Appendix A: The Free Software Tracker The free software Tracker [1, 2] allows the analysis of the particle (or body) motion frame by frame, which is called video analysis. Several types of movement filmed by digital cameras or webcams can be analyzed. Through the use of this technology, physics teachers and students have the opportunity to develop experiments and lab activities of high academic quality using simple and ordinary materials. It is reasonably easy to learn how to use the software, which makes it relatively simple to employ in obtaining relevant information in physics experiments. The software Tracker was created in partnership with the Open Source Physics (OSP). OSP is a worldwide community that offers free resources for the teaching of physics and computer modeling (Fig. A1). 1. Installation The free software Tracker requires Java and Quick Time. Tracker also supplies its own open-source video engine Xuggle. QuickTime is also supported on Windows and OS X. You can access the webpage of the free software Tracker at the link: http://physlets.org/tracker/ (Copyright (c) 2016 Douglas Brown). You can download the free software Tracker accessing the link: http://physlets.org/tracker/installers/installer_help.html 2. Starting the video analysis Opening the video file The icon Open ( ) allows to open both Quick Time video (.mov) and files with the extension.trk, which is used in files saved by the Tracker. Choosing the frames to be analyzed (Fig. A2) In order to choose the first (Start frame) and last (End frame) frames of the interval to be analyzed you must click in Clip Setting ( ), and the dialog box shown in Fig. A3 will appear. Springer International Publishing AG 2017 V.L.B. de Jesus, Experiments and Video Analysis in Classical Mechanics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-52407-8 167
168 Appendix A: The Free Software Tracker Fig. A1 Tracker Graphical interface Fig. A2 Bar used to select the interval of frames to analyze Fig. A3 Clip settings Dialog box. The start and end frames, and also the step size, can be chosen in order to start the video analysis. The acquisition rate (frame rate) and the interval between frames are informed You can also choose the Step size to be used during analysis. The Frame rate informs the acquisition rate (frames per second) of the movie. Usually, the Tracker software makes an automatic reading of the acquisition rate. In any case, it is interesting to know the correct acquisition rate of the video. One way to be sure about it is to undertake the following strategy: at the start of the recording, put a digital clock in front of the camera for about 5 s, so it is possible to measure, during the video analysis, the correct acquisition rate of the recorded video, and compare it with the Frame rate informed in the Clip Settings dialog box. You can change the video display from Frame to Time and vice versa, as shown in Fig. A4.
Appendix A: The Free Software Tracker 169 Fig. A4 Dialog box showing how to change the display (frame; time; step) of the video Fig. A5 The ruler is used as the scale and it is set at the same plane of the motion of the steel ball. The coordinate axes have their origin at the beginning of the movement How to calibrate the scale The icon ( ) has the function to calibrate the scale once the distance between two points is known. Drag the ends of the calibration stick to such positions and inform the known length. Thus, it is essential to include in the video one object whose size is previously known, such as, for example, a ruler (see Fig. A5). Remember to make sure that the object is placed in the same focal plane of the motion to be analyzed. To add the coordinate axes (x, y), the button ( ) should be used. You can set the origin of the coordinate system and the degree of inclination (see Fig. A5). The scale s and reference frame s origin and angle uniquely define the coordinate system used to convert pixel image positions to scaled coordinates. Tracking objects. Click the Create button ( ) and choose a track type from the choices menu, as shown in Fig. A6. Most moving objects can be tracked using a Point Mass track. To start tracking an object, mark its position on every frame selecting the SHIFT key and simultaneously click on the object (its center of mass or any
170 Appendix A: The Free Software Tracker Fig. A6 Choices menu of track type objects Fig. A7 Data can be exported by selecting the desired data in the table, then right-click with the mouse and choose Copy Data from the menu other point of interest). After marking the position of the object, the program automatically steps (using the chosen Step size in Clip Settings Fig. A3) through the video clip. Don t skip frames, because, if you do so, velocities and accelerations cannot be determined. Data can be selected and easily exported from the data table by copying it to the clipboard and pasting it into any electronic worksheet. To copy, select the desired data in the table then right-click with the mouse and choose Copy Data from the menu (See Fig. A7).
Appendix A: The Free Software Tracker 171 The Tracker program allows us to visualize the graph of each coordinate as a function of time or any other variable. For teaching purposes, you may want to copy the data that corresponds to the spatial and time coordinates, and analyze them in any other graph or electronic worksheet program. If you wish to obtain values of velocity or acceleration, for example, it is also interesting to perform the calculation on your own in any electronic worksheet. This is a basic guide for beginners on video analysis using the Tracker software. You do not need to be an expert on Tracker to perform interesting experiments and video analysis. The free software Tracker was used in this textbook in order to obtain the position and time data, and the analysis and graphs where performed in any electronic worksheet or other software. Depending on the number of points obtained by video analysis, it is also possible to perform the data analysis manually.
Appendix B: Graphs Introduction In several areas, it is crucial to know how to construct and analyze graphs. Despite technological advances, such as the use of calculators and computers, it is important to know how to make basic calculations using only pencil and paper. Similarly, producing a graph on a millimeter paper is important for the educational experience. A graph made on a millimeter paper should have the following basic characteristics: The coordinate axes should be drawn 1 cm away from the edge, leaving room to write the scale next to each axis. Axes should have equally spaced marks, with their corresponding values written beside them. It is essential to write the name of the variable, and its respective unit in each axis. The experimental data points should be marked on the millimeter paper, and it is not recommended to write or mark the corresponding values on the scale. Remember, the coordinate axes and their respective scales have already been represented! After marking the experimental data points, it is important to analyze their distribution. Thereafter, such procedure will depend on each case. For example, if the data points distribution appears to have a linear tendency, a linear fit can be used, by calculating the slope of the best fit to the experimental points, and so determining the intersection on the vertical axis. In some cases, it is helpful to write the adjusted function on the graph. Certainly, the basic characteristics presented above suggest only a particular style, and they are not rigid and infallible rules. Springer International Publishing AG 2017 V.L.B. de Jesus, Experiments and Video Analysis in Classical Mechanics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-52407-8 173
174 Appendix B: Graphs A Brief Example Here, the construction of the graph of the physical quantity Y as a function of the physical quantity X using the experimental data shown in Table B1 is suggested. We have the freedom to choose if the plot will be drawn in the portrait or landscape orientation, but it is important to respect the following convention: The horizontal axis should contain the INDEPENDENT variable. The vertical axis should contain the DEPENDENT variable. For example, when a vehicle moves along a road, the position (s) depends on the time (t). After all, time passes regardless of our will and what we can do is simply write down the values of the vehicle position corresponding to certain instants of time. In the graph of the position, s, as a function of time, t, time will occupy the horizontal axis (abscissa axis) and the position will occupy the vertical axis (ordinate axis). In the case presented in Table B1, it was asked to construct the graph of the physical quantity Y as a function of the physical quantity X. Therefore, as the phrase already tell us that the variable Y depends on X, then the variable Y should occupy the vertical axis, while the variable X, the horizontal axis. We still have to decide whether the graph is oriented as portrait or landscape. As previously mentioned, we are free to choose either orientation. But we can use a method to decide which would be the best option, and making a compromise between the best use of the paper area and the choice of a scale of easy reading of the experimental points. If we use a millimeter graph of a size 18 cm 28 cm and choosing the portrait orientation, leaving 1 cm margin for each coordinate axis, we would have to use the following scales: In the HORIZONTAL AXIS (17 cm in length) we have 8 u.a. to distribute in 17 cm of the coordinate axis. Then, we should determine which value of N a.u. corresponds to 1 cm: Table B1 Experimental data of the physical quantities Y and X X (a.u.) Y (a.u.) 0.0 2.0 1.2 5.6 2.0 8.0 3.3 11.9 4.2 14.6 5.1 17.3 6.0 20.0 6.5 21.5 7.1 23.3 7.6 24.8 8.0 26.0 The abbreviation a.u. corresponds to Arbitrary Unit
Appendix B: Graphs 175 8 a:u: $ 17cm N a:u: $ 1cm Result: N ~ 0.47 a.u. for each 1 cm of the axis length. As this number is not good to use as a scale of easy assimilation, we must choose a higher and simpler number. In this case, the number is 0.5 a.u. to 1 cm of the axis length. In the VERTICAL AXIS (27 cm in length) we have to distribute 26 a.u. in 27 cm of the coordinate axis, then: 26a:u: $ 27cm N a:u: $ 1cm Result: N ~ 0.96 a.u. for each 1 cm of the axis length. As this number is also not good to use as a scale, we must choose a higher and simpler number. In this case, the number is 1.0 a.u. to 1 cm of the axis length. But we could also decide to construct the graph using the landscape orientation. In this case, the new values would be: In the HORIZONTAL AXIS (27 cm in length) we have 8 a.u. to distribute in 27 cm of the coordinate axis, then: 8a:u: $ 27cm N a:u: $ 1cm Result: N ~ 0.30 a.u. for each 1 cm of the axis length. The most convenient number would be 1 cm to 0.5 a.u. of the axis. In the VERTICAL AXIS (17 cm in length) we have 26 a.u. to distribute in 17 cm of the coordinate axis, then: 26a:u: $ 17cm N a:u: $ 1cm Result: N ~ 1.53 a.u. for each 1 cm of the axis length. The most convenient number would be 1 cm to 2.0 a.u. of the axis length. But how should we decide between the two orientations? Would they be equivalent? To help us decide, we can observe the percentage concerning the better use of axis length in each case. For example, in the case of the portrait orientation choice, the horizontal and vertical axes are around 94 and 96% occupied by the experimental values, respectively. In order to obtain these numbers, we need only to divide the scale value obtained for the exact distribution by the value chosen to facilitate the marking and reading of data points. Using as an example the horizontal axis, we have 0.47/0.50 ¼ 0.94 ¼ 94%. Proceeding in the same way for the case of the landscape orientation choice, it is found that the horizontal and vertical axes are around 60 and 76% occupied by the experimental values, respectively. It is easy
176 Appendix B: Graphs to see that in this case the portrait orientation is much more advantageous than the landscape orientation. In situations where the occupation rate are very close, and there is no significant advantage in choosing either orientation, the experimentalist is free to choose the orientation that suits him/her best. The procedure for the construction of a graph consists in the following steps: Figure B1 shows the plot of the coordinate axes and the chosen scale. Note that the coordinate axes must have their variables indicated, as well as their respective units. For a better presentation of the graph, we should avoid writing the scale every centimeter of the coordinate axes. You can choose to write the scale every 2 cm (as we did in our example) or every 5 cm. It is interesting to make a small mark in every centimeter, guiding the reader s eyes while he/she observes the graph. Each side of the square of 1 cm should now be interpreted differently when viewed horizontally and vertically. Its equivalence in length can only be understood in this way when they represent the real dimensions of the paper. In our example, the side parallel to the vertical axis corresponds to 1.0 a.u. of the variable Y while the horizontal side corresponds to 0.5 a.u. of the variable X, as presented in Fig. B2. Similarly, the smallest unit that corresponds to a square of 1 mm side, the side parallel to the vertical axis corresponds to 0.1 a.u. of the variable Y while the horizontal side corresponds to 0.05 a.u. of the variable X, as shown in Fig. B3. Figure B4 shows that the accuracy of the dot that represents the experimental point in the graph cannot be better than the half of the smallest part represented in the millimeter paper, i.e., it is not better than about 0.05 a.u. concerning the variable Y and it is not better than about 0.025 a.u. concerning the variable X,as shown in Fig. B4. After constructing the coordinate axes and their respective scales, the data points shown in Table B1 are plotted in the portrait oriented graph. Observing the arrangement of the experimental data points shown in Fig. B5, it is possible to imagine that a visual linear fit (using a ruler and based on reasonability) can be used to mathematically describe the variable Y as a function of the variable X. This can be done using a ruler, and the result is shown in Fig. B6. The slope can be obtained by choosing any two points that belongs to the drawn straight line, which can be mathematically described by the equation Y ¼ ax þ b. Figure B7 shows the chosen points. The result is: a ¼ ΔY ΔX ¼ Y 2 Y 1 20:0 11:0 ¼ X 2 X 1 6:0 3:0 ¼ 3:0 The linear coefficient can be obtained by observing the value of the variable Y where the drawn straight line cross the vertical axis at X ¼ 0 or by substituting the value obtained for the slope, as well as the coordinates of a point belonging to the drawn line, and solving for b. In this case, the linear coefficient, b, is 2.0. It is important to emphasize that the chosen points belonging to the drawn line must maintain a reasonable distance from each other. Mathematically, we know
Appendix B: Graphs 177 Fig. B1 Tracing the coordinate axes and their respective scales. Note that the coordinate axes are named and their units are in parentheses
Fig. B2 Note that each side of the square of 1 cm side should now be interpreted differently when viewed horizontally and vertically. In our example, the side parallel to the vertical axis corresponds to 1.0 a.u. of the variable Y while the horizontal side corresponds to 0.5 a.u. of the variable X Fig. B3 Note that each side of the smallest square, of 1 mm side, should now be interpreted differently. In our example, the side parallel to the vertical axis corresponds to 0.1 a.u. of the variable Y while the horizontal side is 0.05 a.u. of the variable X Fig. B4 The accuracy of the dot representing the experimental point in the graph is not better than the half of the smallest part of the millimeter paper, i.e., it is not better than about 0.05 a.u. of the variable Y and it is not better than about 0.025 a.u. of the variable X. The diameter of the dot shown in the figure should not be interpreted in the same way when vertically or horizontally projected
Appendix B: Graphs 179 Fig. B5 The experimental data shown in Table B1 are plotted in the portrait oriented graph. It is possible to imagine that a visual linear fit (using a ruler and based on reasonability) can be used to mathematically describe the variable Y as a function of the variable X
180 Appendix B: Graphs Fig. B6 Manual linear fit using a ruler that any two points belonging to a straight line can provide both the slope and the linear coefficient. The problem here has nothing to do with the mathematical definition, but with a practical issue. If we choose two points very close to each other, any error of reading, even a minor one, can lead us to mathematically get
Appendix B: Graphs 181 Fig. B7 Obtaining the slope and linear coefficients of the proposed linear fit of the experimental data. It is easy to see that the linear coefficient is equal to 2.0 (value of the variable Y when the variable X is equal to zero)
182 Appendix B: Graphs values for the slope and linear coefficient corresponding to a straight line that will definitely not be related to the straight line that was drawn. Thus, the mathematical equation that describes the dependence of the variable Y as a function of the variable X can be written as: Y ¼ 3:0X þ 2:0 The obtained coefficients have units according to the units of the variables represented in the coordinate axes.
Appendix C: Access to the Videos Discussed in this Book The videos are available here: http://bit.ly/2fzyh4s The reader is encouraged to download the videos and conduct his/her own video analysis. Enjoy it! Springer International Publishing AG 2017 V.L.B. de Jesus, Experiments and Video Analysis in Classical Mechanics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-52407-8 183
References 1. http://physlets.org/tracker/, accessed em 06/10/2016 2. D. Brown, A.J. Cox, Innovative uses of video analysis. Phys. Teacher 47, 145 150 (2009) Springer International Publishing AG 2017 V.L.B. de Jesus, Experiments and Video Analysis in Classical Mechanics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-52407-8 185
Index A Absolute error, 21 Acceleration measurement, 59 Acceleration of gravity, 16, 61 air resistance, 36 38 Arduino acquisition board, 14 average velocity, 31 estimation of velocity, 32 free fall, 13 15, 17, 19, 20, 22, 26, 29, 30, 32, 33, 35 39 friction, 38 graph and parabolic fitting, 35 instantaneous velocity, 15, 31 least-squares method, 24 26 length serves, 31 linear coefficient, 34, 35 linear fit, 21 linear function, 33 linearization, 15 linearized graph, 33 maximal possible uncertainty, 33 measurements, 18 metallic sphere, 14, 16, 17, 22 Moon, 36, 38 photosensors, 16 18 random errors, 30 reference point, 31 relative error, 21, 35 resolution, 16 sensibility coefficient, 33 significant digits, 19, 21, 32 software Tracker, 30 32 standard values, 21 systematic errors, 30 Taylor series, 33 Torricelli equation, 22 uncertainty of measurement, 32 velocity data, 34 Acceleration vector, 42, 71 Air flow unit, 3 Air resistance, 36 38, 84 Air table, 2, 3, 9 Air track, 55 57, 64, 65 Amplitude, 50 Angular acceleration, 52, 63 Angular frequency, 72 Angular variation, 44 Angular velocity, 44 46, 48 50, 52, 53, 63, 108, 115, 121 Arduino acquisition board, 14 Aristotle, 13 Astronomy, 41 Average acceleration, 43 Average velocity, 31, 41, 47 B Bi-dimensional collision assumptions, 159 didactical apparatus, 156 elastic and inelastic groups, 155 energy conservation, 157 impact parameter of the collision, 160 investigation, 158 launching speed, 158 linear momentum, 155, 157 particle system, 155 projectile and target spheres, 159, 160 range vectors, 158, 162, 163, 166 scattering angles, 157, 163, 164 Springer International Publishing AG 2017 V.L.B. de Jesus, Experiments and Video Analysis in Classical Mechanics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-52407-8 187
188 Index Bi-dimensional collision (cont.) spheres, 156 steel sphere (projectile) and the marble sphere (target), 164, 165 Torricelli equation, 160 trigonometric relation, 157 two identical spheres, 161 164 velocity vector, 157, 158 vertical displacement, 160 Brazilian National Observatory, 21 C Cartesian components, 103 Cartesian coordinates, 104, 107, 108 Center of mass (CM), 101 104, 106 111, 113, 115, 116, 119 127, 134 137, 139 141, 144 Centripetal acceleration, 44, 95 Centripetal and tangential accelerations, 71, 83 Centripetal and vertical accelerations, 93 Centripetal net force, 94 Centro Industrial de Equipamentos de Ensino e Pesquisa (CIDEPE), 156 Ciruclar motion acceleration vector, 42 amplitude, 50 analysis of experimental data, 46 53 angular acceleration, 52 angular variation, 44 angular velocity, 44 46, 48 50, 52, 53 astronomy, 41 average acceleration, 43 average velocity, 41, 47 Cartesian coordinates, 46, 47 centripetal acceleration, 44 Copernicus, 41 eternalism, 41 Euclid, 41 experimental development, 45 Inertia, 41 instantaneous acceleration vector, 44 instantaneous angular velocity, 42, 45, 47 least-squares method, 50 linear velocity, 52 Moon, 41 period, 49 Plato, 41 polar coordinate system, 47 significant digits, 47 software Tracker, 46 48 tangential acceleration, 44, 50 uniform circular motion, 41 unitary vectors, 41 vector difference, 41 velocity coordinates, 51 velocity vector, 42, 43 versor, 41 Coefficient of restitution completely inelastic collision, 143, 144 damping and no damping cases, 146, 148 experimental development, 144 145 free software Tracker, 147 graph of velocity, 149 151 impression, 146 mean value, 147, 149 mechanical energy (gravitational potential energy), 149 perfectly elastic collision, 143, 144 snapshots, free software Tracker, 145 time intervals, 152 time measurement, 149 timing, 149 transversal impulse, 146 weighted mean values, 147 Commercial/homemade equipment, 156 Completely inelastic collision, 143, 144 Conical pendulum centripetal acceleration, 93, 95 centripetal net force, 94 circular uniform motion, 93 cylindrical coordinates, 93 dynamometer, 95 99 experimental data analysis, 98 99 free-body diagram, 94 measurement of tension, 98 measurement of the pendulum s length, 97 net force and acceleration vectors, 93 pulley, 97 randomic error, 98 tension s measurement, 99 100 weight and tension, 93 Copernicus, 41 Cosmological models, 41 Cycloid, 113, 114 Cylindrical coordinates, 93 D Damping, 87 Didactical equipment, 14 Dynamics acceleration measurement, 59 acceleration of gravity, 61 air track, 55 57, 64, 65 angular acceleration, 63 angular velocity, 63 experimental development, 55 57 friction, 55 kinematic model, 57, 58
Index 189 kinetic energy, 66 linear fit, 59 maximum possible uncertainty, 58 mechanical energy conservation, 65 67 momentum of inertia, 55, 62, 63, 65 net force, 61 photosensor, 56 58 potential energy, 65, 66 pulley, 55, 56, 59 67 relative error, 61, 64 66 relative uncertainty, 56 resolution, 56 second law of Newton, 61, 63 type A and B uncertainty, 57 Dynamometer, 95 99 E Effective radius of rotation, 135 Elastic collision, 155 Electronic worksheet, 85, 103 Elliptic integral, 84, 85, 87 Errors random errors, 2, 30 relative error, 33, 35, 38, 61, 64 66 systematic errors, 2, 30 Eternalism, 41 Euclid, 41 F Force net force, 8, 9 resultant, 9, 10 Free fall, 13 15, 17, 19, 20, 22, 26, 29, 30, 32, 33, 35 39 Free fall mouse trap, 26 Free-body diagram, 70, 94 Friction, 1, 2, 9, 38, 55 G Galilei, G., 1, 13 Geometric arguments, 120, 121 Gravity acceleration, 13 26 H Horizontal launch analysis of experimental data, 131 133 data analysis, 129 didactical equipment, 129 equations of motion, 129 experimental procedure, 131 launching platform, 130 132 length measurements, 130 linear uniform movement, 129 mechanical energy, 129, 138 142 sphere motion, 129 systematic errors, 131 two-dimensional collision, 130 uniformly accelerated motion, 129 Huygens, C., 89 I Impact parameter of the collision, 160 Inelastic collisions, 156 Inertia, 41 air flow unit, 3 air table, 2, 3, 9 collision and scattering angle of 18.4, 4 5 collision and scattering angle of 81.0, 5 6 drop position dependence on time, 9 first Law of Newton, 1 friction, 1, 2, 9 inertial system, 1 law of inertia, 3 net driving force, 8 net force, 9 Newton s Second Law, 8, 9 Newton s Third Law, 9 OSP, 2 puck, 2 9 random errors, 2 reference system, 1 resultant force, 9, 10 ruler and protractor, 7 software Tracker, 2,4,5,9,10 state of inertia, 1, 3, 8, 9 systematics errors, 2 uniform linear motion, 1 velocity vectors, 7, 8 Inertial system, 1 Instantaneous acceleration vector, 44 Instantaneous angular velocity, 15, 31, 42, 45, 47 K Kinematic model, 57, 58 Kinetic and rolling frictions analysis of experimental data, 124 127 angular velocity, 119, 121 CM, 119 deformable wheel movement, 119 experimental development, 123 124 forces acting on deformable wheel, 120
190 Index Kinetic and rolling frictions (cont.) geometric arguments, 121 kinetic frictional force, 121 kinetic friction coefficient, 121, 123 normal force and the horizontal displacement, 120, 122 normal friction force, 120 pure rolling, 119 rigid body, 119 rolling friction coefficient, 120 123 rolling resistance coefficient, 120, 121 rolling resistance torque, 119 rolling with slipping, 123 rolling without slipping, 121, 123 static friction coefficient, 120 static friction force, 119 121 tire, 119 with and without slipping, 123 Kinetic energy, 66 Kinetic friction coefficient, 121, 124, 127, 140 L Launching platform, 130 134, 138 Law of inertia, 1, 3, 7, 119 Least-squares method, 24 26, 50 Linear momentum, 155, 157, 162, 163, 165 Linear velocity, 52 Linearization, 15 Log-log graph, 76 79 M Maximal possible uncertainty, 18, 19, 33, 58 Mean value, 147, 149 Mean velocity value between the times, 103 Mechanical energy conservation, 65 67, 130, 133 137, 139 Momentum of inertia, 55, 62, 63, 65, 134 Moon, 36, 38, 41 N Napierian logarithm, 77 Net force, 8, 9, 61 Newton, I., 1 Newton s first law, 1 Newton s second law, 8, 9, 71 Newton s third law, 9, 155 Non-linear equation, 84 Normal friction force, 120 O Open Source Physics (OSP), 2 P Particle system, 155 Pendulum acceleration vector, 71 angular frequency, 72 centripetal and tangential accelerations, 71, 83 conical, 93 100 damping, 87 data analysis, 85 92 elliptic integral, 84, 85, 87 experimental data, 73 76 free-body diagram, 70 function procedure, 89 length measured, 72 log-log graph, 76 79 mass, 71, 84 motion, 72, 84 non-linear equation, 84 oscillation, 72, 89, 91 period, 72, 84, 86, 91 polar coordinate, 70 simple harmonic motion, 72, 84 software Tracker, 86, 89 systematic error, 86 theoretical prediction and experimental data, 89 time interval, 86 uncertainties, 72 Perfectly elastic collision, 143, 144 Period, 49, 72 Photosensor, 16, 17, 56 58 Plato, 41 Polar coordinate system, 47, 70, 107 Potential energy, 65, 66 Puck, 2 9 Pulley, 55, 56, 59 67, 97 Pure rolling absolute value of velocity vector, 103, 113 angular velocity, 108, 115 Cartesian components, 103 Cartesian coordinates, 104, 107 109, 112 114 CM, 101 cycloid, 113, 114 linear variation, 109 magnitude of velocity vector, 113 mean velocity value between the times, 103 measurement of time, 103 model, 139 movements of rolling bodies, 101 polar coordinate system, 107, 110, 111 position measurements, 102 reference frame, 111 reference system, 102
Index 191 rolling without slipping, 102 smartphone, 102 software Tracker, 102, 103 static friction force, 101, 109, 116, 117 torque, 101, 109, 116 uniform linear motion, 116 velocity of point, 101 R Random error, 2, 30, 98 Range vectors, 158, 162, 163, 166 Reference frame, 111 Reference system, 1, 102 Relative error, 21, 33, 35, 38, 61, 64 66 Relative uncertainty, 16, 17, 56 Resolution, 16 Resultant force, 9, 10 Rolling friction coefficient, 120 127 Rolling resistance coefficient, 120, 121 Rolling resistance torque, 119 Rolling with slipping, 123, 126 Rolling without slipping, 102, 121, 123, 124 Rotation kinetic energy, 62 S Scattering angles, 157, 163, 164 Scott, D., 36 Second law of Newton, 61, 63, 94 Sensibility coefficient, 33 Significant digits, 19, 21, 32, 47 Simple harmonic motion, 72, 84 Simple pendulum, 69, 72, 73, 77, 83, 85, 89, 92 Slipping motion, 124 Smartphone, 102 Software Tracker, 46 48, 89, 124, 138 State of inertia, 1, 3, 7, 9, 10 Static friction coefficient, 120, 121 Static friction force, 101, 109, 116, 117, 119 121 Stopwatch resolution, 16 Systematic errors, 2, 30 Rension vector, 94 Tension s measurement, 99 100 Time interval, 86 Time measurement, 89 Tire, 119 Torque, 101, 109, 116 Torricelli equation, 22 Tracker software, 2, 4, 5, 9, 10, 29 33, 37, 38, 46 48, 86, 102, 103, 124 Translational kinetic energy, 140 Trigonometric identity, 115 Two-dimensional collision, 130 Type A uncertainty, 17, 57 Type B uncertainty, 17, 18, 57 U Uncertainty maximum possible uncertainty, 18, 19 relative uncertainty, 16, 17 type A uncertainty, 17 type B uncertainty, 17, 18 Uniform circular motion, 41 Uniform linear motion, 116 Unitary vectors, 41 V Vector difference, 41 Velocity coordinates, 114 Velocity vector, 7, 8, 42, 43, 115 Versor, 41 Video analysis, 29 acceleration of gravity (see Acceleration of gravity) circular motion, 41 53 horizontal launch, 129 142 inertia, 1 10 kinetic and rolling frictions, 119 127 mechanical energy, 129, 138 142 pendulum, 83 92 pure rolling, 101 117 T Tangential acceleration, 44, 50 Taylor series, 33 W Weighted mean values, 147