Physics. in the Laboratory. Robert Kingman. PHYS151 General Physics Third Edition Fall Quarter 1998

Transcription

1 Physics in the Laboratory PHYS151 General Physics Third Edition Fall Quarter 1998 Robert Kingman

2 The author expresses appreciation to the Physics faculty and many students who have contributed to the development of the laboratory program and this manual. Special recognition is acknowledged to professor Bruce Lee for the many years that he taught the General Physics course and the introductory laboratories. Professors Margarita Mattingly and Mickey Kutzner have made substantial contributions to the laboratory and to individual experiment instructions. This manual has become a reality because of the efforts of Joseph Soo and Tiffany Karr for rewriting, editing and taking and including the photographs and the outstanding editorial assistance of Anita Hubin. Copyright 1998 by Robert Kingman

3 Preface It is the purpose of the science of Physics to explain natural phenomena. It is in the laboratory where new discoveries are being made. This is where the physicist is making observations for the purpose of identifying the patterns which may later be fit to mathematical equations. Theories are constructed to describe patterns observed and are tested by further experiment. Therefore, it is imperative that students in an introductory Physics course are introduced to both the existing theories in the classroom and to the ways of recognizing natural patterns in the laboratory. In addition, the effort put into the laboratory experiments will ultimately reward the student with a better understanding of the concepts presented in the classroom. This manual is intended for use in an introductory Physics course. Prior experience in Physics is not a requirement for understanding the concepts outlined within. The book was written with this in mind and therefore every experiment contains a Physical Principles section which outlines the basic ideas used. The analysis of the data may be done on the computer with the aid of Science Workshop and Graphical Analysis programs. These tools are very useful in the generation of graphs and curve fitting.. It is important to keep in mind that the best way to become familiar with new software is to use it a lot, trying more than the minimum required for the completion of the laboratory. The student is required to keep a laboratory journal in which the raw data will be recorded as well as the analysis, any graphs and calculations. The lab write-ups need to be done in ink. A good report should contain the date and time the experiment was performed, the title of the experiment, the name(s) of the partners, the objective(s), a sketch of the apparatus properly labeled, a brief summary of the procedures, all the performed analysis and a conclusion. The purpose of the conclusion is to allow the student to comment on the experiment. In addition, a discussion of the errors and where they might have been introduced, suggestions for modifying the experiment to reduce the possibility of errors and overall suggestions for improving the experiment need to be addressed here. In conclusion, we hope that the experiments in this manual will enhance your understanding of the concepts presented in class and will add pleasure to your journey through this exciting field of Physics. Any comments you may have about the laboratories presented in this book are welcomed and encouraged. Since this is a first edition, we hope that you will overlook any missspellings, omis ions, er r ors and inconsistencies and report such to the author. i

4 Table of Contents Preface i Experiment 1 Uniform Motion - Graphing and Analyzing the Motion Experiment 2 Uniform Acceleration Experiment 3 Vector Addition of Forces Experiment 4 Force and Acceleration - Newton s Second Law Experiment 5 Conservation of Mechanical Energy Experiment 6 Inelastic and Elastic Collisions Experiment 7 Momentum Change and Impulse Experiment 8 Torque and Angular Acceleration Experiment 9 Conservation of Energy of a Rolling Object Experiment 10 Rotational Equilibrium - Torques ii

5 General Physics Experiment 1 Part I: Uniform Motion - Graphing and Analyzing the Motion Objectives: < To observe the distance-time relation for motion at constant velocity. < To make a straight line fit to the distance-time data. < To interpret the slope as the velocity of the motion. < To observe that the average mean square error is smallest for the closest fit. Equipment: < Motion sensor < Pasco 1.2 m track and dynamics cart < Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis software Physical Principles: The position of an object moving along a line is indicated by its displacement. The displacement is ±1 times the Figure 1: Object with displacement +2 from origin. distance of the object from a reference point called the origin, the numbers being positive on one side of the origin and negative on the other side. Denoting the displacement as x and the time as t x'v@t%x o (1) In a graph of x (on vertical axis) versus time (on horizontal axis) the velocity of the motion v is equal to the slope of the line. The initial position, the location at the beginning when time is zero, is x o. This value is where the line crosses the vertical axis and is called the intercept. The best fit of a straight line to a data set is the one with the smallest value of the average square deviation. Experiment 1 Page 1

6 The slope is given by v'slope' rise run ' )x )t ' x 2 &x 1 t 2 &t 1 (2) It is often possible and convenient to take x 1 and t 1 to be zero. Predictions: Draw a rough graph in your journal of what you think the motion will be. Plot the displacement x Figure 2 Slope, v, and intercept, x o. versus the time t. Do this when the cart starts at an initial position of 50 cm and travels for a time of 2 seconds at a speed of 50 cm/s. Will the curve be a straight line or a curved line? If it is straight will it slope up or down. If it is a curved line will it curve up or down? Explain why you think it will behave this way. Do this for two cases and label the graph for each. The two cases are: 1. Motion toward the origin, 2. Motion away from the origin. Procedure: Setup: Plug the motion sensor s phone plugs into digital channels 1 and 2 with the yellow banded plug into channel 1. Place the motion sensor about 40 cm from the end of the track opposite the bumper with the center of the sensor about 12 cm above the track. Align the sensor so that the sound waves will travel directly along the track. Place the cart on the track at the end near the sensor. Data Collection: Double click the left mouse button on the physics labs folder to open it if necessary (it is usually open). Double click on the scwkshp icon in the folder to open Science Workshop. See Figure 3 below. Click and drag the phone plug icon to digital channel 1, choose Motion Sensor. Click on the REC button and at the same time push the cart away. Wait until data collection stops. Drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2. Click on the rescale icon (fourth from the left in the lower left of the graph window). Drag the Table icon onto the Motion Sensor icon. Experiment 1 Page 2

7 Figure 3 Science Workshop window. Click on the clock to the right of the E at the upper left of the Table window to display the times. Click just above the time-distance data columns to select all of the data or click and drag to select the portion of the data that is valid. Under the Edit menu, choose copy to store the data temporarily in the Window s clipboard. Graphing Data: Double click on the VernierGA icon in the physics folder to open the graphing analysis program, click on OK and click on the restore (upper right center icon). Click on the row 1, x data position. Under the Edit menu option choose paste data to copy your data from temporary storage in the clipboard. Analyzing Data: Note that the displacement is plotted vertically (y-axis) and the time data is plotted horizontally (x-axis). Click on the graph of your data on the right to select the graph. Choose Analyze from the main menu and click on Manual Curve Fit. Select the Stock function M*x + B to select a linear (straight-line) model. (According to Eq. (1), the x here corresponds to your time values, the M corresponds to your velocity values, and B corresponds to your beginning location x o ). Experiment 1 Page 3

8 Figure 4 Graphical Analysis window. Change the values in the intercept box B = at the lower left and the slope box M = to vary the intercept and slope of the model line. Note values of the Mean Square Error at the lower right of the graph for each value of slope M and intercept B. Do this until the model line visually fits most closely to the data and then make further adjustments until the Mean Square Error is as small as possible. Record the values of the slope M, intercept B and the Mean Square Error. Click on OK-Keep Fit. In the Main menu click on Analyze and choose Automatic Curve Fit, click on the Stock function M*x + B and click on OK. Click on OK-Keep Fit and record the values of the slope M, intercept B and Mean Square Error. Click on the Linear Regression icon (the rightmost icon under the Data menu) to obtain again a linear fit to the data. Record the regression coefficient. A value close to one indicates a close fit to the line. Compare these values with those obtained in your manual fit Click on the graph title and change the title to Displacement versus Time. Click in the text window and enter your name, experiment name, date and experiment details, ie motion away from detector. Choose File in the main menu, then Print, click on Selected Display and click on OK. How does your observed curve compare with your predicted curve? What is the speed of the cart? How far from the detector is the cart when the detector begins measuring its motion. What does the value of the Mean Square Error indicate? Motion in opposite direction: Experiment 1 Page 4

9 Return to the Science Workshop window and repeat the experiment placing the cart on the end opposite the motion sensor and pushing it toward the sensor. At the same time click on the REC button. Repeat the analysis above. Part II: Walking Motion - Distance Versus Time Graphs Objectives: < To observe the distance-time relations for a variety of walking motions. < To determine from the slope of the distance versus time graph the velocity of the motion at various points. Equipment: < Motion sensor and reflector board < Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis software Predictions: A General Physics student stands in front of a motion detector for three seconds. Then the student backs slowly away from the detector at constant speed for a time of four seconds reaching a distance of two meters from the detector. The student stops for five seconds and then walks toward the detector with constant speed for six seconds reaching a distance of.6 meters from the detector. Finally the student stands at this point for two seconds. Draw a rough graph of what the distance versus time and velocity versus time graphs would look like. Title the graph and label the axes indicating distances in meters and times in seconds. Procedure: Mount the motion sensor on a rod so that it is about five feet above the floor. Align the sensor so that you can walk away from the sensor to a distance of two meters using the reflector to send the ultrasound waves back toward the sensor. Direct the sensor along the path that you will walk. Select the Science Workshop window to activate it. Click on the REC button while your lab partner tries to duplicate the motion that you drew in the prediction section. Click and drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2. Click on Position and then click on Velocity to Experiment 1 Page 5

10 display these graphs. Click on the rescale icon (fourth from the left in the lower left of the graph window). Click and drag the Table icon onto the Motion Sensor icon. Click on the clock to the right of the E at the upper left of the Table window to display the times. Click just above the time-distance data columns to select all of the data or click and drag to select the portion of the data that is valid. Choose Edit and Copy to copy this data to the clipboard. Click on the Vernier Graphing window to activate it. Click on the row 1, x data position, and Paste the data from the clipboard. To change the label for the X axis to Time click on Graph, click on Column Appearance and double click on X. Type Time in the New Name box, click on the New Units box and enter seconds. Click on OK to accept these labels. In the same way change the Y label to Displacement with the units of meters. Click and drag (right on the graph!) to select a region where the velocity is approximately constant. Click on the Linear Regression icon (seventh from the left in the tool bar) to determine the speed in that region. Record the velocity from the slope listed as M =... Repeat this for other regions of interest. To observe the slope at each point click on the slope icon (fifth from the left on the tool bar). Move the mouse along the curve to observe the changing instantaneous velocity (slope at a point). Click in the text window and enter your name, experiment name, date and experiment details, ie motion away from detector. Print the Selected Display. Compare your actual walk with what you had drawn in the prediction section. If you have time make your own walk plan, draw a rough graph of it and repeat the experiment. You might wish to try moving with a constant acceleration as well. What can you conclude about how the sensor is responding? Why is the velocity curve not smooth? Experiment 1 Page 6

11 General Physics Experiment 2 Part I: Uniform Acceleration - Distance, Velocity, and Acceleration Versus Time Objectives: < To observe the distance-time, velocity-time, and acceleration-time relations for a cart moving up and down an inclined track. < To determine from the slope of the distance versus time graphs the velocity of the cart at various points. < To determine from the slope of the velocity versus time graphs the acceleration of the cart at various points. < To compare distance traveled by the cart with the area under the distance-time graph. Equipment: < Motion sensor and reflector 2x4 < Pasco dynamics track and cart < Lab jack < Computer with Signal Interface, Science Workshop and Vernier Graphical Analysis software Physical Principles: The position of an object moving along a line is indicated by its displacement. The displacement, x, is ±1 times the distance of the object from a reference point called the origin, the numbers being positive on one side of the origin and negative on the other side. When the velocity changes in time (acceleration) the graph of x versus t is no longer a straight line. However the instantaneous velocity of the motion, v, is equal to the slope of the tangent line at that time. The initial position, the location at the beginning when time is zero, is x o. For a constant acceleration the relation between x and t is x'x o %v 1 2 a@t 2 (1) where a is the acceleration and v o is the initial velocity. Experiment 2 Page 7

12 The relation between velocity, v, and time is v'v o %a@t (2) which is the equation of a straight line with a slope, a, and intercept, v o. Predictions: A physics student pushes the dynamics cart up the inclined track and observes its distance-time motion with a motion sensor. Draw a rough graph of what the distance versus time and velocity versus time graphs would look like. Title the graph and label the axes indicating distances in meters and times in seconds. Is the velocity zero at any point? Is the acceleration zero at any point? Figure 1 Graph of velocity vs. time. Procedure: Setup: Plug the motion sensor s phone plugs into digital channels 1 and 2 with the yellow banded plug into channel 1. Place the 2x4 reflector upright on the dynamics cart and secure it with a rubber band. Elevate one end of the track by placing the lab jack under one end. Set the lab jack to its lowest position. Place the cart on the low end of the track. Mount the motion sensor on a stand so that it is about eight inches above the table top. Align the sensor so that it is directed down along the track and toward the 2x4 reflector on the cart. Data Collection: Double click the left mouse button on the physics labs folder to open it if necessary (it is usually open). Open Science Workshop. Click and drag the phone plug icon to digital channel 1, click on Motion Sensor and then OK. Click on Sampling Options, set the sampling rate to 10,000 Hz and the sampling time to 3 seconds. Press RETURN and click on OK to accept these values. Click on the REC button while your lab partner gives the cart a quick thrust up the track. Experiment 2 Page 8

13 BE CAREFUL NOT TO SEND THE CART OFF THE TOP END OF THE TRACK. Click and drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2. Click on Position, then click on Velocity and click on Acceleration to display these graphs. Click on the rescale icon (fourth from the left in the lower left of the graph window). Click on the Restore icon in the upper right of the Science Workshop window to fill your computer screen with the Science Workshop window. Acceleration from a quadratic fit to the distance-time data Click and drag to select a region where the distance-time data is in the shape of a parabola. Do not include the points near the top where the curve begins to flatten. Click on the E to the right of this graph and drag the mouse down to Curve Fit. Then click on Polynomial Fit. The constant a 3 should be ½ of the acceleration as you can see from Eq. (2). Record this valueof a 3 and multiply it by 2 to obtain the acceleration from this fit. Acceleration from the slope of the velocity-time data Click and drag to select a region of the velocity-time graph where the velocity is approximately constant. Click on the E to the right of this graph and drag the mouse down to Curve Fit. Then click on Linear Fit. Record the acceleration from the slope listed as a 2 =... and note that it is from the slope of the velocity-time graph. Acceleration from the mean of the acceleration-time data Click and drag to select a region of the acceleration-time graph where the acceleration is approximately constant. Click on the E to the right of this graph and drag the mouse down to Mean. Record the acceleration from the mean of the y data and note that it is from the slope of the velocity-time graph. Compare the three values for the acceleration that you obtained from the three graphs. Compare the graphs that you obtained with those that you drew in your predictions. Click on the lower left icon in the graphing window and change the title of the graph to Distance, Velocity, and Acceleration vs Time by (enter your name) and press RETURN. Click on File and then Print to print the Graph window. What are the answers to the questions in the predictions section? Comparison of the velocity to the slope of the distance-time graph Click and drag to select about three data points on the distance-time graph. Click on the E to the right of this graph and drag the mouse down to Curve Fit. Then click on Linear Fit. Record the velocity from the slope listed as a 2 =... and the time at the midpoint of the small time interval. This is an approximation to the slope of the distance-time curve at Experiment 2 Page 9

14 the midpoint. Click on the exam icon nest to the E at the lower left of the Graph Display window and move the cross hair so that it is on the velocity-time graph at the midpoint time. Record the velocity and time values and compare the velocity value with that obtained from the slope of the distance-time graph at that time. Comparison of distance traveled to the area under the velocity-time graph Click on the zoom icon (bottom left) then click and drag on the velocity-time graph to zoom in on a region that includes all the positive velocity data. Click and drag on the positive velocity data, then click on the E at the right and click on Integration. The area under this region of the velocity-time graph is displayed. Click on the cross-hair examine icon at the bottom left and move the mouse so that the cursor is at the left edge of the gray shaded region in the velocity-time graph and the cross is on the distance-time graph. Read the initial position from the distance (y) axis and record this value in your journal. Repeat this process at the right edge of the gray shaded region. Compare the difference of these two values, the distance traveled with the area under the curve displayed as area =.... General Physics Experiment 2 Part II Uniform Acceleration - The Acceleration of Gravity Objectives: < To test the hypothesis that the acceleration of gravity is approximately constant and to measure its value. Equipment: < Timer photogate < Motion Sensor < Computer with Science Workshop and Vernier Graphical Analysis software < Ball Physical principles: A specific case of the equation (1) is the free fall of a body initially at rest. Since the body is initially at rest, v o becomes zero. If we assume that d o is also zero, the equation becomes Experiment 2 Page 10

15 d' 1 2 (3) where the acceleration is now symbolized by g, the acceleration of gravity. Velocity can be found as a function of time according to equation (2). v'v o %gt (4) It is obvious from equation (4) that on a graph of velocity vs. time, the slope represents the acceleration. In this experiment such a method is utilized to estimate the value of g, the acceleration of gravity. Procedure: The Acceleration of Gravity with a Motion Sensor and Falling Ball Plug the motion sensor into channels 1 (plug with yellow band) and 2 on the signal interface box. The sensor should be attached to a stand 1b meters above the floor. On the screen, click on the digital plug icon and drag it over digital channel 1 of the signal interface box, select motion sensor and click on OK. Enlarge the graph to the desired size. Now click on the recording options button, make sure that periodic sampling rate is set to 10,000 Hz and stop condition to time-2seconds, then click on OK. Hold the ball directly beneath the motion sensor so that it is almost touching it. Release the ball and as it is falling click on the record button. In the graph display window click on the rightmost button of the upper row of buttons in the bottom left hand corner of the window to make the graph zoom in on the data. You will notice that there are a few regions where the graph increases steadily in a smooth, sloped line. Highlight one of these regions by clicking at one end of it and dragging to the other, creating a box around it. Next click on the button with the E symbol on it. A new box will appear on the right side of the graph window. Click on the E button in this window, then select curve fit, then linear fit. The acceleration of gravity will be given by this curve fit s slope (a2). Record your value for the acceleration due to gravity and compute the percentage of error using g = 9.81 m/s 2 as the standard: %Err' g&g standard g standard 100%. Experiment 2 Page 11

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17 General Physics Experiment 3 Vector Addition of Forces Objective: < To test the hypothesis that forces combine by the rules of vector addition and that the net force acting on an object at rest is zero. Equipment: < Pasco force table with four pulleys < Hooked weight set < Dual Range Force Sensor with force table bracket < Ruler, protractor, right triangle Physical principles: Definitions of Sine, Cosine, and Tangent of an Angle Consider one of the acute (less than 90E) angles, a, of the right triangle shown in figure 1. As a result of where they reside, the three sides of the triangle are called the opposite side, adjacent side and hypotenuse. The two sides that make up the right angle (exactly 90E) are always the adjacent side and the opposite side. As a result, the length of the hypotenuse is always greater than the length of each of the other two sides but less than the sum of the lengths of the other two sides. The size of the angle a can be related to the length Figure 1 The sides of a right triangle. of the three sides of the right triangle by the use of the trigonometric functions Sine, Cosine and Tangent, abbreviated sin, cos and tan, respectively. They are defined as shown below. Experiment 3 Page 13

18 sin(a)' opposite hypotenuse cos(a)' adjacent hypotenuse (1) tan(a)' opposite adjacent Vector Addition Polygon method - Vectors may be added graphically by repositioning each one so that its tail coincides with the head of the previous one (see figure. 2). The resultant (sum of the forces) is the vector drawn from the tail of the first vector to the head of the last. The magnitude (length) and angle of the resultant is measured with a ruler and a protractor, respectively. Note: In order to measure the angle, a set of axes must first be defined. X axis Figure 2 Vector addition by the polygon method. Component method - Vectors may be added by selecting two perpendicular directions called the X and Y axes, and projecting each vector on to these axes. This process is called the resolution of a vector into components in these directions. If the angle a that the vector makes from the positive X axis, is used (see figure 3), these components are given by F x 'F@cos(a) F y 'F@sin(a) (2) a Fx = F cos(a) Figure 3 Finding the two perpendicular components of a vector. F X axis The X component of the resultant is the sum of the X components of the vectors being added, and sim Experiment 3 Page 14

19 ilarly for the Y component. R x ' j F x R y ' j F y (3) The angle that the resultant makes with the X axis is given by a'arctan R y R x (4) and the magnitude is given by R' R 2 x %R 2 y (5) Equilibrium Conditions Newton's second law predicts that a body will not accelerate when the net force acting on it is zero. So, for an object to be at rest, the resultant force acting on it must be zero. In equation form, the above statement can be written j P F'0 (6) Thus, if four forces act on an object at rest, the following relationship has to be satisfied. An equivalent statement is so that PF 4 PF 1 % PF 2 % PF 3 % PF 4 '0 (7) PF 4 '&( PF 1 % PF 2 % PF 3 ) (8) is equal in magnitude and opposite in direction to the resultant of the other three forces. Procedure: Set up the following situations so that in each case the magnitudes of the forces are unequal. Callibration of the force sensor. Attach five strings about 25 cm long to the white ring and tie loops on the other ends. Connect the din plug from the force sensor into analog channel A of the Science Workshop Interface. Open the Science Workshop software, click and drag the din connector Figure 4 Sample setup of three forces in equilibrium Experiment 3 Page 15

20 icon onto channel A and double click on Voltage Sensor. Double click on the voltage icon below channel A Enter 0 in the Low Value box on the left. Click on the Low Value Read to enter the voltage for zero force. Support 400 g of hooked weights from the end of one of these strings and connect a second one to the force sensor(see figure 4). Be sure that the pulley plane and string are perpendicular to the end of the force sensor. Enter 3.92 in the High Value box on the left. Click on the High Value Read to enter the voltage for the 3.92 N force. Click on OK to accept the calibration. Click and drag the digits icon onto the voltage icon below channel A. Double click on the large digits and click on the Digits Right box. Enter 3 to display three digits to the right the decimal. Click on OK. The value of the force is displayed. a) Support hooked masses of 300 g and 400 g from strings over the pulleys so that F 1 = 2.94 N, F 2 = 3.92 N and the angle between forces F 1 and F 2 is 90E (see figure 5). The force F 3 is the value displayed by the force sensor. Enter in table 1 the magnitude and directions of each of the three forces. Select your X axis to be along the line of force F 1. Make a sketch in your journal showing these forces as arrows and write the values of each force alongside its arrow. Add the vectors Figure 5 Procedure 1.a) setup. F 1 and F 2 graphically (polygon method) and enter the values in table 1. Complete table 2 using the same data and add the vectors F 1 and F 2 using the component method. Note that if x is taken to be along F 1 its x component equals F 1 and its y component is zero. Then the y axis is along F 2 and the x component of F 2 is zero and the y component equals F 2. Compare the magnitude of the resultant with that of the force, F 3 for both solutions. Using a protractor, measure a 3 and compare it with the similarly measured angle of your graphical addition and your trigonometrically computed angle. Do your measurements satisfy the requirements of Newton's second law? b) Repeat as outlined in part (a) using the component method only, but with the angle between F 1 and F 2 at about 120E. Do your measurements satisfy the requirements of Newton's second law? Experiment 3 Page 16

21 2. Repeat step 1a, using only the component addition method with 4 forces(see figure 6). Draw the forces F 1, F 2, F 3, and F 4 approximately as illustrated. Find and add the components of F 1, F 2, and F 3. Compute the magnitude and direction of the sum of these forces and compare your result with a 4 and F 4. Do your measurements satisfy the requirements of Newton's second law? Figure 6 Procedure 2 setup. 3. Repeat as in step 2 using 5 forces extended approximately as illustrated in figure 7. Do your measurements satisfy the requirements of Newton's second law? Figure 7 Procedure 3 setup (extra credit). Experiment 3 Page 17

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23 Recording data: Part 1a. Table 1 Polygon Method Force Magnitude (N) Angle (E) Force 1 Force 2 Force 3 Resultant of 1 & 2 Table 2 Component Method Direction Force 1 Force 2 Resultant X Y Magnitude of resultant = Angle of resultant = Part 1b. Table 3 Component Method Direction Force 1 Force 2 Resultant X Y Magnitude of resultant = Angle of resultant = Experiment 3 Page 19

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25 Part 2 Table 4 Component Method Direction Force 1 Force 2 Force 3 Resultant X Y Magnitude of resultant = Angle of resultant = Part 3 (Optional - Extra credit) Table 4 Component Method Direction Force 1 Force 2 Force 3 Force 4 Resultant X Y Magnitude of resultant = Angle of resultant = Experiment 3 Page 21

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27 General Physics Experiment 4 Force and Acceleration - Newton's Second Law Objective: < To observe the relationship between force and acceleration and to test the hypothesis that the force is equal to the mass times the acceleration. Equipment: < Track with cart, accessory weights < Smart pulley timer < Table clamp < Triple beam balance < Slotted weights, one 10 g, and three 20 g Physical principles: A net force, F, applied to an object with a mass, M, will cause the mass to accelerate with an acceleration, a. Newton's law of motion asserts that the net force is directly proportional to the acceleration produced. The proportionality constant is denoted by the inertial mass, M. In equation form, this law can be written as F'M@a (1) Figure 1 Free body diagram of the hanging mass. When an object with a mass M, on a smooth horizontal surface, is connected by a string over a pulley to another mass m, a tension is created in the string. This tension is the force that accelerates the object on the surface. From the free body diagram shown in figure 1, it can be deduced that the total force acting on the mass is the tension in the string minus the force of gravity. Assuming that the mass of the hanging weight is m, and its acceleration is a, the following equation can be written. The acceleration of gravity is symbolized by g. m@g&t'm@a (2) Experiment 4 Page 23

28 Equation (1) can be solved for tension to yield the following equality. (3) Since the tension is constant in the string, the cart and the mass hanging on the string have the same acceleration. Thus, Newton s law of motion for the cart is Figure 2 Free body diagram of the cart. T'M@a (4) Procedure: Place the cart on the track and level the track so that the cart does not accelerate in either direction. With a table clamp, position the smart pulley (see figure 3) at the aisle end of the track. Measure and record the mass of the cart, M cart. Measure and add the masses of the two blocks to the mass of the cart. Record the total mass, M total. Connect a string to the paper clip or wire loop on the front of the cart and place it over the pulley at the end of the track. Make a loop on the other end of the string and slip a 10 g slotted Figure 3 Smart pulley. weight into it. The length of the string should keep the mass about 5 cm above the floor when the cart is at the track bumper. Following the procedure below, obtain the value of the acceleration. Run Science Workshop. Plug in the smart pulley on the screen by clicking on the digital plug icon, dragging it over digital channel 1 and selecting smart pulley (linear). Click on OK. Click on the recording options button, set periodic samples to 10,000 Hz, and click on OK. Make a graph of velocity versus time by clicking on the graph icon, dragging it over the smart pulley icon, selecting velocity, and clicking on OK. Start statistics by clicking on the E button in the graph window, then click on the E button in the new window annd select curve fit and then linear fit. Position the cart so that the small slotted weight is just below the smart pulley. Experiment 4 Page 24

29 Release the cart, click on the REC button, when click on Stop just before it reaches the end of the track. Record the value of the slope (a2) from the statistics section of the graph window in the column entitled a in Table 1. This value represents the acceleration of the cart system. Take a series of seven (7) more measurements each time increasing the mass at the end of the string by 10 g. Be sure to delete the previous run between measurements by clicking on Run #1, pressing the delete key on the keyboard and clicking on OK. Record the acceleration for each mass in Table 1. Analysis of Data: Complete column 3 of both tables by calculating the values for g-a. For the value of g use the accepted value of 9.81 m/s 2. Compute the values for T by using equation (3). Use Graphical Analysis to plot a graph of tension T vs. acceleration a. To fit the data to a straight line click on the Regression line icon below and to the right of the Data menus item. Rrecord in your journal the slope M and the value of the correlation coefficient COR. As equation (4) predicts, this slope should be very close to the mass of the cart (M cart ) in the case of Table 1. In the case of Table 2 the slope should be very close to the total mass (M total ). Calculate the percent error for both cases by using %Err' *slope&m* M 100 (5) In your conclusions you should: Discuss the percent error that you calculated for both graphs. Interpret the value of the Correlation Coefficient. Examine how the presence of constant frictional force would affect the results of the experiment. Speculate on the origin(s) of error. Mention what you learned in this experiment. Include any additional comments that you think are essential. Experiment 4 Page 25

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31 Recording data: M cart = M total = M cart + M blocks = Table 1 Cart without additional mass data m a g-a T 10g 20g 30g 40g 50g 60g 70g 80g Slope of the Tension vs. Acceleration line = %Err = Table 2 Cart with two blocks data m a g-a T 10g 20g 30g 40g 50g 60g 70g 80g Slope of the Tension vs. Acceleration line = %Err = Experiment 4 Page 27

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33 General Physics Experiment 5 Conservation of Mechanical Energy Objective: > To measure kinetic and potential energies of a simple pendulum and test the hypothesis that the total mechanical energy is conserved for a system involving only conservative forces. Equipment: > Mass suspended from string approximately 1.5 m in length > Meter stick > Video camera on tripod > Computer with video capture board > Videopoint software Physical Principles: Kinetic Energy A body which has a mass m and moves with a speed v has energy by virtue of its motion. This energy is called kinetic energy and is defined as KE' 1 2 mv 2. (1) At any instant of time in the swing of a pendulum, the velocity vector, v, can be broken up into its x and y components, v x and v y, respectively (see figure 1). The magnitude of the velocity vector is related to the components by the Pythagorean theorem v 2 ' v 2 x %v 2 y. (2) Figure 1: Velocity components for a swinging pendulum. Experiment 5 Page 29

34 Gravitational Potential Energy A body moving in a force field has energy by virtue of its position. This energy is called potential energy. The potential energy of an object at a point B with respect to a point A is the work which must be done to move the object from A to B. In the case of a pendulum the work is done against gravity, so that the gravitational potential energy is given by PE'mgy (3) where y is the height of the pendulum bob above some reference level. Conservation of Total Mechanical Energy When no non-conservative forces (e.g., frictional forces) are present, the total mechanical energy is conserved, that is, E Total ' KE%PE'constant. (4) When the pendulum bob is suspended from a string, it will come to rest with the string at the vertical position (equilibrium). When displaced slightly and released, the pendulum will oscillate about the equilibrium position. At the top of its swing, the pendulum will be briefly at rest with zero kinetic energy and a maximum in potential energy. As the pendulum swings through the equilibrium position, the kinetic energy is a maximum and the potential energy is a minimum. At any point in its swing, the sum of the kinetic and potential energies remains constant. The effect of frictional forces is to cause the total energy to gradually decrease as it dissipates into heating the surroundings. Procedure: I. Video Capture Set the pendulum against a light colored background and lay a meter stick directly underneath the bob to indicate the scale of distances in the video picture. Double click on the video capture icon on the computer screen. From the menu choose Options, then Video Source. Specify S-Video as your video input. Toggle the preview button on (it looks like a video camera with film showing p ). Make sure the video camera is directly facing the pendulum, not leaning at an angle, Experiment 5 Page 30

35 connected to the S-Video input of the video capture board, and on. Set the pendulum into vigorous oscillation and note the video in the preview screen. Make sure that you are far enough from the pendulum to take in the full swing. Click on the video capture button (it looks like a video camera with movie film). Click on the STOP button after a couple of full swings. (Don t record for more than a few seconds or the file will become unmanagably large!) Save the video file with your name in the physvid folder by clicking on the icon that looks like a diskette. The downarrow on the right enables you to access the physvid folder as part of the C: drive. The file is in AVI format (with a.avi extension in the name) but needs to be in Quicktime format (with a.mov extension) for analysis. The conversion is performed by choosing the avi-quick conversion icon on the computer desktop. The source movie is the AVI movie you created and the target movie is a file of the same name but with a.mov extension to the name. Double-click on the avi-quick conversion icon. Browse for your source movie (it should be in the physvid folder of the C: drive. The downarrow can be used to indicate an AVI file for the source. Browse in a similar way for the target file but choose the MOV format. Click on the Start button to initiate the conversion process. Close the conversion program window. Open the physvid folder on the desktop and double-click on the MOV file to see that you captured a full swing. Drag and drop your MOV file onto the folder shared by the Network. Collecting Data from Video Return to your computer station in the laboratory and click on the videopoint program icon. Close the introductory window. Open your movie from the shared Network folder. Indicate that you will be tracking the location of only 1 object. Starting near the top of a swing, center the cursor on the pendulum bob and click. The movie automatically advances to the next frame and records the location of the bob in the table. If the point you clicked on is not marked with a red marker, type Ctl-T. After taking data on the location of the bob for an entire swing, you must scale the picture using the ruler icon in the left column. Choose 1 meter for the length, then click on the two ends of the meter stick in your video. Experiment 5 Page 31

36 Analyzing Data: Start the Vernier Graphical Analysis program. Change the titles of the first two columns to t and x by double-clicking on the titles. Add a third column by clicking on the new column icon (second from right end). Title the new column y. Copy and paste the three columns of data from the table in Videopoint to the table in Graphical Analysis. Examine plots of x vs. t and y vs. t to see that they are reasonable. Calculate the x-component of velocity by taking the derivative (slope) of the x data with respect to time. Select the last icon on the right to create a new column that is to be computed from data in the x column. Call the new column vx (x-component of velocity). The functional form for this column is derivative ( x, t ). Calculate the y-component of velocity in a new column in a similar manner. Calculate the kinetic energy per unit mass (1/2mv 2 )/m by creating a column (titled KE ) defined by the relation 0.5*( vx ^2+ vy ^2). Calculate the potential energy per unit mass (mgy)/m by creating a column (titled PE ) with 9.80* y. To make it more tidy, you can scan through your values in the y column to find the lowest equilibrium point and subtract that value, y o, from g(y-y o ) in the potential energy column. This assigns y o as the point of zero potential energy. Calculate the total mechanical energy per unit mass E tot /m by creating a column (titled E ) which sums columns KE and PE. Plot columns KE, PE, and E on the same graph. Print your Graph! Draw a rectangle about the points on the graph of total energy and choose analyze, then statistics from the menu. The standard deviation is an indicator of the amount of spread in the total energy from the mean value. Calculate the percent error from the equation %Err' stdev(e) mean(e) 100% Further Investigation: What is the period of the oscillation? How does the variation of the total energy compare with the variation of the kinetic energy? How does the variation of the total energy Experiment 5 Page 32

37 compare with the variation of the potential energy? Experiment 5 Page 33

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39 General Physics Experiment 6 Inelastic and Elastic Collisions Objectives: < To observe the conservation of momentum during collision processes. < To test that in elastic collisions the kinetic energy is conserved. < To test that in inelastic collisions the kinetic energy is not conserved. Equipment: < Two Pasco photogate timers < Pasco interface and personal computer < Two carts and blocks < Triple beam balance Physical Principles: It can be said that the impulse acting on an object is equal to the change in momentum of the object. In mathematical form, this can be written as PF@)t')Pp'PI (1) where I is the vector impulse, the product of the force and the time that the force acts on the system. When the force is varying in time, this expression gives the impulse imparted in a short time, and the total impulse is just the vector sum of these or the area under the force versus time graph. When the system consists of several parts, the force in equation (1) is the vector sum of the individual forces and the momentum is the vector sum of the moments of all parts of the system. From equation (1) it can be deduced that if there is no force acting on the system (constant velocity), the initial and final momenta must be equal, to make the change ()) in momentum zero. When two objects collide with no external force acting on the system and the total kinetic energy KE of the setup is conserved, it is said that an elastic collision has occurred. The total KE of the system is the sum of the KE of all the moving parts. An inelastic collision is defined as a collision when the total KE is not conserved. In general, an inelastic collision occurs when the objects attach to each other. In the case of one dimensional motion, that is all motions occur along a line, and with no net external force acting on the system, the initial and final momenta for the case of inelastic collision, can be equated. Experiment 6 Page 35

40 m 1 %m 2 '(m 1 %m 2 )@v f (2) If one of the carts is initially at rest (say m 2 ), then the equatin (2) can be rewritten as m 1 '(m 1 %m 2 )@v f (3) The initial KE of the system consists of only the initial KE of m 1 and is KE i ' 1 2 m 2 1 (4) The final KE can be related to the initial KE by a series of steps involving equations (3) and (4), as follows. KE f ' (m 1 %m 2 )@v 2 f 2 '(m 1 %m 2 )@ m (m 1 %m 2 ) 2' 1 2 ( m 1 m 1 %m 2 )@m 2 1 ' m 1 m 1 %m i (5) It can be seen that the initial and final KE are not equal. Procedure: Place a cart on the track and level the track so that the cart does not roll in either direction. Place the photogates at 70cm and 140 cm along the track and plug them into slots 1 and 2 on the Pasco computer interface. The left photogate should be connected to slot 1. Also, connect the two photogates to the interface on the screen by dragging the digital plug icon to channels 1 and 2 and selecting photogate and solid object. Adjust the photogate heights so that the beam is blocked by the blocks on the cart when the blocks are placed on their sides so that they are taller. Measure and record the lengths of the blocks on the carts. Make sure that you place 1 block on the right cart and record it as L 2 and 2 blocks on the left cart and record it as L 1. Under recording options, set the periodic sample rate at 10,000 Hz. Finally, make a table for each gate by dragging the table icon over its respective icon and selecting time to clear gate. In each of the following collisions, make sure that the carts are moving freely before they enter the photogates and that the collisions occur when the carts are entirely between the Experiment 6 Page 36

41 two photogates. Use the triple beam balance to determine the masses of the carts, including blocks (see figure 1) and record these values as m 1 and m 2 (m 1 >m 2 ). 1. Inelastic collisions, m 1 > m 2, cart 2 at rest: Place cart 2 at rest and midway between the photogates with the velcro end to the left. Click on record, then push cart 1 towards the photogate and cart 2. Let the combined setup (cart 1 and 2) go all the way through the second photogate and then click on stop. Make sure the collision occurs between the two photogates. On the table for the first photogate, there is one value displayed. This is the time it took cart 1 to pass through the first photogate. Record this value as t I. On the second table there are two values displayed. The first one is the time it took cart 2 to pass through he second photogate and should be recorded as t f. Now click on Run #1 and delete it by pressing the delete key on the keyboard. 2. Elastic collisions, unequal masses m 1 >m 2, cart 2 at rest: Turn cart 2 around so that the non velcro end faces cart 1. Place cart 2 between the photogates, click on record, and then send in cart 1 from the left. Cart 2 will pass through the right photogate (stop it and remove it as soon as it passes completely through the photogate) followed a short time later by cart 1. Click on stop after cart 1 passes through the second photograte. Record the one value displayed in the table for photogate 1 as t 1, the first value in the table for photogate 2 as t 2 and the second value in this table as t 3. Experiment 6 Page 37

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43 Recording Data: L 1 = m 1 = L 2 = m 2 = Inelastic collision Compute and record the initial and final kinetic energies in joules by filling in the table below. Compare the ratio of Ke f /KE i from the table to the prediction of equation (5). Compute, record, and compare the initial and final momenta of the carts in Newton seconds. t 1 t f v 1 v f p i p f KE i KE f Elastic collision Complete the following tables. Compare the initial and final momenta in Newton seconds and the initial and final kinetic energies in joules for the collision. t 1 t 2 t 3 v 1 v 2 v 3 p 1 =p i p' 1 p' 2 p' f KE 1 =K E i KE' 1 KE' 2 KE f Compare the total initial momentum with the total final momentum and also the total initial kinetic energy with the final total kinetic energy. For the elastic collision of two equal masses qualitatively analyze what happens. Observe the final velocity of cart 2 and compare that with the initial velocity of cart 1. Does cart 1 Experiment 6 Page 39

44 keep moving after the collision? Explain by using theory the observations that you saw. Experiment 6 Page 40

45 General Physics Experiment 7 Momentum Change and Impulse Objective: < To measure the change in momentum of a hot wheels car in a collision and the impulse acting on the car during the collision and to compare these values. Equipment: < Pasco photogate timer < IP 18 power supply < Right angle clamp < Weights used for anchor < Hot wheels car and track assembly < Vernier caliper < Pasco interface and personal computer Physical Principles: Momentum and Impulse A body with mass m and velocity v has momentum by virtue of its motion. This momentum is given by p'm@v (1) Starting from Newton's second law of motion a relationship between force, momentum and time can be found, as summarized in equation (2) PF'm@Pa'm )Pv )t ' )(m@pv) ' )Pp )t )t (2) so that the force has the meaning of the change in momentum over elapsed time. Multiplying Equation (2) by )t gives PF@)t')Pp'PI (3) where I is the vector impulse, the product of the force and the time that the force acts on Experiment 7 Page 41

46 the system. When the force is varying in time, this expression gives the impulse imparted in a short time, and the total impulse is just the vector sum of these or the area under the force versus time graph. When the system consists of several parts, the force in equation (2) is the vector sum of the individual forces and the momentum is the vector sum of the moments of all parts of the system. Procedure: Impulse and momentum for the hot wheels Position the hot wheels track with the bar end overhanging the edge of the table by a few centimeters. Secure it by placing one end of a lead brick on the stabilizing board on the other end. Clamp the force transducer on the top of the U-bar with the end of the gauge beam about.5 centimeters above the track, so that the car's foam bumper is centered on the plastic wedge on the transducer beam (see figure 1). Be certain that the power supply for the transducer is providing 9 volts 1 (see figures 2 and 3). Turn the battery switch on the transducer to off and plug the signal cable into the output jack. Insert the din plug into the A channel of the interface box and in Science Workshop, plug in the transducer by clicking on the analog plug and dragging it over analog channel A, selecting voltage sensor and clicking on OK. To calibrate the force transducer position the force transducer so that it is clamped to the vertical part of the bar and the beam is parallel to the ground with the string hanging straight down. Double click on the voltage sensor icon. Set the low value at zero and click on Read. Hand a 1kg mass from the string and when it has stopped moving, set the high value at 9.8 and click on Read then on OK. Return the force transducer to its original position. Click on the REC OPT button, set the periodic samples rate at 5,000 Hz, and click on OK. Figure 1 Force transducer. Figure 2 IP18 power supply. 1 To measure the voltage with the volt meter just set the dial in the center to the 20 V position. Connect the positive terminal of the power supply to the terminal labeled V on the meter and connect the negative terminal of the power supply to the common terminal of the meter. Experiment 7 Page 42

47 Data collection. Set the photo gate in front of the transducer beam so that the car passes completely through the beam before it collides with the transducer beam. Plug the photogate into digital channel 1 and plug it in on the screen by clicking on the digital plug icon, dragging it over digital channel 1, selecting photogate solid object and clicking on OK. Click on the record button and release the car from a point near the top of the track. Click on stop after it has hit the transducer beam and come back trough the photogate. Click on the table icon, drag it over the photogate icon and select time to clear gate. Record t i and t f, the first and second values in the table. Now create a graph of force versus time by clicking on the graph icon and dragging it over the voltage sensor icon. Click on the upper rightmost button in the lower lefthand corner of the graph window to fit the graph to the data. Zoom in on the jump in the graph by clicking repeatedly on the magnifying lens with the plus sign in it in the bottom right corner of the window. Continue to zoom in until the jump in the graph is wide enough to be clearly seen. To find the area under this jump, click on the statistics (E) buttons and select integration. Highlight the region right around the jump by clicking at one corner of it and dragging the rectangle until it encompasses the whole jump. The value now given for area is the impulse. Using the vernier calliper measure and record the length L of the car vane. Using a triple beam balance, measure and record the mass m of the car. Analysis: Complete Table 1 and compare the change in momentum ()p = p f - p i ) computed from the mass and velocities with the impulse or area under the force versus time graph as computed by the Science Workshop program and read off the computer screen. Calculate the error according to the equation Recording Data: %Err' impulse&)p 100 (4) impulse p i '&m L t i p f 'm L t f t i t f p i p f )p=p f - p i Impulse Experiment 7 Page 43

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49 General Physics Experiment 8 Torque and Angular Acceleration Objective: < To observe the relationships between the torque and angular acceleration and the angular impulse and angular momentum. Equipment: < Rotating table < Ring and disk < Slotted and hooked weight sets < Pulleys < Pasco Photogate timer < Vernier caliper Physical Principles: Consider the setup shown in figure 1. A net force of m@g&t acts on the mass m hanging on the string with a tension T. According the Newton s Second Law equation (1) Can be written. m@g&t'm@a (1) Solving equation (1) for T yields Figure 1 Main setup for the experiment. T'm@(g&a) (2) This tension acting tangent to the rotating table drum with radius r produces a torque J'T@r'm@r@(g&a) (3) There is a frictional torque J f 'm where m o is the mass on the string required to keep the table rotating without acceleration. Newton's law for rotational motion asserts that Experiment 8 Page 45