Tool to read the data points. t 0 = s datum 1 time = ± s datum 1 position = ± m datum 2 time = ± s datum 2 position = ± m 11 From your data in 10, calculate the slope at t 0 and its uncertainty. To find the slope, you will need a time difference t and a distance difference x. You will also need the uncertainty in t and the uncertainty x in x in order to calculate the uncertainty in the slope. For the uncertainty in the distance interval and the time interval, you will need the result that the uncertainty in the sum or difference of two independent quantities x and y, Z Ax B y, is t = ± s x = ± m Z A B 2 x 2 y [U7] 12. The slope can now be calculated as the quotient of x and t and the uncertainty in the slope as the uncertainty in the quotient. The uncertainty in the quotient of two independent quantities Z = x/y is calculated using [U6b] from the Appendix B: Uncertainties t 2 2 a b x y Z Z [U6b] x y In our case, slope = 2 2, and the uncertainty is x t slope = ± m/s slope slope x t x t Physics Autumn 2006

7 13. Compare the value of the slope from 12 with the velocity at t 0 read from the v(t) graph. Note that the times for the velocity data points are midway between those for the position points, so to get v(t 0 ) from the graph, take the values immediately above and below t 0 and average them. Note: whenever the word compare appears, a calculation must be done. For example, if D 1 and D 2 are two measurements of the same quantity, Criterion for agreement: D 1 D Use your value for t ) from Section 8 above for the uncertainty in v(t 0 ). v i ( 0 [U8] slope = ± m/s (from Section 13 above) v(t 0 ) = ± m/s (from v(t) graph) 14. Use the same method you used in to find the slope of the v(t) graph at some time t 0 for which the graph is relatively smooth. Find the uncertainty in the slope. Zoom-select the data around t 0. Record and label the data necessary below. Show your work. t 0 = s point 1 t= ± v= ± point 2 t= ± v= ± slope = ± m/s Compare the slope from 14 with the acceleration at t 0 from the a(t) graph. In order to determine a, place the horizontal axis of Smart Tool to so that it best fits the acceleration data (the acceleration should be a constant, but will have large fluctuations. Use a = ±0.15 m/s 2. a(t 0 ) = ± m/s 2 Consider the case of a< Place two blocks under the leveling screws at one end of the track so that the acceleration will be negative, a<0. Obtain x(t), v(t) and a(t) graphs(don tlethecartcolidewiththedetector). Zoom-select the relevant part of the x(t) graph and print it for both partners. Determine the slope and the uncertainty in the slope for a time t 0 near the center of the graph. This time a method will be used that is different than that used in With the ruler provided, draw a line tangent to the x(t) graph at t 0. [Note: the line should extend across the graph so that an accurate value can be found for the slope of the line]. Record the coordinates of two points, (x a,t a ) and (x b,t b ) near opposite ends of the tangent line. t 0 = s x a = t a = x b = t b = slope = (m/s) Physics Autumn 2006

8 17. To find the uncertainty in the slope you must draw two more lines. The first is the line tangent to the x(t) graph at t 0 with the most negative slope that you believe still fits the data. The second is the line tangent to x(t) graph at t 0 with the least negative slope that you believe still fits the data. Find the two slopes as you did in 16.The uncertainty in the slope is then one-half the difference between these two slopes. max negative slope: x c = t c = x d = t d = slope = m/s min negative slope: x e = t e = x f = t f = slope = m/s slope = m/s slope(t 0 ) = ± m/s 18. Compare the slope at t 0 to the velocity at t 0 obtained from the v(t) graph. For the uncertainty in the velocity, use your value from How, if at all, do the graphs in 10 change when an additional block is placed under the same end of the track as in 10? Sketch your answers below. x(t) v(t) a(t) Physics Autumn 2006

9 20. Draw similar sketches corresponding to one block placed under the opposite end of the track. x(t) v(t) a(t) 21. Identify that inclined track set-up (one block or two, at same end as the motion detector or at the opposite end), which produces each of the accelerations below. Smaller negative acceleration Larger negative acceleration Smaller positive acceleration Larger positive acceleration Vertical Motion Click the Open Activity option of DataStudio or the file menu at the top of the page and open the appropriate file, Lab 1-Ball. 22. Place the motion detector on the lab table facing upward and practice throwing the Nerf ball upward above it. Try not to detect your hand while the Nerf ball is in flight. After throwing the ball, move your hand out from above the detector and catch the ball on its way down. You will find it easiertoacquiregodataifyousethedetectorforabroadbeamandon tatemptotos the ball too high. 0.3m to 0.8m is fine. Hold the ball over the detector, start the data collection and toss the ball upward. If you move deliberately, most of your data may not be useful. This is not a problemasyoucanselecthe god partforanalysis.practiceseveraltimesuntilyouget god data. Zoom-select the good part of the x(t) graph. Maximize the graph and print it. Physics Autumn 2006

10 23. Describe the shape of the x(t) graph. 24 Zoom-select, maximize and print the portion of the v(t) graphcorespondingtothe god portion of the x(t) graph. 25. Indicate on the v(t) graph when the ball is moving upward, when it is moving downward and when it is at the top of it trajectory. 26. Describe the motion of the ball when it is on its way up, on its way down and when it is at the top of its trajectory in terms of its position, velocity and acceleration. up: down: top: 27. Determine the slope of the v(t) graph when the ball is on the way up, when it is on the way down and when it is at the top. Use a method similar to the one you used in 4. Select and record the timesforwhichyouwouldliketheslope.pickthetimeforthe up measurementwherethe trajectoryisaproaching,butnotathetop.forthe down measurement,pickatimenearthe bottom of the trajectory. Use Smart Tool to measure the time and velocity (four digits after the decimal) for two data points, one just before the chosen time, v b,t b, and one just after, v a,t a. Record the values below and use them to calculate the slopes. t up = s t b = s v b = m/s up: t b = s v b = m/s slope up = m/s 2 t top = s t b = s v b = m/s top: t b = s v b = m/s slope top = m/s 2 t down = s t b = s v b = m/s down: t b = s v b = m/s slope down = m/s 2 Physics Autumn 2006

11 28. Compute the average of the three slopes, the uncertainty in the individual values of the slope and the uncertainty in the average. As described in Section 5 above, the uncertainty in the average of n measurements is less than the uncertainty in any individual measurement. If each value of slope has the same uncertainty, a, then the uncertainty in the average slope is n i a ai with n = 3 from the three measurements: a = m/s 2 and the final result of the measurement is a ave = ± m/s 2. If the three values of a are considered to be independent measurements, then the uncertainty in their mean is smaller than the uncertainty in an individual measurement by 1/3. The v(t) graph should be a nearly straight line with a negative slope. The slope could be determined by hand fitting a straight line to the data, but in this case we will let the computer do the work. 29. Zoom-select the straight-line portion of the v(t) graph. In the toolbar above the graph, select Fit and from the Fit menu, select linear. A straight line will appear through the data with the slope of the line and its uncertainty in a box. Both should have 4 digits past the decimal. If not, adjust the number of digits by double clicking on the linear fit icon in the Data column on the left of the screen, selecting the Numeric tab and adjusting the data to four digits as you did in 4. Drag the fit box so it does not cover the data. Print the graph and record the value of a and its uncertainty below. Compare the value to local g = 9.80 ± 0.01 m/s 2. a fit = ± m/s 2 REFERENCE Salviati:... Very well; up to this point you have explained to me the events of motion upon two different planes. On the downward inclined plane, the heavy moving body spontaneously descends and continually accelerates, and to keep it at rest requires the use of force. On the upward slope, force is needed to thrust it along or even to hold it still, and motion which is impressed upon it continually diminishes until it is entirely annihilated... Now tell me what would happen to the same movable body placed upon a surface with no slope upward or downward. from Dialog by Galileo Galilei (1632) Galileo realized that, although he could not directly measure the position versus time of a freely falling object accurately because the full acceleration of gravity quickly produced velocities too large for him to measure with the clocks available to him (his pulse and a Chinese-style water clock). He could measure the uniformly accelerated motion of a ball rolling down a very smooth track and found that all uniformly accelerated motion behaves in the same way! By using an inclined track, he was able to reduce the acceleration sufficiently that he could make more reliable position versus time measurements. In this lab you have used a modern version of Galileo's method to study motion for two different cases: 1) almost uniform velocity and 2) uniform acceleration. Physics Autumn 2006

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