Physics. in the Laboratory. Robert Kingman and Gary W. Burdick. PHYS130 Applied Physics for the Health Professions First Edition Spring Semester 2001

Transcription

1 Physics in the Laboratory PHYS130 Applied Physics for the Health Professions First Edition Spring Semester 2001 Robert Kingman and Gary W. Burdick January 15, 2001

2 The authors express 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 2001 by Robert Kingman and Gary Burdick

3 Preface It is the purpose of the science of Physics to explain natural phenomena. It is in the laboratory where new discoveries are made. This is where the physicist makes observations for the purpose of identifying patterns which later may be fit by 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 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 form a permanent record of your work, and must be done in ink. Attached is a copy of the Lab Evaluation form which should be placed at the end of each laboratory writeup. This will form the outline which the lab instructor will use for grading. Copies of this evaluation form are attached to the end of your lab manual. Each laboratory writeup should have the following sections: 1. Layout: Where you write the laboratory title, the date and time of the laboratory section, and the name(s) of your lab partner(s). 2. Preliminaries: Where you make a brief statement of the laboratory objectives in your own words, a description of the methods to be used in order to achieve the stated objectives, a labeled sketch of the setup to be used, and completed predictions (if required). 3. Data: Here is where you record the actual procedures as they are performed. Record the actual time that you start each procedure section, and include data as it is taken. If the data is recorded by the computer, include computer graphs of raw sample data. 4. Results: Include here all equations used in the calculations, computer graphs of analyzed data, and fitting results. Include analyses of the error in your results. 5. Conclusions: This should include a more general statement of whether your measurements confirm the stated objectives, what fundamental physical laws were illustrated by the experiment, and how the experimental error could have been reduced in the experiment. Also include here a constructive critique of the lab, stating what went well, what didn t, and how the laboratory could be improved. i

4 6. Abstract: This is a formal statement of what this laboratory experiment was all about. Included in this paragraph should be something about the objectives, results, and conclusions of the laboratory. 7. Certification: You must get the signature of the lab instructor in your journal before leaving the lab. This will indicate how much of the writeup you did in the laboratory period, and how much you did outside the lab. One point extra credit will be given if you complete the laboratory writeup (including the evaluation form) before the end of the laboratory period and get your lab assistant s signature after the evaluation form. You must also sign the evaluation form. Your signature certifies that this laboratory writeup represents a true and accurate presentation of your work. 8. Bonus: Extra credit (up to 10% of the laboratory) will be given in selected labs for the completion of the further investigation section. This is in addition to the one point you can receive for completing the writeup during the laboratory time. 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. We hope that you will overlook any missspellings, omis ions, er r ors and inconsistencies and report such to the authors. ii

5 Applied Physics for the Health Professions Lab Evaluation Form Name: Box # Lab date/time: Completion date/time: Signature: Pgs Pts /1 Layout: Title, Date, Time, Partner(s) /3 Preliminaries: Statement of objectives, methods to be used, predictions (if required), labeled sketch of setup /5 Data: Record of procedures as they are performed, actual time record for each procedure, raw data /6 Results: Calculations, computer graphs and data fits, analysis of results, error analysis /4 Conclusions: What physical laws were discovered, were the objectives met, what effects (if any) were the result of possible errors, critique of lab (what went well, what didn t, how it could be improved) /1 Certification: Signature of lab assistant before leaving lab, inclusion of signed evaluation form. Your signature certifies that this laboratory writeup represents a true and accurate presentation of your work. /0 Bonus: Extra credit for promptness, further investigation Total: / 20 Lab Assistant: Date Scored: iii

6 Table of Contents Preface i Experiment 1 Uniform Motion Experiment 2 Uniform Acceleration Experiment 3 Forces in Equilibrium Experiment 4 Force and Acceleration - Newton s Second Law Experiment 5 Torque and Angular Acceleration Experiment 6 Rotational Equilibrium - Torques Experiment 7 Conservation of Mechanical Energy Experiment 8 Elastic and Inelastic Collisions Experiment 9 Conservation of Energy of a Rolling Object Experiment 10 Resonant Vibrations of a Wire Experiment 11 Current, Voltage and Power - Ohm s Law Experiment 12 Series and Parallel DC Circuits Experiment 13 Magnetic Field, Force on a Current Experiment 14 Motional EMF iv

7 Applied Physics for the Health Professions Experiment 1 Uniform Motion - Graphing and Analyzing Motion Objectives: < To take data using a computer and signal interface < To use computer software programs to interpret data < To observe the distance-time relation for motion at constant velocity 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 the positive or negative of the distance of the object from a reference point called the origin, the numbers being positive to the right of the origin and negative to the left. Denoting the displacement as x and the time as t, x ' vt%x o (1) The initial displacement x o is the starting position when time t equals zero. This value is where the line crosses the vertical axis and is called the intercept. In a graph of displacement x (on the vertical axis) versus time t (on the horizontal axis) the velocity of the motion v is equal to the slope of the line, v ' slope ' rise run ' )x )t ' x 2 &x 1 t 2 &t 1 (2) The best fit of a straight line to a data set is the one with the smallest value of the average square deviation. Figure 1 Slope, v, and intercept, x o. Page 1

8 Prediction: Consider the following two cases: A. Motion away from the origin Cart has an initial position at +50 cm, travels for a time duration of 2 seconds, at a constant velocity of +50 cm/s B. Motion towards the origin Cart has an initial position at +50 cm, travels for a time duration of 2 seconds, at a constant velocity of!50 cm/s Draw graphs in your journal of what you think the motion will be for these two cases, plotting the displacement x versus the time t. Then answer the following questions for both cases: 1. Will the graph be straight or curved? 2. Will the graph slope up or down? 3. If it is curved will it curve up or down? Explain your reasoning for each of these answers. Procedure: Setup motion sensor: 1. Plug the motion sensor s phone plugs into digital channels 1 and 2 with the yellow banded plug into channel Mount the motion sensor at the end of the track opposite the bumper. 3. Align the sensor so that the sound waves will travel directly along the track. 4. Use the screw adjustment on the bottom end of the track to level the track. Setup Science Workshop: (See Figure 2 below.) 1. If necessary, double click the left mouse button on the physics labs folder to open it. 2. Double click on the science workshop icon in the folder to open Science Workshop. 3. Click on the maximize icon to maximize the display. 4. Click on Sampling Options, change the sampling rate to 10,000 Hz, and the stop time to 3 sec. Press RETURN and click on OK to accept these values. 5. Click and drag the phone plug icon to digital channel 1, choose Motion Sensor. Change the trigger rate to 50 Hz. 6. Drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2. Click on position to display the position vs. time graph. 7. Drag the Table icon onto the Motion Sensor icon. Click on position to display the position vs. time data. Click on the clock to the right of the E at the upper left of the Table window to display times. 8. Move the graph and table windows so that all three windows are accessible. Page 2

9 A. Motion away from the origin Data Collection: 1. Place the cart on the track at the end near the sensor. 2. Click on the REC button as your lab partner pushes the cart away. Wait until data collection stops. 3. On the graph window, Click on the rescale icon (fourth icon from the left in the lower left corner). 4. Click on the lower left icon in the graphing window and change the title of the graph to Distance, vs. Time by (enter your name) and press RETURN. Click on File and then Print to print the Graph window. This graph presents the raw data for your journal data section. Figure 2 Science Workshop window. Open Graphical Analysis: 1. In order to further analyze the data, we will use the more powerful analysis tools contained in the Graphical Analysis program. Double click on the Graphical Analysis icon in the physics folder to open the graphing analysis program, click on OK and maximize the display. 2. Since you will be plotting time (T) vs. distance (X), double click on the X at the top of the first data column to change it to a T, and double click on the Y to change it to X. Page 3

10 Transferring data to Graphical Analysis: 1. In Science Workshop: Click and drag your cursor to highlight a rectangle of the best data on the graph display. Then click once on the table display to select the highlighted data on the table. Under the Edit menu, choose copy to store the data temporarily in the Window s clipboard. 2. In Graphical Analysis: Click on the upper left data position. Under the Edit menu option choose paste to copy your data from the clipboard. Note that the displacement is plotted vertically (y-axis) and the time data is plotted horizontally (xaxis). Analyzing Data: (See Figure 3 above) Manual Fitting: 1. Click on the graph of your data on the right to select the graph. 2. Choose Analyze from the main menu and click on Manual Curve Fit. 3. Select the 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 at time equal zero). 4. 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. 5. Note values of the Mean Square Error at the lower right of the graph for each value of slope M and intercept B. 6. 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. 7. Record the values of the slope M, intercept B and the Mean Square Error in your journal. 8. Click on OK-Keep Fit. Automatic Fitting: 1. Choose Analyze from the main menu and click on Automatic Curve Fit. 2. Select the function M*x + B to select a linear (straight-line) model. 3. Record the values of the slope M, intercept B and the Mean Square Error in your journal. 4. Click on OK-Keep Fit. Page 4

11 Figure 3 Graphical Analysis window. Analyze Results: 1. Compare the values obtained in your manual fit, with the automatic fit values. 2. Click on the graph title and change the title to Displacement versus Time. 3. Click in the text window and enter your name, experiment name, date and experiment details, ie motion away from detector. 4. Choose File in the main menu, then Print to print the entire screen. Questions (to be included in your Results): 1. How does your observed curve compare with your predicted curve? 2. What is the speed of the cart? 3. How far from the detector is the cart when the detector begins measuring its motion? 4. What does the value of the Mean Square Error indicate? B. Motion towards the origin 1. 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. 2. Repeat the Data Collection, Transferring data to Graphical Analysis, Analyzing Data, and Questions sections above. Further Questions: 1. How are the results for parts A and B similar? 2. How are the results for parts A and B different? Page 5

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13 Applied Physics for the Health Professions Experiment 2 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 velocity-time graph. Equipment: < Motion sensor and reflector < 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 the positive or negative distance of the object from a reference point called the origin, the numbers being positive to the right of the origin and negative to the left. When the velocity changes in time, 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 Figure 1 Graph of velocity vs. time. Page 7

14 velocity. The relation between velocity v and time t is v ' v o %a@t (2) which is the equation of a straight line with a slope, a, and intercept, v o. According to Galileo, the acceleration of an object down a slope is equal to the component of the constant gravitational acceleration directed down the slope. That is, a ' gsin(2) (3) where 2 is the angle between the track and the horizontal. Prediction: Consider the following scenerio: A physics student pushes the dynamics cart up the inclined track and observes its distance-time motion with a motion sensor. Draw graphs in your journal of what the distance vs. time and velocity vs. time graphs would look like. Title the graph and label the axes indicating distance in meters, velocity in meters/second, and time in seconds. Then answer the following questions: 3. Will the distance vs. time graph be a straight line or curved? 4. Will the velocity vs. time graph be a straight line or curved? 5. Is the velocity zero at any point? If so, where? 6. Is the acceleration zero at any point? If so, where? Explain your reasoning for each of these answers. Procedure: Setup motion sensor: 1. Plug the motion sensor s phone plugs into digital channels 1 and 2 with the yellow banded plug into channel Place the reflector upright on the dynamics cart and secure it with a rubber band. 3. Elevate one end of the track by placing the lab jack under one end. Set the lab jack to its lowest position. 4. Place the cart on the low end of the track. 5. Mount the motion sensor at the upper end of the track. 6. Align the sensor so that it is directed down along the track and toward the reflector on the cart. Page 8

15 Setup Science Workshop: 1. Open Science Workshop. 2. Click on Sampling Options to change the sampling rate to 10,000 Hz, and the stop time to 3 sec. 3. Click and drag the phone plug icon to digital channel 1, choose Motion Sensor. Change the trigger rate to 50 Hz. 4. Drag the Graph icon onto the Motion Sensor icon below digital channels 1 and 2. Click on position, velocity, and acceleration to display the three graphs: position vs. time, velocity vs. time, and acceleration vs. time. On the graph window, click on the E icon (lower left corner) to display statistics. 5. Maximize the display and move the graph window so that all windows are accessible. Data Collection: 1. 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. 2. Click on the Rescale icon (fourth from the left in the lower left of the graph window). 3. 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. This graph presents the raw data for your journal data section. Analysis: A. Acceleration On the graph display, click and drag the curser to select regions of good data for analysis. You should select the same (or nearly the same) starting and ending times in each of the three graphs, and the data in the velocity vs. time graph should be approximately a straight line. Acceleration from distance-time data Click on the E to the right of the position vs. time graph and drag the mouse down to Curve Fit. Then click on Polynomial Fit. This will give a fit of the y-axis (position) versus the x- axis (time) given by the equation: y ' a 1 % a 2 x % a 3 x 2 (4) x ' x o %v 1 2 a@t 2 (5) which when compared to Eq. (1), yields the equivalences: y = position x, a 1 = initial position Page 9

16 x 0, a 2 = initial velocity v 0, x = time t, and a 3 = 1/2 acceleration a. Thus, we have a = 2 a 3. Record the value of a 3, and two times a 3 (which equals a) in your journal. Acceleration from velocity-time data Click on the E to the right of the velocity vs. time graph and drag the mouse down to Curve Fit. Then click on Linear Fit. This will give a fit given by the equation: y ' a 1 % a 2 x (6) v ' v o %a@t (7) which when compared to Eq. (2), yields the equivalences: y = velocity v, a 1 = initial velocity v 0, a 2 = acceleration a, and x = time t. Thus, we have a = a 2. Record the value of a = a 2 in your journal. Acceleration from acceleration-time data Click on the E to the right of the acceleration vs. time graph and drag the mouse down to Mean. Repeat to select Standard Deviation. The mean of the y data gives the acceleration directly, with the plus-or-minus error in the acceleration given by the standard deviation of the y data. Record the acceleration and plus-or-minus error in the acceleration in your journal. Acceleration from prediction Measure the length L and height H of the track. Since sin(2) = H / L, calculate the acceleration from Eq. (3) as: a ' g sin(2) ' g H L (8) where g = 9.81 m/s 2. Record the calculated value of acceleration in your journal. Compare the three values for the acceleration that you obtained from the three graphs. Calculate the percent difference between the average of the three acceleration values obtained from the graphs with the value obtained through calculation, using the expression, %Err ' a calculated &a measured a measured 100%. Page 10

17 B. Velocity Velocity from distance-time data 1. Click and drag to select about three data points on the distance-time graph. 2. 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. 3. Click on the examine icon next 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 from the cross-hair and compare the velocity value with that obtained from the slope of the distance-time graph at that time. Calculate the percentage difference. 4. This shows that the velocity is equal to the slope of the distance vs. time graph. In calculus, we call this the Derivative. C. Displacement Distance traveled from velocity-time data 1. 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. Record this value in your journal. 2. Click on the cross-hair examine icon at the bottom left and move the mouse so that the cursor is vertically 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. The difference between these two values is the distance traveled. 3. Compare the distance traveled with the area under the curve. Calculate the percentage difference. 4. This shows that the displacement is equal to the area under the velocity vs. time graph. In calculus, we call this the Integral or Integration. Conclusions: In your conclusions, compare the graphs that you obtained with those that you drew in your predictions, and write down the correct answers to the questions in the prediction section. Explain any differences between your answers now and what you wrote at the beginning of the lab. Page 11

18 Further Investigation: Measuring the value of the acceleration of gravity directly using a falling ball. From Eq. (3) we have a = g sin(2). If 2 = 90E, then sin(2) = 1, and a = g. We can thus measure g directly if we let a ball drop vertically downward, rather than having the object go down a slope. In this experiment, attach your motion sensor, pointed vertically downward, to a horizontal rod at the top of a vertical stand about one meter above the table. Make sure that there is room for a ball to bounce on the table directly under the motion sensor. Hold the ball directly beneath the motion sensor about 50 cm above the table. Release the ball and as it is falling click on the record button. You should record two or three bounces of the ball. In the graph display, select a region of the position vs. time graph that is approximately parabolic, and determine the acceleration of the ball using a polynomial fit on the data. Then select a region of the velocity vs. time graph and determine the acceleration of the ball using a linear fit on the data. Then select a region of the acceleration vs. time graph and determine the acceleration using the mean of the data. Record these three values for the acceleration due to gravity, and calculate their average. Compute the percentage error using g = 9.81 m/s 2 as the standard: %Err ' g&g standard g standard 100%. Print out the graph display, and explain the setup, calculations, and conclusions in your journal. Page 12

19 Applied Physics for the Health Professions Experiment 3 Forces in Equilibrium 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 (Newton s First Law). 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, 2, 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 2 can be related to the length 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. sin(2)' opposite hypotenuse cos(2)' adjacent hypotenuse (1) tan(2)' opposite adjacent Page 13

20 Vector Addition Graphical 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 Component method Figure 2 Vector addition by the polygon Vectors may be added by selecting two method. 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 Fx = F cos(2) components are given by F x 'F@cos(2) F y 'F@sin(2) (2) 1 F The X component of the resultant is the sum of the X components of the vectors being added, and similarly for the Y component. The angle that the resultant makes with the X axis is given by Figure 3 Finding the two perpendicular components of a vector. X axis 2'arctan F y F x (3) and the magnitude is given by F' F 2 x %F 2 y (4) Page 14

21 Newton's first 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 (5) Equilibrium Conditions Thus, if three forces act on an object at rest, the following relationship has to be satisfied. PF 1 % PF 2 % PF 3 ' 0 (6) An equivalent statement is PF 3 '&( PF 1 % PF 2 ) (7) so that forces. PF 3 is equal in magnitude and opposite in direction to the resultant of the other two Prediction: Suppose you have one force, of magnitude 3.0 N, directed in the positive x direction (2 1 = 0E), and a second force, of magnitude 4.0 N, direction in the positive y direction (2 2 = 90E). In your journal, draw a graph that includes these two forces (to scale), the vector sum of these two forces, and the needed force that would be equal in magnitude and opposite in direction to the resultant of these two forces. What is the magnitude and direction of? F 3 Procedure: Setup Science Workshop: 5. Connect the din plug from the force sensor into analog channel A of the Science Workshop Interface. 6. Open Science Workshop. 7. Click on Sampling Options to change the sampling rate to 1000 Hz. 8. Click and drag the analog plug (right hand side) icon to analog channel A, choose Force Sensor. 9. Double click on the Force Sensor icon below analog channel A, to calibrate. 10. Enter 0 in the Low Value box on the left. With no tension in the force sensor string click on the Low Value Read to enter the voltage for zero force. 11. Support 500 g of hooked weights from the end of one of these strings across a pulley and connect a second one to the force sensor. The tension force equals the weight of the mass, which is (0.500 kg)(9.8 m/s 2 ) = 4.90 N. Position this pulley opposite the force sensor. Be sure that the pulley plane and string are perpendicular to the end Page 15

22 of the force sensor. Enter 4.9 in the High Value box on the left. Click on the High Value Read to enter the voltage for the 4.9 N force. Click on OK to accept the calibration. 12. 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 2 to display two digits to the right of the decimal. (hundreths of a Newton force). Click OK. 13. Click on the monitor (looks like play) icon to get a continuous display of the force. Figure 4. Pasco Force Table with Dual Range Force Sensor balancing two forces. Data collection: Set up the following situations so that in each case the magnitudes of the forces are unequal. In each case, adjust the position of the force sensor in both the angular and radial direction so that the knot in the strings is exactly over the cross-hairs in the center of the force table. The pulleys should be adjusted so that the strings are exactly horizontal and as close to the force table as possible without actually touching the table. a) Support hooked masses of 200 g (F 1 = kg * 9.8 m/s 2 = 1.96 N) and 300 g (F 2 = kg * 9.8 m/s 2 = 2.94 N) from strings over the pulleys so that the angle between forces F 1 and F 2 is 90E (see figure 5a). The force F FS is the value displayed by the force sensor. Enter in your journal the magnitude and directions of each of the three forces, F 1, F 2, and F FS. Figure 5. Force table setup with two forces balanced by the force sensor. Table 1: Initial forces and force sensor measurement Part 1a Mass Force = M*g Angle 2 Force 1 Force 2 Force Sensor Page 16

23 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. Draw to scale two vectors for F 1 and F 2 and add the vectors graphically. Use at least 1/2 of a page for the graphical solution in order to improve the accuracy of your measurement. Then compare the results from the force sensor with the graphical measurements. Table 2: Comparison of Graphical solution with Force Sensor measurement Part 1a Force Angle Force Sensor + 180E Graphical Solution % Error / Angle Error For the force sensor, add 180E to the angle measured in Table 1 in order to compare it with the resultant force calculated graphically. Then calculate the % error of the force, and the angle error (2 FS + 180E- 2 GS ). b) Repeat the measurement with forces F 1 and F 2 120E apart (see Figure 5b), and write the results in your journal. Table 3: Initial forces and force sensor measurement Part 1b Mass Force = M*g Angle 2 Force 1 Force 2 Force Sensor Make a sketch in your journal showing these forces as arrows and write the values of each force alongside its arrow. Use the component method to add forces F 1 and F 2. Then compare the results from the force sensor with the component method calculations. Table 4: Component method calculation Part 1b F x = F cos 2 F y = F sin 2 Force 1 Force 2 Resultant Force Page 17

24 Resultant force magnitude = F 2 x % F 2 F y =, Angle = tan &1 y = F x Table 5: Comparison of Component solution with Force Sensor measurement Part 1b Force Angle Force Sensor + 180E Component Solution % Error / Angle Error c) Repeat the measurement with forces F 1 and F 2 60E apart (see Figure 5c), and write the results in your journal. Table 6: Initial forces and force sensor measurement Part 1c Mass Force = M*g Angle 2 Force 1 Force 2 Force Sensor Make a sketch in your journal showing these forces as arrows and write the values of each force alongside its arrow. Use the component method to add forces F 1 and F 2. Then compare the results from the force sensor with the component method calculations. Table 7: Component method calculation Part 1c F x = F cos 2 F y = F sin 2 Force 1 Force 2 Resultant Force Resultant force magnitude = =, Angle = = F 2 x % F 2 y tan &1 F y F x Page 18

25 Table 8: Comparison of Component solution with Force Sensor measurement Part 1c Force Angle Force Sensor + 180E Component Solution % Error / Angle Error 2. Support three different masses in an arrangement approximately as shown in Figure 6. Using the component addition method, 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 2 FS + 180E and F FS. Figure 6 Force table setup with three forces. Table 9: Initial forces and force sensor measurement Part 2 Mass Force = M*g Angle 2 Force 1 Force 2 Force 3 Force Sensor Make a sketch in your journal showing these forces as arrows and write the values of each force alongside its arrow. Use the component method to add forces F 1, F 2, and F 3. Then compare the results from the force sensor with the component method calculations. Table 10: Component method calculation Page 19

26 Part 2 F x = F cos 2 F y = F sin 2 Force 1 Force 2 Force 3 Resultant Force Resultant force magnitude = F 2 x % F 2 F y =, Angle = tan &1 y = F x Table 11: Comparison of Component solution with Force Sensor measurement Part 2 Force Angle Force Sensor + 180E Component Solution % Error / Angle Error Conclusions: In your conclusions, discuss whether your measurements satisfy the requirements of Newton s First Law. Further Investigation: As a further investigation, repeat the measurements and calculations you did in step 2 using five different forces. Compare the measured force sensor result with the calculated component method solution for the five forces. Do these results satisfy the requirements of Newton s First Law? Page 20

27 Applied Physics for the Health Professions 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 (Newton s Second Law). Equipment: < Track with cart, accessory weights < Smart pulley timer < Force Sensor < Triple beam balance 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 Figure 1 Free body diagram of the hanging mass. F'Ma (1) 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 force of gravity minus the tension in the string. 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. mg&t'ma (2) Equation (2) can be solved for tension to yield the following equality. T'm (g&a) (3) Page 21

28 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 T'Ma (4) Procedure: Figure 2 Free body diagram of the cart. Setup Science Workshop: 1. Open Science Workshop. 2. Plug in the smart pulley into digital channels 1 and Drag the digital plug icon over digital channel 1 and select smart pulley (linear). Double click on the smart pulley to calibrate it. Follow the instructions of the lab instructor, or use the average calibration value of Plug the force sensor into analog channel A. 5. Drag the analog plug icon over analog channel A and select force sensor. Double click on the force sensor to calibrate it. Set the low value to zero, and click on low value read with no mass hanging from the force sensor. Set the high value to 4.9N and click on the high value read with a 500g mass (weight = 4.9N) hanging from the force sensor. 6. Click on Sampling Options to change the sampling rate to 1000 Hz and the stop time to 5 sec. 7. 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. Add the force sensor graph to the graph window by clicking on the multiple graph icon (lower right icon in the lower left corner of the graph) and selecting analog A and force sensor. 8. Start statistics by clicking on the E button in the graph window, then click on the E button beside the smart pulley graph and select curve fit and then linear fit. Click on the E button beside the force sensor graph and select mean and standard deviation. Setup system: 1. Measure and record in your journal the mass of the free-hanging pulley. This mass will be added to the masses of the weights hung on the pulley during the experiment. 2. Measure and record in your journal 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. 3. Place the cart on the track and level the track so that the cart does not accelerate in either direction. 4. With a table clamp, position the smart pulley (see figure 3) at the aisle end of the track. Run the cable through the second, free-hanging pulley and back up to a force sensor mounted vertically. Page 22

29 Data Collection: Part 1: 1. Hang 30 g mass from the free-hanging pulley, and position the cart so that the free-hanging pulley is just below the smart pulley. Click on the REC button as your lab partner releases the cart. 2. Reproduce Table 1 (below) in your journal. Click and drag the curser to select the best data from the velocity vs. time graph. The statistics section will fit a straight line to the selected data. The slope of the line, a 2, is the acceleration of the cart. Record the value of the acceleration in the column titled a in Figure 3 Smart pulley. Table 1. Click and drag the curser to select the same data on the force vs. time graph. Record the mean value of the force in the T measured column of Table Take a series of seven (7) more measurements, increasing the mass by 10 g each time. Record effective mass (m + m pulley )/2 and the acceleration and force for each mass in Table 1. Table 1 Cart without additional mass (m+m pulley )/2 a g-a T calc = m(g-a) T measured T measured - T calc 4. Complete the table by calculating the other columns. For the value of g use the accepted value of 9.81 m/s 2. Print out one of the data graphs and place in the data section of your journal as a sample raw data measurement. Page 23

30 Part 2: 1. Place the two blocks on the cart and repeat the experiment. 2. Fill in a second table in your journal in a similar manner to Table 1 as given above. 3. Print out one of the data graphs and place in the data section of your journal as a sample raw data measurement. Analysis of Data: 1. Use Graphical Analysis to plot a graph of measured tension T (y axis) vs. acceleration a (x axis) for the data in Table 1. Click on the y axis and change the setting to Autoscale at 0. Repeat for the x axis. This will include the origin (0,0) point in the graph. Put a title and label the axes of the graph. 2. To fit the data to a straight line click on Analyze and Automatic Curve Fit and select function y = M*x + B to select a linear (straight-line) function. Since we have T given as y, and a given as x, the slope M should be the same as the mass M in Eq. (4): T'Ma (4) Theoretically, the value of the intercept B should be zero, since no T 0 appears in Eq. (4). However, T 0 can be thought of as the tension required to cause the cart to move with zero acceleration, that is, the tension required to overcome kinetic friction. Thus the intercept B provides a measure of the friction of the system. Print your graph. 3. Record in your journal the slope M and the intercept B. 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* 100 (5) M 4. Repeat the above analysis for Table 2. In your conclusions you should: 1. Discuss the percent error that you calculated for both graphs. 2. Interpret the value of the intercept and discuss how the presence of constant frictional force would affect the results of the experiment. 3. Speculate on the origin(s) of error. 4. Discuss the difference between the measured and calculated tensions. 5. Mention what you learned in this experiment. Page 24

31 Applied Physics for the Health Professions Experiment 5 Torque and Angular Acceleration Objective: < To observe the relationships between the torque and angular acceleration and the moments of inertia. Equipment: < Rotating table < Ring and disk < Hooked weight sets < Pasco Smart Pulley < Vernier caliper Physical Principles: Figure 1 Main setup for the experiment. Consider the setup shown in figure 1. A net force of mg&t acts on the mass m hanging on the string with a tension T. According to Newton s Second Law, Solving equation (1) for T yields mg&t'ma (1) T'm (g&a) (2) This tension acting tangent to the rotating table drum with radius r produces a torque J'Tr'mr(g&a) (3) Newton s Second Law for angular motion states J ' I " (4) where the angular acceleration " is related to the acceleration a by " = a / r and r is the table drum radius. The moment of inertia I is the sum of all the m r 2 where m is the mass and r is the perpendicular distance of the mass to the axis of rotation and the sum includes all the small pieces of the object (an infinity of them). For example, the moment of inertia of a system composed of two masses is the sum of the moments of inertia of each piece I tot 'I 1 %I 2 (5) The following expressions describe the moments of inertia about the axis of symmetry of several familiar objects that have a mass M. Page 25

32 I disk ' 1 2 MR2, I ring ' MR 2 (6) where R is the outer radius of the disk, or the average of the inner and outer radii of the ring. Procedure: General Measurements and Setup Using the vernier caliper measure the diameter, d drum, of the drum on which the string is wrapped. Compute the radius of the drum, r drum, and record these values. This radius will be used to calculate the torque acting upon the rotating platform by means of Eq. (3). Run Science Workshop. Plug the smart pulley into the digital slot #1 on the Pasco computer interface and do the same on the screen by clicking on the digital plug, dragging it over digital channel 1, selecting smart pulley (linear) and clicking on OK. Calibrate the smart pulley by double clicking on the smart pulley icon and entering a radius of Create a graph of velocity vs. time by dragging the graph icon over the smart pulley icon and selecting velocity. Click on the statistics buttons E and select curve fit and linear fit. The curve fit will give an equation y = a 1 + a 2 x, corresponding to the equation v = v 0 + a t. Thus, the acceleration will be displayed as a 2. Make sure, in reading the value for acceleration, that you have selected only the good data that appears as a straight line in the graph. Take care that the rotating platform does not rotate except when taking acceleration data! Keep the string taut with care that it does not become wrapped around the axle! Inertia of the rotating platform Hook a 50 g mass to the loose end of the string that is wrapped around the rotating platform drum and drape the string over the smart pulley timer. Rotate the platform so that the mass is just below the top of the smart pulley. Release the platform and click on Record after it has started to rotate. Make sure the string is short enough so that the weight will not hit the floor. Record the value for the acceleration of the string in a table (similar to Table 1 below) in your journal. Repeat using 60 g, 70 g, 80 g, 90 g, and 100 g. Open Graphical Analysis and plot the values for torque J on the y-axis vs. angular acceleration " on the x-axis. Fit a linear curve fit to the data, and record the values of the slope M and intercept B in your journal. The slope M = J / " is simply the moment of inertia I of the rotating platform, and the intercept B is that torque needed to overcome friction (i.e., the frictional torque). Page 26

33 Table 1: Inertia of the rotating platform m [kg] a [m/s 2 ] F=m(g - a) [N] J = F r drum [Nm] "= a/r drum [rad/s 2 ] Inertia of the disk Place the disk on the rotating platform and by following the procedure of the previous experiment take a series of readings of the acceleration, a, for masses on the string of about 50, 100, 150, 200, 250, and 300 grams. Record the results in Table 2, analogous to Table 1. Table 2: Inertia of the disk and rotating platform m [kg] a [m/s 2 ] F=m(g - a) [N] J = F r drum [Nm] "= a/r drum [rad/s 2 ] Plot the values for torque J on the y-axis vs. angular acceleration " on the x-axis using Graphical Analysis. Fit a linear curve fit to the data, and record the values of the slope M and intercept B in your journal. Record the difference between the moments of inertia recorded in your second graph minus that recorded for your first graph. This difference is your measured value for the moment of inertia of the disk by itself. Page 27

34 Measure the radius of the disk, and record the mass value stamped on the disk. Using these values, and Eq. (6), compute a calculated value for the moment of inertia of the disk. Compare the calculated moment of inertia for the disk with the measured value, and record the percent error of difference: %Err ' I calc & I meas 100 I calc Inertia of the ring With the ring on the platform, determine the moment of inertia of the system as done for the disk. Use masses of 50 g, 100 g, 150 g, 200 g, 250 g, and 300 g, and generate a Table 3 for the inertia of the ring, analogous to Table 2 for the disk. Plot the values for torque J on the y-axis vs. angular acceleration " on the x-axis using Graphical Analysis. Fit a linear curve fit to the data, and record the values of the slope M and intercept B in your journal. Record the difference between the moments of inertia recorded in this graph minus that recorded for your first graph. This difference is your measured value for the moment of inertia of the ring by itself. Measure the average radius of the ring, and record the mass value stamped on the ring. Using these values and Eq. (7), compute a calculated value for the moment of inertia of the ring. Compare the calculated moment of inertia for the ring with the measured value, and record the percent error of difference. Inertia of the ring and disk together With both the ring and the disk on the platform determine the moment of inertia as done in the previous measurements. Use masses of 100 g, 200 g, 300 g, 400 g, 500 g, and 600 g, and generate a Table 4 for the inertia of the disk plus ring, analogous to the previous tables. Plot the values for torque J on the y-axis vs. angular acceleration " on the x-axis using Graphical Analysis. Fit a linear curve fit to the data, and record the values of the slope M and intercept B in your journal. Record the difference between the moments of inertia recorded in this graph minus that recorded for your first graph. This difference is your measured value for the moment of inertia of the ring plus the disk. Compare the measured moment of inertia with the sum of the calculated moments of inertia for the ring and disk, and record the percent error of difference. Conclusions: In your conclusions, include a discussion of the source of the intercept values B in each of your calculations. Page 28

35 Applied Physics for the Health Professions Experiment 6 Rotational Equilibrium - Torques Objective: < To test the hypothesis that a body in rotational equilibrium is subject to a net zero torque and to determine the typical tension force that the biceps must produce. Equipment: < Arm assembly < Table clamp, two rods, and a swivel clamp < Hooked weight set < Force sensor < Computer with Pasco interface and Science Workshop Physical Principles: A torque is produced about an origin when a force acts at a point of a body in a direction other than the direction of the origin from this point. Torques tend to make a body rotate about an axis. Those torques that rotate a body in a clockwise direction are called clockwise and are usually described by negative numbers. Those that rotate a body in a counterclockwise direction are called counterclockwise and are usually described by positive numbers. Figure 1 Torque generation. The torque generated by a force F acting at a point which is a distance, d, from the origin is defined as J'F z d (1) Figure 2 Generation of a counterclockwise torque. Page 29

36 F z is the component of the force perpendicular to the line from the origin to the point of action of the force and therefore is given by F z 'F sin(a). In figure 1 a is the angle between this line and the direction of the force. The distance d is called the length of the lever arm. If a body is in rotational equilibrium the net torque acting on it must be zero. Procedure: Setup Science Workshop, and calibrate the force sensor as done in previous laboratories. Refer to the figures for further assistance. Record the mass m arm and length l arm of the wooden forearm. Also calculate the weight m arm g of the wooden forearm. The length of the forearm lever arm l arm should be taken to be the length from the center of the joint screw to the center of the hole on the opposite end that contains the thread supporting the load. Record the distance d cg of the forearm center to the elbow joint and the distance d m of the attachment point to the joint. The distance of the forearm center to elbow d cg should be taken to be the length from the middle of the forearm to the center of the joint screw. The distance of muscle attachment d m should be from the center of the hole with the upper support string to the center of the joint screw. Setup 1 With the upper arm approximately vertical adjust its position in the clamp so that the lower arm is horizontal (see figure 3). You may wish to sight along the horizontal wall joints to level the forearm. Measure and record the angle b, and record the tension from the scale. Then make the following calculations: 1. Compute the clockwise torques: Torque due to the Load = W load l arm Figure 3 Diagram of Setup 1. Torque do to the Forearm = W arm d cg 2. Compute the counterclockwise torques: Torque due to support = T d m sin(b) 3. Compare the total clockwise torque with the counterclockwise torque by computing the percent error. This is done by dividing the difference of the two torques (clockwise and counterclockwise) by their average and multiplying by 100. Page 30

37 Setup 2 Position the upper arm at an angle so that the angle b is about 40E and the lower arm is again horizontal. Repeat the observations and calculations in setup 1. Setup 3 Position the upper arm vertically and the lower arm at an angle of about 40E above the horizontal. Repeat the calculations of setup 1 noting that the clockwise torque calculations must include the additional factor of cos(a) where a is the angle of the forearm above the horizontal. 1. Compute the clockwise torques: Torque of the Load = W load l arm cos(a) Torque of the Forearm = W arm d cg cos (a) 2. Compute the counterclockwise torques: Torque due to support = T d m sin(b) 3. Compare the total clockwise torque with the counterclockwise torque by computing the percent error. This is done by dividing the difference of the two torques (clockwise and counterclockwise) by their average and multiplying by 100. Figure 4 Diagram of Setup 2. Figure 5 Diagram of Setup 3. Estimating the tension in a muscle Using the results from setups 1, 2, and 3, explain what positions of arm and forearm allow you to lift a weight in your hand most easily. Try this with a 1 kg mass. Estimate the muscle tension when you are raising a weight of about 20 N in your hand when the upper arm and forearm are nearly horizontal. Make reasonable estimates of the angles and lengths and neglect the weight of your forearm. Page 31

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39 Applied Physics for the Health Professions Experiment 7 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 mv2 (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 Fig. 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. 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 Page 33

40 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. Prediction: Consider a pendulum swinging with an amplitude of 20 cm. Draw graphs in your journal of position vs. time, velocity vs. time, and acceleration vs. time for one complete cycle of oscillation. Where on the cycle will the following occur: 1. Where will the displacement (position) be equal to zero? 2. Where will the displacement (position) have maximum magnitude? 3. Where will the velocity be equal to zero? 4. Where will the velocity have maximum magnitude? 5. Where will the acceleration be equal to zero? 6. Where will the acceleration have maximum magnitude? Explain your reasoning for each of these answers. Procedure: Video Capture 1. 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. 2. Double click on the Digital Video Producer Capture icon on the computer screen. Page 34

41 3. From the menu choose Capture / Settings and set stop time to 5 seconds. 4. Make sure the video camera is directly facing the pendulum, not leaning at an angle, connected to the S-Video input of the video capture board, and on. Set the pendulum into 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. 5. From the menu choose Capture / Video Capture. Select OK just after your lab partner releases the pendulum. At 30 frames per second, you will get about 150 data frames in two full swings of the pendulum to analyze. 6. The video file will automatically be saved to a file called ball991.avi in the physvid folder. Change the file name in the physvid folder to yourname.mov (make sure the extension is.mov or else the Videopoint program will not be able to recognize it). Collecting Data from Video 1. Return to your computer station in the laboratory and click on the videopoint program icon. 2. Close the introductory window. 3. Open your movie from the shared Network folder. 4. Indicate that you will be tracking the location of only 1 object. 5. From the menu choose Movie / Double Size in order to better see the pendulum bob. 6. Starting with the first frame, 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. 7. After taking data on the location of the bob for all 150 points, 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. 8. From the menu choose Movie / Normal Size so that you can see the data table. All the data should now be entered in the table. Now you will need to copy the data to Graphical Analysis in order to analyze the data. Transferring data to Graphical Analysis 1. Start the Vernier Graphical Analysis program. 2. Change the titles of the first two columns to t and x by double-clicking on the titles. 3. Add a third column by clicking on the new column icon (second from right end). Title the new column y. 4. 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. Print out these plots and include in the data section of your journal. Page 35

42 Analyzing Data: 1. 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 ). 2. Calculate the y-component of velocity in a new column in a similar manner. 3. 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). 4. Calculate the potential energy per unit mass (mgy)/m by creating a column (titled PE ) with 9.81* 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 y in the potential energy column 9.81*( y -y o ). This assigns y o as the point of zero potential energy. 5. Calculate the total mechanical energy per unit mass E tot /m by creating a column (titled E ) which sums the kinetic and potential energy columns KE + PE. 6. Plot columns KE, PE, and E on the same graph. Double click on any data point to get Graph Options. Turn on the Legend, and select Column Appearance to use different symbols for KE, PE, and E. 7. Select all the data 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% 8. Select the Analyze / Automatic Curve Fit from the menu. Select Perform fit on E to perform a linear fit. Compare the results of the linear fit with the mean and standard deviation determined from the statistics fit. Title and print your Graph! Questions: 1. What is the period of the oscillation? 2. How does the variation of the total energy compare with the variation of the kinetic energy and potential energy? 3. Was the total energy conserved during the two swings of the pendulum? Page 36

43 Applied Physics for the Health Professions Experiment 8 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. From equation (1) it can be deduced that if there is no external force acting on the system, the initial and final momenta must be equal, making the change )p in momentum zero. When two objects collide with no external force acting on the system, the total initial and final momenta of the system will be equal. If 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 scalar 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 collide and stick to each other. In the case of one dimensional motion, that is, all motions occur along a line, the conservation of momentum states that, m 1 v 1i %m 2 v 2i ' m 1 v 1f %m 2 v 2f (2) For an inelastic collision, the final velocities are equal, and Eq. (2) simplifies to, m 1 v 1i %m 2 v 2i '(m 1 %m 2 ) v f (3) Page 37

44 A further simplification is possible if one of the carts (say m 2 ) is initially at rest, m 1 v 1 '(m 1 %m 2 ) v f (4) The initial KE of the system now consists of only the initial KE of m 1 (since the initial KE of m 2 is zero) and is KE i ' 1 2 m 2 1i (5) The final KE can be related to the initial KE by a series of steps involving equations (4) and (5), as follows. KE f ' (m 1 %m 2 ) 2 KE f KE i 2 f ' (m 1 %m 2 ) m 2 1i 2 (m 1 %m 2 ) ' ( m 1 m 1 %m 2 )@m 2 1i ' m 1 m 1 %m i It can be seen that the initial and final KE are not equal, with a ratio given by, m 1 m 1 %m 2 (7) (6) Procedure: Place a cart on the track and level the track so that the cart does not roll in either direction. Place the photogates about 50 cm apart along the track and plug them into slots 1 and 2 on the Pasco computer interface. The photogates should be far enough apart so that the collision process occurs entirely between the gates. 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. Measure and record the lengths of the blocks on the carts. Make sure that you place 2 blocks on the left cart and record it as L 1 and 1 block on the right cart and record it as L 2. 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 velocity. 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 photogates. Use the balance to determine the masses of the carts, including blocks and record these values as m 1 and m 2 (m 1 >m 2 ). Page 38

45 1. Inelastic collisions Place cart 2 at rest 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 velocity of cart 1 as it passed through the first photogate. Record this value as v 1I. On the second table there are two values displayed. The first one is the velocity of cart 2 as it passed through the second photogate, and the second number is the velocity of cart 1 as it passed through the second photogate, which should be nearly identical to that of cart 1. Reproduce Table 1 (below) in your journal and record the data. Table 1: Inelastic collision cart 1 m v i v f p i p f KE i KE f cart both Repeat the experiment, reversing carts 1 and 2 to get a second set of data, this time with the lighter cart hitting the heavier cart. Table 2: Inelastic collision m v i v f p i p f KE i KE f cart cart 2 both Compute and record the initial and final momenta and kinetic energies by filling in the tables in your journal. Compare the ratios of p f / p i. Also compare the ratios of KE f /KE i for each run with the prediction of equation (7). 2. Elastic collisions 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 (remove it from the track after it passes completely through the photogate) followed a short time later by cart 1. Click on stop after cart 1 passes through the Page 39

46 second photograte. Reproduce Table 2 (below) in your journal and record the data. Record the one value displayed in the table for photogate 1 as v 1i, the first value in the table for photogate 2 as v 2f and the second value in this table as v 1f. Compute and record the initial and final momenta and kinetic energies by filling in the table in your journal. Compare the ratios of p f / p i. Also compare the ratios of KE f /KE i for each run. Table 3: Elastic Collision cart 1 m v i v f p i p f KE i KE f cart total Repeat the experiment, reversing carts 1 and 2 to get a second set of data, this time with the lighter cart hitting the heavier cart. Table 4: Elastic Collision m v i v f p i p f KE i KE f cart cart 2 total Questions: 1. Was momentum conserved in the inelastic collision? 2. Was Kinetic Energy conserved in the inelastic collision? 3. Was momentum conserved in the elastic collision? 4. Was Kinetic Energy conserved in the elastic collision? 5. Compare the actual loss of Kinetic Energy in the inelastic collision with the prediction of Eq. (5), calculating the percentage error. Page 40

47 Applied Physics for the Health Professions Experiment 9 Conservation of Energy of a Rolling Object Objective: < To test the principle of conservation of energy in the case of rolling motion for three objects with different moments of inertia. Equipment: < Racquetball sphere < Superball < PVC tube < Track < Motion sensor connected to a PC through a data acquisition board < Triple beam balance < Vernier caliper < Lab jack Physical Principles: A very important concept in physics is the Law of Conservation of Energy. According to this principle, the difference between the final and initial energy of an object is the work done on the object itself. Recall that the work done on an object is given by the product of the force acting on the mass and the distance it moved. If there is no net force acting on the body, and friction is neglected, the total work done on the object is zero and, as a result, the final and initial energy of the object must be equal. In other words, the total energy of a system stays constant. At any given time, total energy is the sum of two components. One of the components, caused by motion, is the kinetic energy KE, and the other is a result of position and is called the potential energy PE. It follows that the law of conservation of energy can be written mathematically as KE i %PE i 'KE f %PE f (1) The PE of the object is given by mgh. However, it must be recognized that the KE consists 1 of two parts. The first is given by and is called the translational Kinetic Energy. 2 mv2 If an object is spinning in one place, with no net translational movement, the velocity v is zero, and it therefore has no translational Kinetic Energy. However, it still has a KE caused Page 41

48 by rotation. This rotational Kinetic Energy is given by PE = m g h0 m g d sin( a) (2) 1 2 I T2 where I is the inertia of the object and T is the rotational speed. Consequently, the total KE is the sum of the translational and rotational Kinetic Energies. In the case of an incline, the PE can be related to d, the distance measured from a reference point above the object (see figure 1) by The total energy is given by E tot '&mgdsin(a)% 1 2 mv2 % 1 2 I T2 (3) If the object rolls without sliding, the linear velocity v and the angular velocity T can be related by v'r T (4) Figure 1 Diagram of the incline used. The inertia of the object must be known in order to calculate the KE. The three inertias used in this experiment are given below. I solid ball ' 2 5 MR2 I spherical shell ' 2 3 MR2 I ring ' MR 2 (5) Procedure: Setup: Motion Sensor Set up the equipment as shown in figure 2. Make sure that the angle of the motion Track sensor is adjusted in such a way that it is pointing parallel to the track. Run Science Workshop. Connect the two connectors of the motion sensor to the Figure 2 Setup of the equipment. Signal Interface box, the one with the yellow band to the digital input 1 and the other to digital input 2. Place the cursor on the phone plug icon, hold the left mouse button down and drag the cursor to the digital input 1 on the Science Workshop window. Click on the Motion Sensor option and click on OK. Click on the REC OPT button, set the periodic Page 42

49 samples rate to 50,000 Hz and set the stop condition to 2 seconds. Data collection: Place the PVC tube between the two edges of the track, close to the top, so that it can roll down the incline. Move the motion sensor or the tube so that the distance between them is at least 40 cm. At this point you are ready to take data. Let go of the tube and click on the REC button to start recording the data. The velocity graph should be straight line and the distance graph should be parabolic. If they are not, the angle of the motion sensor needs to be adjusted and the experiment redone. It will take you several trials to get a good set of data. Once you have a good set, record the mass and radius of the tube and complete the analysis of this data. Repeat the data collection using the spherical shell (racquetball) and a solid sphere (superball). Analysis: Open the calculator window which can be found in the experiment window. Enter the equation for kinetic energy using the mass, radius, and velocity that you measured. To enter the velocity values from the experiment click on the input button and select velocity. Enter kinetic energy for the calculation name, KE for the short name, then click on the equals button. Now click on new and enter the equation for the potential energy using the mass, angle and distance that you measured. Get the symbol for distance in the same manner as you did for kinetic energy. Name the calculation potential energy, the short name PE and click on the equals button. Finally enter the equation for the total energy, the sum of the kinetic and potential energies. To reference these values click on the input button and select calculator and then KE or PE. Enter energy for the calculation name and E for the short name, then click on equals. Graph the kinetic energy, potential energy and total energy versus time by clicking on the down arrow to the left of the graph in the graph display window and selecting calculations. The total energy should be constant with time. Calculate the percent error by the formula. %Err' *stdev(etot)* 100 (6) max(ke) Page 43

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51 Applied Physics for the Health Professions Experiment 10 Resonant Vibrations of a Wire Objectives: < To determine the relation between the frequencies of resonant vibrations of a wire and the physical parameters of tension, length, and mass per unit length of the wire. < To determine the relation between the velocity of waves on a wire and the tension and mass per unit length of the wire. Equipment: < Pasco Digital Function Generator-Amplifier < Banana leads, 2, 1.5 m long, alligator clips, 2 < Large magnet < Table clamps, 2 < Pulley, rods, weight set < Wires with m/l about.02,.005,.002,.001,.0005,.0003 kg/m < Graphical Analysis software Physical Principles: The relation between frequency f, speed of wave propagation v, and wavelength 8, is given by f ' v 8 (1) The velocity of a wave in a string or wire will depend on the tension T and the mass per unit length (µ = m/l) of the string. When a string under tension is pulled sideways and released, the tension in the string is responsible for accelerating a particular segment back toward its equilibrium position. The acceleration and wave velocity increase with increasing tension in the string. Likewise, the wave velocity v is inversely related to the mass per unit length of the string. This is because it is more difficult to accelerate (and impart a large wave velocity) to a massive string compared to a light string. The exact relationship between the wave velocity v, the tension T, and the mass per length µ, is given by v ' T µ (2) If a stretched string is clamped at both ends, traveling waves will reflect from the fixed ends, creating waves traveling in both directions. The incident and reflected waves will combine according to the superposition principle. As an example, when the string is vibrated at exactly the right frequency, a crest moving toward one end and a reflected trough will meet Page 45

52 at some point along the string. The two waves will cancel at this point, which is called a node. The resulting pattern on the string is one in which the wave appears to stand still, and we have what is called a standing wave on the string. At the nodes, there is no motion in the string. The points which vibrate with maximum amplitude are called anti-nodes. If the string has length L and is fixed at both ends, the condition for achieving standing waves is that the length of the string be equal to a half-integral number of wavelengths. L ' n 8 n 2 or f n ' v 8 n ' n 2L 1 ' n 8 n 2L Combining these three results in equations (1), (2) and (3) gives T µ as the basic model for a resonantly oscillating wire, where µ = m/l. Procedure: Setup Secure the two table clamps on opposite ends of the table about 20 cm from the edge. Mount the pulley and rod in the table clamp away from the wall and clamp the 30 cm rod in the other. Select Figure 1 Experimental setup. a wire with a mass of about.005 kg/m that is about 2 m long with loops on each end. Support the weight hanger with total mass of about 500 g on the pulley end and place the other loop on the rod. Connect the leads from the 8 S terminals of the Pasco function generator to each end of the wire. Make certain that the wire is bare at the ends and any enamel is removed. Set the generator frequency range switch to the position and the slide switch to the sine wave position. Turn the amplitude all the way up (see figures 1 and 2). Figure 2 Pasco Signal Generator. Part 1: Relationship between resonant frequencies and the number of nodes Vary the frequency of the signal generator output until the string vibrates resonantly. Record the frequency f, the number n of anti-nodes (points of maximum displacement), and the value f / n. Repeat this process for n increasing from one to about five and put the data in a table. Use Graphical Analysis to plot a graph of f vs. n. From equation (4), (3) (4) Page 46

53 f n ' 1 2L T µ n (5) and the quantity in parenthesis will be the slope of the graph. Print the graph. Analysis: 1. Compute the percentage error between the theoretical value and the plotted slope of the graph. 2. The slope of the graph is the calculated value of the fundamental frequency f 1. The y-intercept, b, which should be zero, gives a measure of the uncertainty in your determination of the fundamental frequency. Compute the value of 100% * b /slope and record the fundamental frequency and uncertainty. 3. Compare the two different calculated percentage errors. Explain why they are similar (or different). 4. What is the velocity v of the wave in this experiment? Calculate the velocity v two different ways: First, using equation (2), v ' T/µ. Second, using equation (4), v = 2 L f 1. Compare the two values by calculating the percentage error. Part 2: Relationship between resonant frequencies and tension Using the same string from part 1, vary the frequency until resonance occurs for five tension values of between 1 and 15 Newton (hanging masses of between 100 and 1500g). Use the same number of anti-nodes (e.g. three) in each step. Record the values for m, T = mg, f n, T, and v ' 2Lf n /n in a table. Use Graphical Analysis to construct a graph of f vs. T. From equation 4, f n ' n 2L µ T (6) and the quantity in parenthesis will be the slope of the graph. Print the graph. Also use Graphical Analysis to construct a graph of v vs. T. From equation 2, v ' 1 µ T (7) and the quantity in parenthesis will be the slope. Print the graph. Analysis: 1. Compute the percentage error between the theoretical value and the plotted slope of the f n vs. T graph. 2. Compute the percentage error between the theoretical value of 1/ µ and the plotted Page 47

54 slope of the v vs. T graph. Part 3: Relationship between mass per unit length and the resonant frequency Holding the tension at about 5N (m = 0.5kg) and using the same number of anti-nodes (e.g. three), select strings with masses per unit length of about 0.02, 0.003, 0.001, , kg/m. For each of these strings with the same tension and mode number, find the frequency of resonant vibration. Record the values for µ = m/l, f, 1/ µ, and v ' 2Lf n /n in a table. Plot a graph of f vs. 1/ µ in Graphical Analysis. From equation 4, f n ' n T 2L and the quantity in parenthesis will be the slope of the graph. Print the graph. Also use Graphical Analysis to construct a graph of v vs.. From equation 2, 1/ µ 1 µ (8) v ' T 1 µ (9) and the quantity in parenthesis will be the slope. Print the graph. Analysis: 1. Compute the percentage error between the theoretical value and the plotted slope of the f vs. 1/ µ graph. 2. Compute the percentage error between the theoretical value of T and the plotted slope of the v vs. 1/ µ graph. Page 48

55 Applied Physics for the Health Professions Experiment 11 Current, Voltage and Power in DC Circuits Objective: < To study the relation between current and voltage in direct current circuits, to make direct measurements of resistance, current and voltage, and to make calculations of power dissipation. Equipment: < Digital multimeters, 2 < DC power supply, IP18, 0-15 V < 6 Selected Resistors, such as 100 S, 300 S, 1000 S, 2200 S, 3300 S, 4700 S < Bread boards, bread board leads < Graphical Analysis software Physical Principles: Current The current in a circuit, measured in Ampere, is defined to be the rate charge passes a point in a circuit. I ' q t (1) Voltage and Ohm s Law The voltage at a point, measured in volts, is the work per unit charge required to move a charge from some arbitrary reference point to that point. Voltage is sometimes called the electromotive force, it "pushes" the current through circuits. When a voltage is applied across a conductor the resulting current is directly proportional to the applied voltage. The proportionality constant is defined by R ' V I (2) where R is called the resistance and is measured in ohms (S). Page 49

56 Electrical Power When current flows in a circuit, the voltage (work/charge) times the current, (charge/time) is the power (work/time) supplied to the material. The electrical power P measured in watts is P ' IV' I 2 R ' V 2 (3) R Resistor Color Code The resistance of a resistor is indicated by the color bands on it according to the formula below, R'N 10 M (4) where N is called the two digit nominal value and M is the multiplier exponent. The values of these parameters are given in the table below. Figure 1 bands. Resistor with color Color Value Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Violet 7 Gray 8 White 9 Gold -1 Silver -2 For example, a resistor with bands [Brown, Black, Red] would be R = 10 x 10 2 S = 1000 S. Some resistors also have a fourth band, which represents precision according to the following table. Color Precision Page 50

57 No band 20% Silver 10% Gold 5% Procedure: Learning the resistor color code Measure the resistance of six resistors with different values and complete table 1 in your journal. In the first four columns record the colors of the resistor bands. Compare the readings from the digital multimeter with the values obtained from the color code. In the last column indicate the percentage difference of the indicated and measured values. Are the percent differences that you found within the precision predicted by the fourth band? Table 1 Colors of the resistor bands Color 1 Color 2 Color 3 Color 4 Indic ated Resistance Multimeter as an ohm- Figure 2 meter. Precision Measured Resistance Percent Difference Page 51

58 Ohm's Law in the case of a constant resistor Connect a 1000 S resistor and two digital multimeters as shown in figures 3 and 4. CAUTION - Do not turn on the power supply until your circuit has been checked by the lab assistant or instructor! Always begin on the largest voltage or current reading scales. The black probe should be in the common receptacle and the red lead should be in the receptacle bordered by the red region, labeled V for voltage measurements and A for current measurements. Take a series of voltage measurements when the current is set to about 0.002, 0.004, 0.006, 0.008, A. Record a table of voltage and current measurements in your journal. Plot a graph of voltage on the y - axis vs. current on the x - axis. The slope of the line should be the resistance R (see equation 2). Compare the slope with the measured value of the resistance, and find the percent error. Figure 3 Circuit for testing Ohm s Law Ohm s Law in the case of constant voltage For the same circuit hold the voltage constant at about 10 volts, and measure the current and voltage for six different resistors. Record a table of current, resistance, and 1/resistance (use the actual measured values for each resistor as determined in part 1). Plot a graph of current versus the reciprocal of resistance and compare the slope to V, finding the percent error. Figure 4 Setup for Ohm s Law for a constant resistor. Calculation of power dissipation Measure and record the resistance of a heater element and calculate the electrical power dissipated by this element when it is connected to a 220 V source (see equation 3). How much would it cost to run this element continuously for a month at 5 cents per kwh? How much power would be dissipated if the element had a resistance of 100 ohms? Page 52

59 Applied Physics for the Health Professions Experiment 12 Series and Parallel DC Circuits Objective: < To study the relation between current and voltage in DC circuits, and to observe resistance, current, and voltage relationships in series and parallel combinations. Equipment: < Digital multimeters, 2 < DC power supply < Resistors, 180 S, 270 S, and 390 S < Bread boards, bread board leads < Computer with Graphical Analysis software Physical Principles: Ohm's Law Figure 1 Resistors in series. For many materials the current resulting when a voltage is applied is directly proportional to the voltage. The proportionality constant is defined by R V I = (1) where R is called the resistance and is measured in ohms. Resistors in Series The effective resistance of the three resistors in figure 1 connected in series is equal to the sum of the resistances of each resistor. R = series R + R R3 (2) Resistors in Parallel The effective resistance of three resistors connected in parallel (see figure 2) is given by the following formula = + + (3) R R R R parallel Kirchoff's Current Law Figure 2 Resistors in parallel. Page 53

60 According to this law, the sum of the currents leaving a branch point is zero. A branch point is a point where more than two conducting paths connect. So, in the case of figure 3 it can be said that I + I + I = (4) Kirchoff's Voltage Law Figure 3 Diagram of a branch point. The sum of all the voltage drops within a conducting loop always adds up to zero. According to the figure 4, the following equation can be written V1 V2 V3= 0 (5) If one proceeds clockwise around the loop the voltage V 1 is designated as a drop (going from + to - through the device) and is positive, the voltage V 3 and V 2 are designated as voltage rises and are subtracted. Figure 4 Diagram of a closed loop. When a power supply is connected across three resistors in series the same current flows through each resistor and the ammeter to measure the current must be in series with the resistors. To measure the voltage across a resistor the voltmeter is placed in parallel with the resistor. The total voltage supplied by the power supply is the sum of the voltages across each resistor. When a power supply is connected across three resistors in parallel the same voltage exists across each resistor and the ammeter to measure the current in a resistor must be in series with it. To measure the voltage across a resistor the voltmeter is placed in parallel with the resistor. The total current supplied by the power supply is the sum of the currents through each resistor. Figure 5 Voltage across series resistors. Page 54 Figure 6 Voltage across parallel resistors.

61 Procedure: Place the ohmmeter leads into the connection holes in blocks A and B. Record the value of the resistance R 1. Repeat for resistors R 2 and R 3. R 1 = R 2 = R 3 = Series Resistors: To connect the three resistors in series place a jumper between blocks B and C and a second jumper between blocks D and E. Place the ohmmeter leads in blocks A and D to measure the resistance between these points. Compare this value to that of R 1 + R 2. Repeat this between busses C and F. Repeat this between busses A and F. Figure 7 Circuit board with three resistors. R AD = R CF = R AF = R 1 + R 2 = R 2 + R 3 = R 1 + R 2 + R 3 = Figure 8 Jumper connections for three resistors in series. Parallel Resistors: To connect the three resistors in parallel place a jumper between blocks A and C and a second jumper between blocks C and E. Similarly place jumpers between blocks B and D and between blocks D and F. Place the ohmmeter leads in blocks A and B to measure the resistance between these points. Compare this value to that of R p where R p is calculated from 1/R p = 1/R 1 + 1/R 2 + 1/R 3. R p meas = R p calc = Figure 9 Jumper connections for three resistors in parallel. Page 55

62 Voltages in Series Resistors: Place jumpers between the blocks as shown in figure 10. Connect the power supply positive lead to block G and the negative lead to block L. Turn on the power supply and set the voltage to 5 V. Set the voltmeter to the 20 V scale. Place the leads of the voltmeter across each resistor and measure the voltages. Place the leads of the voltmeter across the blocks A and D. Compare the values of the voltages across resistors R 1 and R 2 with the measured voltages between blocks A and D. Place the leads of the voltmeter across the blocks C and F. Compare the voltage across blocks C and F with the voltages across resistors R 2 and R 3. Place the leads of the voltmeter across Figure 10 Voltage and current measurements for three resistors in series. the blocks A and F. Compare the voltage across blocks A and F with the voltages across resistors R 1, R 2 and R 3. V R1 = V R2 = V R3 = V AD = V R1 + V R2 = V CF = V R2 + V R3 = V AF = V R1 + V R2 + V R3 = Currents in Series Resistors: Turn off the power supply. Remove the jumper between blocks G and A. Connect the leads of the ammeter to these blocks and set the ammeter to the 20 ma scale. Turn on the power supply. Record the current. Repeat this for the jumpers between the blocks B and C, the blocks D and E and finally the blocks F and L. Record the average of these four values as I ave. I GA = I BC = I DE = I FL = I ave = Multiply each of the resistance values R 1, R 2 and R 3 by the average current I ave and record the results. V R1 = I ave R 1 = V R2 = I ave R 2 = V R3 = I ave R 3 = Compare these results with the measured voltages above. Page 56

63 Voltages in Parallel Resistors: Place jumpers between the blocks as shown in figure 11. Connect the power supply positive lead to buss G and the negative lead to buss L. Turn on the power supply and set the voltage to 5 V. Set the voltmeter to the 20 V scale. Place the leads of the voltmeter across each resistor and measure the voltages. V AB = V CD = V EF = Figure 11 Voltage and current measurements for three resistors in parallel. Currents in Parallel Resistors: Turn off the power supply. Remove the jumper between blocks G and A. Connect the leads of the ammeter to these blocks and set the ammeter to the 20 ma scale. Turn on the power supply and set the voltage to 5 V. Record the current. Repeat this for the jumpers between the blocks B and H, the blocks D and J and finally the blocks F and L. Compare the current I GA to the sum of the currents I BH, I DJ, and I FL. I GA = I BL = I DL = I FL = I BL + I DL + I FL = From Ohm s law predict the current in each resistor and record the results. I 1 = V AB / R 1 = I 2 = V CD / R 2 = I 3 = V EF / R 3 = Compare these currents with the values of I BL, I DL and I FL. Describe in your conclusion what you have learned about how resistors, currents and voltages combine in series and parallel circuits. 5. Complex Circuit Choose three resistors with values about 200 S, 300 S and 500 S. Since these values are not exact, measure the value of each resistor using the meters Figure 12 A simple current network. Page 57

64 provided. Build the circuit in figure 11, and then record the values of V 1, V 2 and I 1, I 2, I 3 and compare with the theoretical values that you can find by using Ohm s Law and Kirchoff s Laws. The values of I 1, I 2 and I 3 are determined theoretically from the three equations I 1 % I 2 % I 3 ' 0 R 1 I 1 & R 2 I 2 & 0 I 3 ' V 1 (7) 0 I 1 % R 2 I 2 & R 3 I 3 ' V 2 The first equation is the current law, equation (4), the second is the voltage law, equation (5) applied to the left side loop and the last equation is the voltage law applied to the right side loop of the circuit shown in figure 11. Solve for the theoretical values of the three currents using a method of your choice and compare the results to your measured values. Page 58

65 Applied Physics for the Health Professions Experiment 13 Magnetic Forces and Ampere s Law Objectives: < To determine the forces on a current carrying coil due to the magnetic field. < To determine magnetic field strengths generated by simple current geometries Equipment: < Power Supplies, Pasco, 8 amp < Lab jacks, clamps and stands < Bell Gauss meters with Hall element probes < Vernier Hall element probes < Solenoid with side slit, coils, 50 turns < Long wire mounted on a meter stick < Round and rectangular coils < Dual range force sensor < Science Workshop and Graphical Analysis software Physical Principles: Magnetic Force The force on a current-carrying wire segment in a magnetic field is given by, F ' BIR sin2 (1) where B is the magnetic field (in units of Tesla), I is the current, R is the length of the wire, and 2 is the angle between the current and the direction of the magnetic field. For current flowing through multiple wires, such as a rectangular coil, the force on a single wire must be multiplied by the number of wires in the coil, N. If B and I are perpendicular to each other, then sin 2 = 1, and the equation simplifies to, F ' NBIR (2) Biot - Savart Law According to the Biot-Savart Law, the strength of the magnetic field at a point P, caused by the current I in a short segment of wire of length L is given by the following formula: B ' µ 0 4B ILsin2 r 2 (3) Page 59

66 In this equation, the angle 2 is measured between the direction of the current flow and the direction of the point P from the current segment and r is the distance to the point. B is perpendicular to the plane of I and r in the direction of advance of a right hand threaded bolt when rotated in the direction of I into r through the angle 2 (not more than 180E). The standard unit of measurement for magnetic field (B) is Tesla (T). Another popular unit for this parameter is Gauss (G) which is given by: 1 Tesla = 10 4 Gauss. In equation (1), the constant µ 0 is given by 4B 10-7 Tm/A. Long Straight Wire: The magnetic field lines produced by a long straight wire (see figure 1) are circles concentric with the current. The magnetic field strength given by B ' µ 0 I (4) 2Br The right hand rule can be used to obtain the direction of the magnetic field. If the right hand thumb points in the direction of current flow, the right hand fingers of a half-closed hand point in the direction of the magnetic field. Figure 1 Measuring the magnetic field around a long straight wire. Solenoid: The magnetic field inside a long solenoid with n = N/L turns per unit length carrying a current Figure 2 Picture of a solenoid. I, is given by B ' µ 0 ni (5) For solenoids as in figure 2, that are not very long, the following formula is used to find the magnetic field: B ' 1 2 µ 0 ni@(cos2 1 %cos2 2 ) (6) The angles 2 1 and 2 2 are those subtended by the radius R of the ends as observed at the point where the magnetic field is measured. They are given by Page ' tan &1 R L 1 and 2 2 ' tan &1 R L 2 Figure 3 Geometry of the short solenoid.

67 where L 1 and L 2 are the distances from the probe to the ends of the solenoid. See figure 3. The direction of the magnetic field is parallel to the solenoid axis. With the fingers of the right hand wrapped around the solenoid in the direction of the current flow, the right hand thumb points in the direction of B inside the solenoid. Figure 5 Pasco Power Supply. Circular Coil: The magnetic field on the axis of a circular coil with a radius R and observed at a distance z along the axis from the plane of the coil is given by µ B ' 0 NIR 2 (7) 2@(z 2 %R 2 ) 3/2 With the fingers of the right hand wrapped in Figure 4 Picture of a circular coil. the direction of current flow, the right hand thumb points along the axis in the direction of the magnetic field. Procedure: Magnetic Force Hang the rectangular coil from the force sensor and attach the ends of the wires of the coil to the Pasco power supply with the banana clips and wires provided. The bottom edge of the coil should be centered between the poles of the permanent magnet. Use the paper guide provided to constrain the coil to remain oriented perpendicular to the magnetic field in the center of the magnet. Take care to see that the coil hangs freely inside the paper guide in order to eliminate frictional forces in the vertical direction. Measure the number of turns, N, of the coil, and the average length, R, of the bottom section of the coil, and record these values in your lab book. Using a Bell Hall element gaussmeter, measure the magnetic field at five equally spaced points along the bottom length of the coil. Record the average of these five values as the average magnetic field. You will compare this measured value with the experimentally Page 61

68 determined value from Equation (2). Open Science Workshop and calibrate the force sensor, using 0g for the minimum and 500g for the maximum weight. Under sampling options, set stop time to 1 second. Drag the graph icon to the force sensor icon and make a graph of force versus time. Click on the sigma icon on the lower left to turn on statistics options, and select mean and standard deviation statistics. Add a 100g stabilizing mass to the bottom of the coil, and take a reading, determining the mean and standard deviation for the 1 second measurement. This is the zero current value of the force, which should be subtracted from all future measurements. Turn the Pasco power supply on, set the current to +2 A, and take a reading of the mean and standard deviation of the force. Repeat for values of +4 A, +6 A, +8 A, -2 A, -4 A, -6 A, and -8 A. Open Graphical Analysis and make a graph of force (y-axis) vs. current (x-axis). Fit the data to a straight line and record the value of the slope. According to Equation (2), the slope will be equal to NBR. Divide the slope by NR to determine the magnetic field B. Compare the determined value with the value measured by the gaussmeter, and compute the percentage error. Magnetic Fields of simple current configurations Figure 6 F. W. Bell gauss meter. Since it is time consuming to set up the apparatus for the magnetic field experiments, you will take turns at stations already prepared. Below are several things that you should remember throughout the experiment. 1. Every time the Bell magnetic field meters are turned on, they have to be recalibrated. The instructions to do this are on the back of the device. 2. Both the voltage and current knobs on the power supply should be set to maximum. 3. Do not leave the power supplies on for too long when they are connected to the wires Page 62

69 since this causes the wires to overheat. 4. Normally, the power supplies display the voltage. If you need to know the current passing through the wire, press the button to the left of the display and hold it in. 1. Straight Wire Secure a 1.5 m lead along the length of a meter stick. Connect the positive lead from the Pasco power supply across one end of the straight wire. Connect the lead from the negative terminal to the common receptacle of the multimeter. Connect a banana lead into the recepticle marked 20 A. Connect the other end of this lead into an alligator clip and connect the clip to the other end of the long straight wire. Be certain that the enamel has been removed from the ends of the wire. Orient the meter stick horizontally along the north-south direction. Clamp the Hall element probe above the wire. The flat face of the probe should be vertical and in the plane of the wire. Place the sensor at a distance r of about 0.5 cm from the center of the wire. Set the voltage switch to the lowest value and rotate the voltage control pot until the current is near but not more than 8 amps. Measure and record in the data recording section the distance r from the wire in meters, the current I in amps and the magnetic field B in tesla. (One tesla is 10,000 Gauss.) Compare the measured value with the value calculated from equation (2). 2. Solenoid Orient the solenoid with its axis horizontally in the east-west direction. Connect the positive lead from the Pasco power supply across one end of the solenoid. Connect the lead from the negative terminal to the common receptacle of the multimeter. Connect a banana lead into the recepticle marked 20 A. Connect the other end of this lead into an alligator clip and connect the clip to the other end of the solenoid. Be certain that the enamel has been removed from the ends of the wire. Insert the Hall element gauss meter probe (see figure 6) into the slit on the side of the solenoid until the probe end is on the solenoid axis. Set the voltage switch to the lowest value and rotate the voltage control pot until the current is near but not more than 8 amps. Measure and record in the data recording section the radius R of the solenoid, number of turns N, the distances of the Hall probe from the ends L 1 and L 2, the number of turns per unit length n = N / (L 1 + L 2 ), the current I in amps and the magnetic field B in Tesla. Compare the measured value with the calculated value for the magnetic field from equation (3). For greater accuracy, use equation (4) to calculate the magnetic field B, using 2 1 ' tan &1 R L and 2 2 ' tan &1 R. Does equation (4) improve the error between 1 L 2 calculated and measured magnetic fields? 3. Circular Coil Double click on the Science Workshop icon to open the program. Click and drag the din connector icon to analog input channel A. Double click on the Magnetic Field Sensor in the Page 63

70 list of sensors. Double click on the magnetic field sensor icon under channel A to open the calibration window. With the probe plane along the earth s magnetic field and no current in the coil set the low value to 0 gauss and click on the read button. Mount the coil coaxially on a meter stick with the meter stick along the coil axis in the east-west direction. Mount the Bell Hall element probe of the gauss meter on the coil axis with the flat face of the probe vertical and perpendicular to the coil axis. Position the probe so that it is a distance of 9 cm from the coil plane. Set the voltage switch to the lowest value and rotate the voltage control pot until the current is near but not more than 8 amps. Read and enter this value of the magnetic field in the Science Workshop calibration window in tesla. Click on the Read button, then click on OK to complete the calibration. Replace the Bell Hall probe with the Vernier Hall probe and take all readings with it. For a series of distances z of the probe from the plane of the coil, measure the magnetic field and complete table 1 in the data recording section. Use values of z of about 9, 10, 11, 14, and 20 cm. Using Graphical Analysis plot 1 a graph of B as the ordinate (y) and as the abscissa (x) and compute a value of z 2 %R 2 3/2 µ µ o from the value of the slope which is 0 NIR 2 as predicted by equation 5. 2 Circular Coil z (z 2 + R 2 ) 3/2 1/(z 2 + R 2 ) 3/2 B Page 64

71 Applied Physics for the Health Professions Experiment 14 Electromotive Forces Objective: < To measure the motional emf induced on a coil passing through a magnetic field. Equipment: < Motion sensor < Pasco 1.2 m dynamics track < Dynamics cart with coil mounting bracket and rectangular coil < Permanent magnet < Banana clips < Bell Gauss meters with Hall element probes < Science Workshop and Graphical Analysis software Physical Principles: When a conducting wire, attached to a circuit, is passed through a perpendicular magnetic field, the induced emf (voltage) on the wire is given by, V ' B R v (1) where B is the magnetic field (in Tesla), R is the length of the wire, and v is the speed of the wire through the magnetic field. For a coil, with N turns, the induced emf is, V ' NBR v (2) If the maximum induced voltage is measured for different velocities of the coil through the magnet, a plot of maximum voltage (y-axis) vs. velocity (x-axis) will yield a slope equal to N B R. Thus, the average magnetic field of the magnet can be determined by dividing the slope of the graph by N R. Page 65

72 Procedure: Mount the rectangular coil onto the dynamics cart and attach the ends of the wires of the coil to the Science Workshop interface with the banana clips and wires provided. The far edge of the coil should be centered vertically between the poles of the permanent magnet. Measure the number of turns, N, of the coil, and the average length, R, of the bottom section of the coil, and record these values in your lab book. Using a Bell Hall element gaussmeter, measure the magnetic field at five equally spaced vertical points along the far edge of the coil. Record the average of these five values as the average magnetic field. You will compare this measured value with the experimentally determined value from Equation (2). Open Science Workshop and plug the motion sensor into digital channels 1 and 2, and the voltage sensor into analogue channel A. Set the motion sensor for 50 measurements per second, and the voltage sensor at 1000 measurements per second. Under sampling options, set the stop time to 3 seconds. Drag the graph icon to the force sensor icon and make a graph of voltage versus time, and a second graph of velocity versus time. Click on the sigma icon on the lower left to turn on statistics options, and select mean and standard deviation statistics. Click on the record icon and tap the launching post on the dynamics cart to launch the cart toward the motion sensor. Make sure to stop the cart before it runs off the end of the track or hits the motion sensor. Figure 1 F. W. Bell gauss meter. On the voltage vs. time graph, click and drag the cursor to select a portion of the top section of the graph that is approximately constant for the maximum voltage. Read the mean and standard deviation for V max from the statistics. Find the percent error of the measurement by taking the standard deviation divided by the mean and multiplied by 100%. On the velocity vs. time graph, measure the velocity of the cart at the point of maximum voltage. Calculate and record the value of V max / v, and compare with the measured value N B R, giving the percent difference. Page 66