Physics. in the Laboratory. Robert Kingman. Applied Physics Second Edition Fall Quarter 1997

Transcription

1 Physics in the Laboratory Applied Physics Second Edition Fall Quarter 1997 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 1996 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 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 Torque and Angular Acceleration Experiment 8 Conservation of Energy of a Rolling Object Experiment 9 Rotational Equilibrium - Torques ii

5 Applied 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 display these graphs. Click on the rescale icon (fourth from the left in the lower left of the graph window). Experiment 1 Page 5

10 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 timedistance 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 Applied Physics Experiment 2 Part I: Constantly Accelerated Motion - 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. The relation between velocity, v, and time is Experiment 2 Page 7

12 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. 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. Experiment 2 Page 8

13 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 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. Experiment 2 Page 9

14 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 =.... Applied 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 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). Experiment 2 Page 10

15 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 Applied 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: < Five spring balances < Five 1-2 kg weights used for anchors < Small washer < Large sheet of paper < Ruler, protractor, right triangle < Scientific calculator (with sin, cos & tan functions) or Mathcad 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 of the three sides of the right triangle by the use of the Figure 1 The sides of a right triangle. 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. 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 Figure 2 Vector addition by the polygon method. Figure 3 Finding the two perpendicular components of a vector. F x 'F@cos(a) F y 'F@sin(a) (2) The X component of the resultant is the sum of the X components of the vectors being added, and similarly for the Y component. Experiment 3 Page 14

19 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 PF 1 % PF 2 % PF 3 % PF 4 '0 (7) PF 4 '&( PF 1 % PF 2 % PF 3 ) (8) so that forces. PF 4 is equal in magnitude and opposite in direction to the resultant of the other three Procedure: Set up the following situations so that in each case the magnitudes of the forces are unequal. 1. Attach three strings about 12 cm long to the small washer and connect the other end to spring balances, to the end connected directly to the center force indicating shaft. Connect string loops, about 8 cm long, to the other end of the Figure 4 Sample setup of three forces acting on a small washer at equilibrium. Experiment 3 Page 15

20 spring balances and wrap these loops around the 1-2 kg weights (see figure 4). You will need to make sure that when there is no load on the spring balance the scale reads zero. If it does not, you will need to adjust it by sliding the metal tab at the top of the device. a) Move the weights so that the angle between forces F 1 and F 2 is 90E (see figure 5). On a paper (as large as.3 by.3 m, if possible) draw lines parallel to its edges and intersecting near its center. These lines will act as the X and Y axes, described in the Physical Principles section. Position the paper so that the origin of the axes is right under the small washer, with the forces F Figure 5 Procedure 1.a) setup. 1 and F 2 along the two lines. Tape the sheet of paper to the table. Use a pencil to mark two points at opposite ends of the string supplying the force F 1. By connecting these two points, draw a line below the string showing the direction of the force. Following the same procedure, draw the direction of the other two forces. Record the weight on each string in Newtons. For those spring balances calibrated in grams, convert the scale readings by multiplying by 9.80*10-3 N/g. Place arrows on your lines in the direction of the force exerted by the spring balances. Select your X axis to be along the line of force F 1. Add the vectors for F 1 and F 2 both graphically (polygon method) and with trigonometry (component method). 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? 2. Repeat step 1a, using only the component addition method with 4 spring balances (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 Figure 6 Procedure 2 setup. Experiment 3 Page 16

21 requirements of Newton's second law? 3. If time permits, for extra credit, 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 Applied 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 object on the surface and the mass hanging on the string have the same acceleration. Thus, Newton s law of motion for the object on the surface 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 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 Figure 3 Smart pulley. Figure 4 Cart on the air track. Experiment 4 Page 24

29 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. 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). Plot a graph of tension T vs. acceleration a using Mathcad. Use the slope() function of Mathcad to compute the slope of the best fitting line. Use the corr() function to find the closeness of the fit. 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 Applied Physics Experiment 5 Conservation of Mechanical Energy Objective: < To measure kinetic and potential energies of a mass suspended vertically on a spring, and to test the hypothesis that mechanical energy is conserved for a system free of frictional forces. Equipment: < Table clamp, 2m rod, 30 cm rod, right angle clamp < Triple beam balance < Computer with Science Workshop, and Pasco interface box < Ultrasound motion sensor < Hooked masses, 1 kg,.1 kg < Tapered spring < Force transducer < IP 18 power supply with red and black 1.5 m long banana leads 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 is energy is called kinetic energy and is given by KE' 1 2 m@v 2 (1) 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 (component of force along displacement multiplied by the displacement) which must be done to move the object from A to B. In the case of a spring, the force produced is the product of a constant k and the amount by which the spring is stretched. Mathematically, this relation can be written as Experiment 5 Page 29

34 (2) The work done by a force is equal to the area under the force versus distance graph. For a spring, this is a straight line through the origin (see figure 1) so that the work done is the area of the shaded triangle. This work done represents an energy called the potential energy PE that is stored in the spring. Mathematically, the PE of a spring can be written as PE' 1 2 x@f' 1 2 x(k@x) PE' 1 2 k@x 2 (3) Figure 1 Plot of spring force versus distance stretched. Energy Balance When no frictional forces are present KE%PE'E'constant (4) where E is a constant called the total mechanical energy. When an object is supported by a vertical spring, it will come to rest at an equilibrium position. If it is moved up or down from this position, it will oscillate. At the top and bottom of its motion, it will be at rest and will have zero kinetic energy. At these two points its energy is all in the form of potential energy. At the equilibrium point x = 0 and the potential energy is zero. The object is moving fastest there and all its energy is in the form of kinetic energy. As the object moves the spring does positive or negative work causing the kinetic energy to change. The sum of the kinetic and potential energies of the object is constant and is equal to the total energy of the object. The effect of frictional forces is to cause the total energy to gradually decrease. Procedure: Use a triple beam balance to measure and record the mass of the spring m s and the mass of the object m (this mass should be 1 kg). Record the effective mass, m e = m + m s /3. Mount the force transducer at the top of the 2 m rod using a right angle clamp. Mount the 2 m rod on the table near one of the aisle corners using a table clamp. Rotate the force Experiment 5 Page 30

35 transducer so that the transducer beam is horizontal. Run Science Workshop and plug the force transducer into analog channel A on the signal interface box. Plug it in on the screen as well by clicking on the analog plug icon, dragging it over analog channel A, selecting voltage sensor, and clicking on OK. Calibrate the force transducer. To do this, double click on the force transducer icon. Set the low value to 0 and click on the Read button. Hang a 1 kg hooked mass (weight of 9.8N) on the transducer, set the high value to 9.8, click on the Read button, then click on OK. Remove the 1 kg mass from the force transducer. Raise the 2 m rod as high as possible. Place the hook at the small end of the tapered spring over the S hook of the force transducer and support the 1 kg mass from the hook at the large end of the spring. Lower the mass to a point near the equilibrium position of the spring and release it. Directly under the mass position the ultrasound motion sensor (see figure 2 ). Plug the motion sensor into channels 1 (yellow banded plug) and 2 on the signal interface box. Also plug it in on the screen by clicking on the digital plug, dragging it over digital channel 1, selecting motion sensor, and clicking on Figure 2 Setup of the OK. Click on the recording options button, set the periodic apparatus. samples rate to 1,000 Hz, the stop condition to time: 5 seconds and click on OK. Displace the 1 kg mass about.2 m, and release it. After several oscillations, click on REC to begin data collection. After data collection has stopped, make graphs of distance versus time and velocity versus time by clicking on the graph icon, dragging it over the motion sensor icon, selecting position or velocity and clicking on OK. Make a force versus time graph by clicking on the graph icon and dragging it over the voltage sensor icon. Then make a force versus distance graph by first making a force versus time graph and then clicking on the clock below the graph and selecting digital 1 and then position. Next calculate the average displacement of the mass. To do this, click on the position versus time graph and click on the E button. Record the value given for mean (y =) as d ave. Next find the value of k by finding the absolute value of the slope of the force versus distance graph. Click on the E key and then the newe key. Select curve fit and linear fit. The slope will be displayed as a2. Now open up the calculator window which can be found in the Experiment menu. Enter the equation for kinetic energy using the mass and velocities that you measured. To get the velocities for the calculation, click on the input button and select digital 1 and velocity. Enter Experiment 5 Page 31

36 kinetic energy for the calculation name, KE for the short name, and click on the = button. Click on the NEW button and enter the equation for potential energy, using the values for mass and distance that you measured. To get distance, click on the input button and select digital 1 and position. Be sure when you enter distance that you factor in the average displacement by entering it as (position - d ave ). Enter potential energy for the calculation name, PE for the short name and click on = and NEW. Finally, enter the equation for total energy which is KE + PE. To get the KE and the PE, click on the input button and select calculation and KE or PE. Click on the = button. Finally, create graphs of kinetic, potential, and total energy. To do this, first make a graph of position versus time by clicking on the graph icon and dragging it over the motion sensor icon and selecting position. Then, click on the downward pointing triangle on the left side of the graph window and select calculation and the type of energy you are graphing. In the total energy graph window, click on the statistics E button and then calculate percent error from the equation %Err' stdev(e) mean(e) 100 (5) Analysis: 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 compare with the variation of the potential energy? Read the maximum velocity from the velocity versus time graph, compute the maximum kinetic energy, and compare it with the total energy. Read the maximum displacement from equilibrium from the displacement versus time graph, compute the maximum potential energy, and compare it with the total energy. Discuss each one of these in your conclusions. Experiment 5 Page 32

37 Applied 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 33

38 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 two Experiment 6 Page 34

39 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 35

40 Experiment 6 Page 36

41 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 keep moving after the collision? Explain by using theory the observations that you saw. Experiment 6 Page 37

42 Experiment 6 Page 38

43 Applied Physics Experiment 7 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 J net 'T@r&m (4) Experiment 7 Page 39