Document 6541308
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Document 6541308
Brock University Physics Department St. Catharines, Ontario, Canada L2S 3A1 PHYS 1P91 Laboratory Manual Physics Department c Brock University, 2014-2015 Copyright Contents Laboratory rules and procedures 1 Introduction to Physica Online 5 1 The pendulum 11 2 Check your schedule! 19 3 Angular Motion 21 4 Collisions and conservation laws 29 5 The ballistic pendulum 35 6 Harmonic motion 41 A Review of math basics 47 B Error propagation rules 49 C Graphing techniques 51 i ii Laboratory rules and procedures Physics Department lab instructors Frank Benko, office B210a, ext.3417, [email protected] Phil Boseglav, office B211, ext.4109, [email protected] This information is important to YOU, please read and remember it! Laboratory schedule To determine your lab schedule, click on the marks link in your course homepage; your lab dates are shown in place of the lab marks and correpond to the section number that you selected when you registered for the course. You cannot change these dates unless you have a conflict with another course and make the request in writing. The schedule consists of five Experiments to be performed every second week on the same weekday. On any given day there are four different Experiments taking place, with up to three groups of no more than two students in a group performing the same experiment. You need to: • prepare for your scheduled experiment. Out of schedule Experiments cannot be accomodated; • be on time. The laboratory sessions begin at 2:00 pm and end no later than 4:45 pm and you will not be allowed entry once the experiments are under way. Lab report format and submission You are required to submit the Discussion component of your Lab Report to www.turnitin.com prior to the lab submission deadline. Instructions for registering and submitting your work are found in your course web page. Be sure to have a working account before you need to use it. After you submit your Discussion to the Turnitin webpage, Turnitin will email to your Turnitin login email address (i.e. [email protected]) a receipt tagged with the submission date and a unique ID number. Include this email as part of your lab report. Submit your report in the clearly marked wooden box across the hall from room MC B210a. Reports are due by midnight one week after the experiment is performed. For example, the report for an experiment performed on a Tuesday is due by midnight on the Tuesday following. Compile your Lab Report as follows: • submit the complete Lab Report in a clear-front document folder. Do not use three-hole Duo-tang folders, envelopes or submit a stapled set of pages; • insert the first lab worksheet so that your name is visible through the folder front cover. Do not include a title page as the first experiment page is the title page; 1 2 • add the other lab worksheets in the proper sequence, followed by printouts and pages of calculations. • At the end of the Lab Report include a complete copy of the email sent by Turnitin, followed by a complete copy of the Discussion that you submitted to Turnitin. This can be a printout from your Word Processor or a copy downloaded from the Turnitin web site. • Note: you should anticipate and be prepared for the likelyhood that Turnitin may not provide an immediate email response following your Discussion submission; this response may take several hours. Submit your work well ahead of the submission deadline. • Note: Late Lab Reports will receive a zero grade, no exceptions. • Note: Lab Reports not formatted as outlined will receive a 20 % grade deduction. • Note: Marked Lab Reports will be returned to you during your next Lab session. Be sure to pick-up your marked labs and review the comments made by the marker. The lab manual Your lab manual is available as a .PDF document in your course webpage. This allows you to print a copy of the experiment that you need for the current lab. It also allows the department to make quick edits to the manual to fix typographical errors, etc. This lab manual contains five experiments. Each experiment consists of three components, and completing the lab means reaching all three of the milestones described below. 1. Pre-lab review questions, to be completed before entry into the lab, are intended to ensure that the student is familiar with the experiment to be performed. A Lab Instructor will initial and date the review page if the questions are answered correctly. The review questions contribute to your lab grade. • You will be required to leave the lab if the review questions are not completed as instructed. Missing your assigned lab date could result in a grade of zero for that Experiment. • Be sure to have a TA check and initial the completed review questions before you begin the lab. Lab reports missing the initials will be subject to a 20 % grade deduction. • In case of difficulties with any of the review questions, a student is expected to seek help from a lab instructor well before the day of the lab. 2. A lab component is the actual performing of the experiment. Marks are deducted for failing to complete all of the required procedures, follow written instructions, answer questions, provide derivations, the improper use of rounding and incorrect calculations. The lab report markers use a standard marking scheme to grade the lab reports. At the end of the lab session, if the lab procedures have been completed as required, a Lab Instructor will also initial and date the front page of your Experiment. • An incomplete lab component will not be initialled; you will need to finish the work on your time and have it signed before submission. A report missing this signature will be subject to a 20 % grade deduction. 3. The final component is the compilation of the experimental data, its analysis, and a critical assessment of the results into a lab Discussion. This component is worth 40 % of the lab mark. 3 The Discussion should consist of a series of paragraphs rather than an itemized list of one-line answers. You do not need to review the theory or reproduce formulas or tables of experimental data contained in the workbook as part of the discussion. You should: • begin the discussion with a tabulated summary of your data, properly rounded according to the associated margin of error; • thoughtfully answer and expand on the given guide questions, outline your observations, summarize the results of the experiment and support your conclusions with data or reasoned arguments; • assess the validity of your results by comparing your values and their associated errors with values estimated from the theory or cited in your textbook or other literature. Suggestions for improving the experimental procedure and a summary of the implications of the obtained results will make the discussion complete. A guide to team collaboration To ensure that the collaborative nature of the experimental team is expressed in a fair and mutually advantageous way for every member of the team: • Come prepared and ready to participate constructively as part of your lab team! • Do not sit idle and expect others to provide you with their data. The data gathering procedures should be undertaken by all the members of the team. While it may not be practical to have every student perform the same reading every time, each member of the team must become familiar with the equipment and perform some of the readings. The lab instructor will ask procedural questions during the lab and you will be expected to know what is going on in the experiment. All measurements are to be made by more than one student. This is a very effective way to verify a measurement; the use of an incorrect value in a lengthy calculation can waste a lot of your team’s lab time and result in an incomplete lab. All labs finish by 4:45pm sharp. • Do your own calculations!. There is sufficient time during the lab for this to be accomplished. As above, this is also a good strategy; comparing the results of several independent calculations can expose numerical errors and lead to the correct result or give you the confidence that your result is indeed correct. To access a calculator on your workstation, type xcalc in a terminal window. • Submit your own set of graphs. Enter your own data, include your name and a description of the plotted data as part of the title. This approach will also expose any errors in the data entry or the computer analysis of the data. Needless to say, the Discussion section of the lab report is not to be a collaborative effort. • Warning: Do not copy someone else’s review questions, calculations or results. This is an insult to the other students, negates the benefits of having an experimental team and will not be tolerated. Any such situations will be treated as plagiarism. You should review in your student guide Brock University’s definition and description of plagiarism and the possible academic penalties. • Warning: Do not allow others to copy the content and results of your calculations or review questions; doing so makes you equally responsible under the definition of plagiarism. Do not feel pressured to allow another student to copy your work; inform the lab instructor. 4 Academic misconduct The following information can be found in your Brock University undergraduate calendar: ”Plagiarism means presenting work done (in whole or in part) by someone else as if it were one’s own. Associate dishonest practices include faking or falsification of data, cheating or the uttering of false statements by a student in order to obtain unjustified concessions. Plagiarism should be distinguished from co-operation and collaboration. Often, students may be permitted or expected to work on assignments collectively, and to present the results either collectively or separately. This is not a problem so long as it is clearly unbderstood whose work is being preented, for example, by way of formal acknowledgement or by footnoting.” Academic misconduct may take many forms and is not limited to the following: • Copying from another student or making information available to other students knowing that this is to be submitted as the borrower’s own work. • Copying a laboratory report or allowing someone else to copy one’s report. • Allowing someone else to do the laboratory work, copying calculations or derivations or another student’s data unless specifically allowed by the instructor. • Using direct quotations or large sections of paraphrased material in a lab report without acknowledgement. (This includes content from web pages) Specific to the Physics laboratory environment, you will be cited for plagiarism if: • you cannot satisfactorily explain to the lab instructor how you arrived at some numerical answer entered in your laboratory workbook; • you cannot satisfactorily describe to the lab instructor how you derived a particular equation in the lab procedure or as part of the review questions; • your data is identical to that of some other student when the lab procedure stated that each student should obtain their own data. The above points are based on the conclusion that if you cannot explain the content of your workbook, you must have copied these results from someone else. In summary, you are allowed to share experimental data (unless otherwise instructed) and compare the results of calculations and derivations for correctness with other members of your group, but the derivation of results must be your own work. Academic penalties A first offence of plagiarism in the lab will result in the expulsion of all parties concerned from that lab session and the assignment of a zero grade for that particular lab. A record will be made of the event and placed in your student file. A subsequent offence will initiate academic misconduct procedures as outlined in the Brock University undergraduate calendar. Introduction to Physica Online Overview Figure 1: Physica Online opening screen Physica Online is a web-based data acquisition and plotting tool developed at Brock University for the first-year undergraduate students taking introductory Physics courses with labs. It is accessible on the 5 6 INTRODUCTION TO PHYSICA ONLINE Web at http://www.physics.brocku.ca/physica/ When you first point your browser to this page, you should see approximately what is shown in Fig. 1. The “engine” behind the Web interface is the program physica written at the Tri-University Meson Facility (TRIUMF) in Vancouver. It is also available in a stand-alone menu-driven Windo$se version from http://www.extremasoftware.com/. First and foremost, Physica Online is a plotting tool. It allows you to produce high-quality graphs of your data. You enter the data into the appropriate field of the Web interface, select the type of graph you want, and make some simple choices from the self-explanatory menus on the page. After that, a single button generates the graph. You can view it, print it, save it as a PostScript file for later inclusion into your lab report. Physica Online is a fitting and data analysis tool. The physica engine has extensive and powerful fitting capabilities. Only a small subset is used in the “easy” default web mode, but it is sufficient for all of the experiments that Brock students encounter in their first-year labs. The full “expert” mode is also available for those needing more advanced capabilities of full physica; some learning of commands may be required. Physica Online is a data acquisition tool. The web interface connects to a LabProTM by Vernier Software or a similar interface device, typically attached to a serial port of one of the thin clients1 (or of some serial-port server). You can then copy-and-paste the returned data into the data field of Physica Online, ready to be plotted and/or analysed. Below, we will follow the approximate sequence of a typical lab experiment. A demo mock-up of one such experiment (RC time constant determination) is available online, even from outside of the lab. It may be useful to open a browser, and point it to the demo RC lab while reading this manual. Acquiring a data set The data acquisition hardware consists of a variety of interchangeable sensors connected to a programmable interface device called a LabPro. This unit samples the sensors and transmits the data to a serial port of a thin client (or a personal computer). To acquire a set of data from a sensor press Get data in the control panel of Physica Online. In the main plot frame to the right, a LabPro frame will open up, similar to the one seen in Fig. 2. In this frame you have to set several options by hand. Begin by identifying the IP address of the thin client to which the LabPro hardware is attached. The thin client is identified as ncdXX, where XX should be set as indicated by the label on the terminal. Usually, it is the one you are sitting at, but sometimes you may need to use the LabPro attached to another thin client. Several groups of students can use the same hardware device to collect data, but not at the same time! In the example shown in Fig. 2, the IP is set to ncd36. Next, specify one or more channels from which the data will be read. There are four available analog channels, Ch1–Ch4, used to attach probes of voltage, temperature and light intensity. The two digital channels, Dig1 and Dig2, are used to connect probes such as photogate timers and ultrasonic rangefinders. More than one channel can be selected; in this case, more than two columns of data will be returned by the LabPro. In the example of Fig. 2, a single voltage probe is attached to Ch1. Select the number of data points to be collected, the delay between successive data points and then initiate the data acquisition by pressing the Go button. Once the data collection begins, a progress message appears in this window indicating the time required to complete the data collection. Be patient 1 Thin clients are desktop devices that provide a display, a keyboard, a mouse, etc., but that do not have a disk or an operating system software. Instead, they connect to one of several possible servers, running whatever operating system that is required. The files and applications run on these servers, and the thin clients, or Xterminals, take care of the interactions with the user. INTRODUCTION TO PHYSICA ONLINE 7 Figure 2: LabPro configuration frame and let the LabPro process complete. If something is wrong and the browser is unable to communicate to the LabPro, it will time out after a few extra seconds of waiting. This may happen if the network is busy or if more than one browser is trying to obtain the data from the same LabPro; check that your IP field is set correctly. Once the data acquisition is complete, you will have two columns of data in front of you. The next step is to select the data using your mouse, and copy-and-paste it to the data entry field below, as shown in Fig. 3. You can also do this by pressing the Copy from above button. You are now ready to plot and fit the data. Graphing your data The data entry field of Physica Online is usually filled through a copy-and-paste operation from the LabPro frame, as described in the previous section. You can also manually enter into this field any other data2 that you wish to graph and analyse. You can press Ctrl+A to select all of the contents in the data window and Ctrl+X to delete those contents. On some machines, you need to use Alt+A and Alt+X. The default settings are appropriate for generating a scatter plot of the data; all you need to do after entering or pasting in the data is to press Draw . You have the option of associating error bars with each data point by entering one or two extra columns into the data field; the third column, if present, would be interpreted as ∆y, and the fourth one, if present, as ∆x. If the error is the same for all data points, you may use the the dx: and dy: fields below the data field. To omit the error bars, set these values to zero (this is the default). There are several graphing options available. A scatter plot graphs a set of coordinate points using a chosen data symbol of a specific size. Check the line between points box to connect the data points with line segments or the smooth curve box to interpolate a smooth curve through the data points. After you made all your selections and pressed Draw , you may see the plot that looks similar to that shown in Fig. 4. 2 Feel free to use Physica Online for preparing graphs for your other lab courses! 8 INTRODUCTION TO PHYSICA ONLINE Figure 3: LabPro data has been copy-and-pasted into the data entry field Figure 4: Physica Online data scatter plot INTRODUCTION TO PHYSICA ONLINE 9 The fit to: y= option allows you to fit an equation of your choice to the data points. A second-order polynomial is pre-entered as the default, but in each experiment this will need to be changed to reflect the expected form of the y(x) dependence. The simple web interface allows up to four fitting parameters A, B, C, D, which is enought for most of the first-year lab data. Much more elaborate fitting is possible from the “Expert Mode” of Physica Online. One essential point about fitting, especially when the fitting equation contains non-linear functions such as sin(x) or exp(x), is to have a good set of approximate initial guesses for all fitting parameters. Examine the scatter plot of your data carefully, estimate the approximate values of all parameters your use in the fitting equation and enter your initial estimates in the fields provided. The default values of A = B = C = D = 1 are almost never going to work. If the fit fails to converge, Physica Online will return a text error message when you press Draw , you should then re-examine whether the fitting equation and the initial guesses for all parameters have been entered correctly. You can fit more than one function to your data set, such as for example, a steady-state straight line followed by an exponential decay. These fits are explained further in the experiment in which they occur. You can also constrain the fit to two separate regions of your data set. In this case, you must copy-andpaste the fitting equation used in the fit to: y= box into the constrain X to: box or the error values will be incorrect. Additional settings allow you to display the plot with the axes scaled in linear or logarithmic units, and the scale limits and increments can be manually set. A grid can be optionally included. The font and size of the text used to label the axes and in the title is also user selectable. Fig. 5 shows a set of values Figure 5: Physica Online fit and plot parameters and settings that Ms. Jane Doe may have used for her data set. When she presses Draw , Physica Online returns the plot shown in Fig. 6. An Expert mode button is available if other, more advanced features of physica are desired. Selecting this mode passes on all the settings from the “Easy Mode” and allows further changes to be made directly to the physica macro script. There is on-line help and tutorials, as well as hardcopy reference 10 INTRODUCTION TO PHYSICA ONLINE Figure 6: Experimental data and the fit, with settings from Fig. 5 manuals if you want to learn how to use these advanced features. For example, you may wish to plot two data sets on the same graph and add a legend. Feel free to change the macro; if you run into difficulties, press Easy mode and start again. You can press Print to redirect the output to a PostScript printer. Only a few printer names are accepted as valid by the web script; your TA will tell you which one to use. If you leave the Print to: field blank, a PostScript file will be sent to your browser; if the browser knows how to display PostScript (though GhostView, or Adobe Acrobat, or a similar external application) it will do so. Otherwise, it should offer you an option to save it as a file; this is useful for later including your plot in a lab report or attaching it to an email. Your browser’s Back button will come in handy on occasion. If you find yourself hopelessly lost, you can also use Reload to bring you back to the starting point, although this will reset the graph settings such as title and axis labels to their default values. first name (print) last name (print) Brock ID (ab13cd) TA initials grade Experiment 1 The pendulum A simple pendulum consists of a compact mass m suspended from a fixed point by a string of length L, as shown in Fig. 1.1. Figure 1.1: Pendulum experimental arrangement The equilibrium position of m is O. Here, the tension P~ exerted by the string on m is exactly equal and opposite to the weight m~g of the mass. This is not true however if m is anywhere else along the arc, say at an angle α (in radians) with respect to O. Then the weight will have a component tangent to the arc, whose magnitude is m~g sin α. If α is small, we may approximate sin α ≈ α, so that the force on m is equal to mgα. This force is a restoring force, as it will drive m back to its equilibrium position O. When a mass is acted on by a restoring force, whose magnitude mgα is proportional to the deviation α from O, then the resulting motion is an oscillation around the equilibrium position. In our case, m will swing back and forth around O. The time for one complete swing is defined as the period T . An equation for acceleration due to gravity g is given by: g= 4π 2 L T2 (1.1) This equation predicts that the period T is independent of the mass m of the ball, and that T is also independent of the angle α through which the ball swings, as long as the approximation sin α ≈ α is valid. For α = 15◦ ≈ 0.2618 radians, there is a difference of approximately 1.2% between α and sin α. 11 12 EXPERIMENT 1. THE PENDULUM Introduction to error analysis The result of a measurement of a physical quantity must contain not only a numerical value expressed in the appropriate units; it must also indicate the reliability of the result. Every measurement is somewhat uncertain. Error analysis is a procedure which estimates quantitatively the uncertainty in a result. This quantitative estimate is called the error of the result. Please note that error in this sense is not the same as mistake. Also, it is not the difference between a value measured by you and the value given in a textbook. Error is a measure of the quality of the data that your experiment was able to produce. In this lab, error will be considered a number, in the same units as the result, which tells us the precision, or reliability, of that experimental result. Note that error value, represented by the Greek letter σ (sigma), is always rounded to one significant digit; the result is always rounded to the same decimal place as σ (see below). Error of a single measurement Consider the measurement of the length L of a bar using a metre stick, as shown in Figure 1. One can see that L is slightly greater than 2.1 cm, but because the smallest unit on the metre stick is 1 mm, it is not possible to state the exact value. We can, however, safely say that L lies between 2.1 cm and 2.2 cm. The proper way to express this information is: L ± σ(L) = 2.15 ± 0.05 cm This expression states that L must be between, (2.15−0.05) = 2.10 cm and, (2.15+0.05) = 2.20 cm, which is our observation. The quantity σ(L) = Figure 1.2: Measurement with a metre stick ±0.05 cm is referred to as the maximum error. This number gives the maximum range over which the correct value for a measurement might vary from that recorded, and represents the precision of the measuring instrument. Propagation of errors In many experiments the desired quantity, call it Z, is not measured directly, but is computed from one or more directly-measured quantities A, B, C, . . . with a mathematical formula. In this experiment, the directly-measured quantities are T , y and b, and the desired quantity g is calculated from g = 4π 2 L/T 2 , with L = y + 12 b. The following rules give a quick (but not exact) estimate of σ(Z) if σ(A), σ(B) etc. are known Always use the absolute value of an error in a calculation . Error rules are tabulated in the Appendix. 1. If Z = cA, where c is a constant, then σ(Z) = |c|σ(A). This is used only if A is a single term. For example, it can be used for Z = 3y, so that σ(Z) = 3σ(y), but not for Z = 3xy. 2. If Z = A + B + C + · · ·, then σ(Z) = σ(A) + σ(B) + σ(C) + · · · . For example, if 1 b 2 1 σ(L) = σ(y) + σ b (See 2. above.) 2 1 σ(L) = σ(y) + σ(b) (See 1. above.) 2 L = y+ then 13 3. To derive an error equation for any relation, rewrite that relation as a series of multiplications, then apply the change of variables method as shown in the Appendix to evaluate the error terms: g= 4π 2 L −→ g = 4π 2 LT −2 −→ g = ABCD, ( letting A = 4, B = B = π 2 , C = L, D = T −2 ) T2 Then and σ(A) = σ(4), σ(g) σ(A) σ(B) σ(C) σ(D) = + + + , |g| |A| |B| |C| |D| (Appendix, Rule 4) σ(π) σ(T ) σ(D) σ(B) = |2| = | − 2| , σ(C) = σ(L), . |B| |π| |D| |T | (Rules 1,6) The quantities 4 and π are constants and have no error (strictly speaking, σ(4) = σ(π) = 0), therefore these terms do not contribute to the overall error. The error equation for g then simplifies to σ(π) σ(g) σ(4) = +2 |g| |4| |π| σ(T ) σ(L) + |−2| + |L| |T | σ(T ) σ(L) σ(g) = +2 . |g| |L| |T | −→ The right hand side of the above equation, called the “relative error” of g, results in a fraction that describes how large σ(g) is with respect to g. The desired quantity, σ(g), is obtained by multiplying both sides of the equation by g: σ(T ) σ(L) +2 σ(g) = |g| |L| |T | . Standard deviation of a series of measurements When the same measurement or calculation is repeated several times, an error estimate can be calculated form the variation in this set of values. This is convenient because on other error values, measured or calculated, need to be known. For example, the five results for g can be used to estimate an error σ(g). To calculate the standard deviation σ(g) for a sample of N values of g: 1. determine the average hgi of N values gi , where i = 1, 2, · · · , N by summing the values and then dividing by the number of values: hgi = 1 (g1 + g2 + · · · + gN ); N 2. for each gi , calculate the deviation ∆gi = gi − hgi from the average hgi; 3. for each gi , calculate the square of the deviation (∆gi )2 to make all the values positive; 4. calculate the variance σ 2 (g) of the sample by summing the squares of the deviations (∆gi )2 and dividing by the number of values minus one, N − 1 σ 2 (g) = 1 2 (∆g12 + ∆g22 + · · · + ∆gN ); N −1 5. undo the previous squaring operation by taking the square root of the variance. This is the root average squared deviation, or standard deviation σ(g), of b: σ(g) = q σ 2 (g) A large σ value indicates a result which is not very precise. The theory of statistics shows that the probability that a further measurement of g falls within the range of σ(g) is 68% and that σ is proportional √ to 1/ N , so σ will decrease as the number of samples N is increased. 14 EXPERIMENT 1. THE PENDULUM Rounding a final result: X ± σ(X) The value of σ(x) is rounded to one significant digit whether it represents a maximum error estimate, calculated error, or standard deviation of a sample. The result corresponding to this error must be rounded off and expressed to the same decimal place as the error. For example, hxi = 25.344 mm and σ(x) = 0.0427 mm. Rounded to one digit, σ(x) = 0.04 mm. Rounded to the same decimal place, hxi = 25.34 mm. The final result is expresses as hxi ± σ(x) = (25.34 ± 0.04) mm. Do not use a rounded off value in further calculations. Use the original unrounded value. Use of a truncated value will decrease the quality of your result. Powers of 10 Express both the result and its error to the same power of 10. This allows the reader to immediately judge how large the error is relative to the result: 1. 2.68 × 10−2 ± 5 × 10−4 should be written as 0.0268 ± 0.0005 or (2.68 ± 0.05) × 10−2 . Note the parentheses, indicating that both the result and the error are to be multiplied by 10−2 . 2. 1.634 ± 3 × 10−3 m should be written as 1.634 ± 0.003 m Format of calculations Record all calculations, in the appropriate space if provided or on a separate sheet of paper. A calculation is performed in three lines. The first line displays the formula used. In the second line, the variables in the formula are replaced with the actual values used in the calculation. The third line shows the final answer formatted according to the section on Rounding above and if any, the units associated with the result. Review questions Using a textbook, the Physics Handbook, the Internet or some other source, find the accepted value of g. Cite your reference so that a reader can find this information. Be specific; do not cite ’my physics book’, ’my buddy’ or ’i remember it from high school’ as your source. Include specific info such as the uncertainty, or error, in the value, and the condition (i.e. sea level) under which it was measured. g = .............................................................................. An error value is only meaningful when expressed with ..... significant digits. The measurement error σ of a scale graduated in minutes and seconds is .................... My Lab dates: Exp.1:...... Exp.3:...... Exp.4:...... Exp.5:...... Exp.6...... I have read and understand the contents of the Lab Outline (sign) ................ CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! Procedure and analysis You will determine the period T of a pendulum of length L by monitoring the distance of the pendulum bob from a computer controlled rangefinder. This device determines the distance to an object by transmitting an untrasonic ping and measuring the time required to receive the echo of the original signal after it is reflected from the object. 15 The transmitted signal propagates in a narrow conical beam and as it is the echo from the nearest object that is accepted, you will need to be sure that you are pinging the pendulum bob and not the pendulum arm or some other structure in the cone of the beam. As the pendulum bob swings toward and away from the rangefinder, the variation in distance over time can be approximated by a sine wave. By analysing the properties of this sine wave, we can calculate the period T of the pendulum. The pendulum is an aluminum ball suspended by a light nylon string from a sliding arm. Assume the mass of the string is negligibly small compared to the mass of the ball. To calibrate the pendulum: 1. Release the string to set the ball on the table. Align the bottom of the sliding arm with the zero mark on the scale. Adjust the string length so that the top of the ball lightly contacts the bottom of the arm, ensuring that the string is not stretched. Secure the string with the clamping nut. 2. Raise the arm away from the ball and carefully reposition the arm until it once again just contacts the top of the ball. If the bottom of the arm is no longer at the zero mark, repeat the sequence of adjustments until the bottom of the arm is in line with the zero mark of the scale and the arm lightly contacts the top of the pendulum ball. All members of the group must check the calibration. The scale is now calibrated to display the length of the string y from the bottom of the sliding arm, the pivot point of the pendulum, to the top of the ball, to a precision of one millimetre (mm). This number is not exactly equal to L since the ball is not a point-mass. The length of the pendulum L is the sum of the length of the string y and half the ball’s diameter b given in Table 1.1: 1 L = y + b. (1.2) 2 Mount the larger ball m1 and calibrate the pendulum. Adjust the sliding arm so that the string length y is approximately 0.3 m. Record the actual length to a precision of 1 mm. • Set the pendulum swinging in a straight line, keeping α less than approximately 15◦ . Wait several seconds to allow for any stray oscillations present in the bob to dissipate. • Shift focus to the Physicalab software. Check the Dig1 box and choose to collect 50 points at 0.1 s/point. Click Get data to acquire a data set. • Click Draw to generate a graph of your data. Your points should display a nice smooth sine wave, without peaks, stray points, or flat spots. If any of these are noted, adjust the position of the rangefinder and acquire a new data set. • Select fit to: y= and enter A*sin(B*x+C)+D in the fitting equation box. Click Draw . If you get an error message the initial guesses for the fitting parameters may be too distant from the required values for the fitting program to properly converge. Look at your graph and enter some reasonable initial values for the amplitude A of the wave and the average distance D of the wave from the detector. The value of B (in radians/s) is given by B = 2π/T since the period of a sine wave is 2π = B ∗ x radians, where x = T is the period in seconds. Estimate the time T between two adjacent minima of the sine wave. Estimate and enter an initial guess for B. (See Appendix A) • Label the axes and identify the graph with your name and the string length y used. Click Print to print your graph. Every student should print a copy of each graph. • Record in Table 1.1 the trial length y and the computer calculated value of the fitting parameter B (radians/second). Do a quick calculation for g to make sure that the value is reasonable. • Repeat the above steps for m1 with y = 0.45, 0.60, 0.75 and 0.90 m. • Mount the second ball m2 , recalibrate the pendulum and verify the calibration, set the string length to approximately y = 0.5 m, and record this one measurement in Table 1.1. 16 EXPERIMENT 1. THE PENDULUM Run, i m (kg) b (m) 1,m1 0.0225 0.02540 2,m1 0.0225 0.02540 3,m1 0.0225 0.02540 4,m1 0.0225 0.02540 5,m1 0.0225 0.02540 1,m2 0.0095 0.01904 y (m) L (m) B (rad/s) T (s) T 2 (s2 ) gi (m/s2 ) Table 1.1: Table of experimental results Part I: Determining g from a single measurement • Use Equation 1.2 and the single measurement of m2 to calculate L and g. Show the calculation below, in three steps as previously outlined and recalling the proper application of significant figures and physical units. L = ............................... g = ............................... ............................... ............................... ............................... ............................... • The measurement errors in y, b, and T , represented by σ(y), σ(b), and σ(T ), respectively, are determined from the scales of the measuring instruments. The micrometer used to measure the ball has a resolution, or scale increment of 0.00001 m. The data logger can measure time with a precision of 0.00002 s. Summarize these measurement errors below: σ(y) = ±.................. σ(b) = ±.................. σ(T ) = ±.................. • Calculate the error σ(L) in the length L and the error σ(g) in g. Express the final values as L ± σ(L) and as g ± σ(g), with the correct units. σ(L) = ............................... σ(g) = ............................... ............................... ............................... ............................... ............................... L = .............. ± ............... g = .............. ± ............... 17 Part II: Determining g from a series of measurements • Use Equation 1.1 to calculate the five measurements of g for m1 . Record the results in Table 1.1 and Table 1.2. Include all the calculations for m1 as part of your Lab Report. • Use Table 1.2 to calculate the average and standard deviation of the sample of g values, then summarize the result in the proper format below: i gi gi − hgi (gi − hgi)2 1 2 3 4 5 hgi = variance = σ(g) = Table 1.2: Calculation template for the standard deviation of g. g = ............. ± ............. Part III: Determining g from the slope of a graph Equation 1.1 can be written as: g T2 (1.3) 4π 2 This is the equation of a straight line Y = mX + b, where in this case Y = L, X = T 2 , b = 0, and the slope m = rise/run = g/(4π 2 ). L= • Using the Physica Online software, enter the five coordinates (T 2 , L) in the data window. Select scatter plot. Click Draw to generate a graph of your data. Select fit to: y= and enter A*x+B in the fitting equation box. Click Draw . The computer will generate a line of best fit through the data points. Print the graph, then record below the slope A and error σ(A) A = ............. ± ............. • Calculate below values for g and σ(g). g = ....................... = ....................... = ....................... σ(g) = ....................... = ....................... = ....................... g = ............. ± ............. 18 EXPERIMENT 1. THE PENDULUM Summary of results Now that you have quantitatively estimated the reliability of your various results for g by application of error analysis, you can make a meaningful comparison of these results with the accepted value of g = 9.80 m/s2 . • In the space below, plot and label your three results for g and the error bars representing the uncertainty in each of these results. Begin by drawing on the given line a scale that is representative of the range in your results. The range of agreement in two results is determined by the region of overlap of their error bars. For clarity, plot your data and error bars above one another. • Which of the error bars overlap the accepted value of g? ............................... • Is there a region where all the error bars overlap? ..................................... • Do any of your error bars overlap the accepted value of g? .............................. IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Discussion Your Discussion should summarize the essence of this experiment in a clear and organized way, so that someone reading (or marking) this report will obtain a good understanding of what was done, how it was done, and of the results that were obtained. Here are some ideas on the issues you should address in your discussion: Begin by tabulating your three experimental values and the accepted value of g. Identify the source of the data and include appropriate units. Give the table a descriptive title. Compare your three results for g. Do your values agree with the accepted value of g. Explain your reasoning. Consider which method is likely to provide the most reliable value for g. Which method is the least reliable? Explain your conclusions. From your graph, determine whether T varies with L. Does T vary with the mass of the ball? Explain. Do your conclusions agree with the predictions of Equation 1.1? Both the averaging method and the line of best fit use the same five experimental values for L and T . What similarities do you think there may be in these two approaches to determining g? If drawing a line through the plotted experimental points, how would you choose where to place the line? How might this choice affect your resulting value of g? What is the resolution, or scale, of the time measurement and how did you determine this? How might the resolution be improved? A Final Note: Have you submitted your Discussion to Turnitin and printed and included the receipt that Turnitin sent you? If not, you will lose 40% of your grade. Also, be sure to submit your Lab Report only in a clear-front folder (no duo-tangs, envelopes or staples) or you will lose 20% of your grade. first name (print) last name (print) Brock ID (ab13cd) TA initials grade Experiment 2 Check your schedule! This is a reminder that there is no Experiment 2 and that you need to check your lab schedule by following the Marks link in your course homepage to determine the experiment rotation that you are to follow. The lab dates are shown in place of lab grades until an experiment is done and the mark is entered. My Lab dates: Exp.3:......... Exp.4:......... Exp.5:......... Exp.6......... Note: The Lab Instructor will verify that you are attending on the correct date and have prepared for the scheduled Experiment; if the lab date or Experiment number do not match your schedule, or the review questions are not completed, you will be required to leave the lab and you will miss the opportunity to perform the experiment. This could result in a grade of Zero for the missed Experiment. To summarize: • There are five Experiments to be performed during this course, Experiment 1, 3, 4, 5, 6. • Everyone does the first experiment on the first scheduled lab session. • The next four experiments are scheduled concurrently on any given lab date. • To distribute the students evenly among the scheduled experiments, each student is assigned to one of four groups, by the Physics Department. The schedule is entered as part of your lab marks. Lab make-up dates: You may perform one missed lab on December 1, 2, 3, 4, 5 from 2-5 pm. N otes : .......................................................................... ................................................................................ ................................................................................ ................................................................................ 19 20 EXPERIMENT 2. CHECK YOUR SCHEDULE! first name (print) last name (print) Brock ID (ab13cd) TA initials grade Experiment 3 Angular Motion Imagine placing a drop of paint near the edge of a rotating wheel. As the wheel rotates with an angular speed ω, the marked point moves along a circular trajectory, its path a certain arc length s. When a wheel turns through a full circle, θ = 2π, this arc length is s = 2πr; for half a circle, θ = π, the arc length is half the circumference of a circle, sπr, etc. For an arc of radius r that spans an angle θ, s = θr. In this way, the radius of the circle acts as a natural unit of angle: one radian is the angle that spans the arc of length s = r. It is natural to express angles in radians: the circumference of a circle, 2πr, corresponds to the angle of 2π radians. Another way of seeing this is to express all lengths in units of r or, equivalently, to choose a unit circle, r = 1. The circumference of a full unit circle is 2π, for the angle Figure 3.1: Rotational coordinates of θ = 2π radians. Note that since the radian is a unit that represents a ratio of two lengths, θ = s/r, it is a dimensionless quantity. To convert between radians and other units of measuring angles, conversion coefficients such as π/180◦ or 180◦ /π are introduced as appropriate. If the particle moves a distance s and sweeps an angle θ in a time t, it has a linear speed v = s/t along the circular path and an angular speed ω = θ/t = (s/r)/t = v/r. The tangential acceleration of the particle is a = v/t and the corresponding angular acceleration is α = ω/t = a/r. The inverse relations are, of course, v = rω and a = rα, so all translational quantities scale with r. The kinetic energy of the particle is then K= mv 2 mr 2 ω 2 Iω 2 = = . 2 2 2 The quantity I = mr 2 represents the rotational moment of inertia of a point mass m placed a distance r from the center of rotation; for extended bodies, adding up contributions from many small point masses that make up the body yields a result that is always proportional to M R2 , where M is the total mass and R is some measure of the size of the body. However, there will be a numerical factor in front of M R2 that will vary with the shape of the body. It has been calculated and tabulated for many common shapes: disks, rods, spheres, etc. The physical meaning of I is that it’s a measure of the body’s resistance to a change in its angular speed, in other words to an applied torque attempting to cause an angular acceleration α. This 21 22 EXPERIMENT 3. ANGULAR MOTION is in analogy to an applied force trying to cause a linear acceleration, where the measure of the body’s resistance is the familiar inertial mass m. If Newton’s Second Law for translational motion is F = ma, or m = F/a, for rotational motion the analogous relation is τ = Iα, or I = τ /α, where τ denotes the applied torque. Torque acting on a rotating body To understand what torque is, let us consider a rigid body with its centre of mass at a point r = 0, constrained to rotate about that point. The radius vector ~r of a point on the body has the origin at r = 0 and ends at the point, a distance r from the origin, as shown in Figure 3.1. If a force F~ is applied at that point, in the plane of rotation of the disc, at a distance r from the centre, and at an angle φ to the radius vector ~r, the magnitude of the turning torque τ produced by this force is: τ = rF sin φ The torque τ depends both on the distance from the centre of rotation and on the direction of the applied ~ into a radial component F~r and a tangential component F ~t so that F ~ = F~r + F~t , force F~ . Decomposing F ~ it can be seen that only the tangential component of F causes torque and affects the magnitude of τ . A force applied through the centre of rotation has a zero tangential component (φ = 0, Ft = 0) and the radial component alone produces no torque and will not cause angular acceleration. As the angle of the force changes, so does the torque experienced by the body; for a given magnitude of the force, F , the maximum torque is produced when F~ = F~t is perpendicular to ~r and τ = rFt = rF . Resisting the force F~ is the total mass M of the rotating body. Suppose that this mass consists of P many particles of mass mi so that M = i mi . In a translational motion, the force acts equally on all the P component particles of the body at once, according to Newton’s second Law F~ = i (mi~a) = M~a. In a rotation, the rotational effect of F~ on a particle is proportional to the distance r from the centre of rotation. While the entire rigid body experiences a single angular accelaration α, common to all its particles, each of them will experience a tangential acceleration ai that is proportional to the distance ri from the centre of rotation. The rotational moment of inertia I of a rigid body composed of many particles is simply the sum of the individual rotational moments of inertia of all particles: I= X mi ri2 (3.1) i For a thin hoop, where all the particles are located at the same common radius away from the centre, ri = R, Eq. 3.1 reduces to I = M R2 ; for other shapes, the calculation may not be so simple. The rotational form of the Newton’s Second Law is τ = Iα: F = X i (mi a) = M a −→ τ= X (mi ri2 α) = Iα. i The torque τ plays the same role for rotational motion that the force F plays for translational motion. Determining the rotational inertia of a disc Consider a homogeneous disc of radius R and mass M constrained to rotate without friction around the centre of mass. A massless string is wrapped around the outer edge of the disc and connected to a mass m that is subjected to the force of gravity Fg , as shown in Figure 3.2. The string experiences a tension T due to the weight of m; at the other end of the string the same tension T acts on the edge of the disc. The displacement of the falling mass is given by Equation 3.2: 23 y = y0 + v0 t + at2 /2 (3.2) When a force is applied, the disc, initially at rest, begins to spin as m falls with linear acceleration a=g− T . m (3.3) The rotational acceleration of the disk is α= τ TR a = = . R I I (3.4) By combining Eqs. 3.3 and 3.4, the rotational inertia I of the disc can be expressed in terms of a as follows: g Figure 3.2: A falling block causes the disc to rotate −1 . (3.5) I = mR2 a Measuring the acceleration of the falling mass, and comparing it to the acceleration of the free fall, yields an operational measurement of the moment of inertia of the disk. Review questions Determine from your textbook or the Internet the equation for the rotational inertia I of a solid uniform disc of radius R, thickness h and mass M . You will use this expression to calculate the theoretical I for the disc used in this experiment. ...................................................................... ...................................................................... Derive equation 3.5. Begin by evaluating the force F on the cradle due to the tension T of the string acting on the edge of the disc. Show a complete, step-by-step solution. ...................................................................... ...................................................................... ...................................................................... ...................................................................... ...................................................................... ...................................................................... CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 24 EXPERIMENT 3. ANGULAR MOTION Procedure and analysis Figure 3.3: Experimental setup The equipment consists of a plastic disc bolted to a metal drum (this combination is called the cradle), that can rotate nearly friction free, around a vertical axis. A torque τ can be applied to the cradle with a string-and-pulley combination, at the end of which a mass m is attached. The pulley is arranged so that the string leaving the drum is perpendicular to the axis of rotation and to the radius vector. Pulley A is part of a photogate system. The C-shaped photogate has an infrared transmitter at one end and an infrared detector at the other end. As the pulley rotates, the spokes interrupt the beam, turning on and off the signal at the detector, thus allowing LabPro to count a series of pulses. A light emitting diode (LED) on the photogate shows the current on/off state at the detector. The pulley has ten spokes and a circumference of 155 mm, so there are twenty pulses transmitted every rotation of the pulley. Each pulse represents a radial distance of 7.75 mm travelled by the string and hence the same distance y moved by the mass m. After the data acquisition is initiated, LabPro waits for the first transition and marks this event with the elapsed time and assigns it an initial distance y = 0. After all subsequent transitions, the new time and total distance travelled are sent. Note that the time t = 0 set by LabPro may not coincide with the start of the motion, therefore: • For the most reliable results, adjust the cradle by monitoring the LED so that a transition will occur as soon as the cradle begins to rotate. Wait for the first data point to be sent, then release the cradle. You should delete any of these (0,0) data points at the start of your data set. Part 1: Determining the rotational inertia of the cradle • Wrap the string, trying not to overlap the strands, around the second smallest pulley (r = 0.02282 m) on the cradle and arrange the string path as shown in Figure 3.3. Place a 10 g weight on the mass holder and note how the weight accelerates as it falls. It should not shake sideways. If it makes sudden vertical jerks, then the string was binding as it unravelled from the pulley and should be wound less tightly. Rotate the cradle to raise the weight to the top of the assembly. 25 • Change focus to the graphing software. Select Dig2, the channel that the photogate should be connected to, then choose to collect 20 points with 0.5 seconds between points. Calibrate the photogate by monitoring the LED as previously described, then release the weight. Press Get data to acquire the data set. As the weights change, you may need to adjust the number of points collected and/or the sample time. Try to stop the weight before it reaches the floor; it will save you the trouble of having to re-spool the string on the pulleys. At the end of the run, review your graphed data; it should be a smooth curve that represents the falling of a mass m under constant acceleration a. To determine a, you will now fit Equation 3.2 to your data set. • Delete any (0, 0) points. Select fit to: y= and enter A+B*x+C*x**2/2 in the fitting equation box. Click Draw to perform a fit of your data. Click Print to generate a representative graph of your data. Fill in Table 3.1, then calculate an average acceleration hai for the various masses. Make a printout of one trial, properly labelled, and include it in your report. m (kg) 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 a1 (m/s2 ) a2 (m/s2 ) hai (m/s2 ) Table 3.1: Acceleration data for unloaded cradle assembly • You may now wish to evaluate the rotational inertia Ic for the unloaded cradle, using Equation 3.5 and the values of a for a given m. This would not likely lead to the correct Ic value because you do not know the actual mass m0 required to overcome the friction of the pulley system, representing the least mass required to keep the cradle rotating at a constant speed, once started by hand, to overcome the static friction of the system. Note that the cradle will likely begin to rotate with only the mass holder attached (m = 0). In this case, to prevent the system from rotating, m0 would need to be negative (i.e. the mass holder would need to be lighter). There is however a method often used that does not require m0 to be known, and m0 is actually obtained from the follwong procedure: begin by rearranging Equation 3.5 so that m is expressed as a function of a: Ic = m T r 2 g g − 1 ≈ mT r 2 a a gr 2 = (m − m0 ) a ! → m = m0 + Ic a. gr 2 (3.6) Here, the equation can be simplified only if a ≪ g so that (g/a − 1) ≈ (g/a). The total mass mT = m − m0 that causes the tension T on the string is the sum of the mass m that you used and the mass m0 of the mass holder. The resulting equation is that of a straight line with slope Ic /(gr 2 ) and y=m intercept at x=a=0 equal to m0 . • Enter into an empty data window your set of (hai, m) coordinates. The scatter plot of your data should show a linear behaviour. Select fit to: y= and enter A+B*x in the fitting equation box. Click Draw and record below the fit parameters A and B along with their appropriate units. A = ............... ± ............... B = ............... ± ............... 26 EXPERIMENT 3. ANGULAR MOTION • Calculate Ic using Equation 3.6 and use the appropriate error propagation rules to evaluate the corresponding uncertainty σ(Ic ): Ic = .............................. σ(Ic ) = .............................. .............................. .............................. .............................. .............................. Ic = ................. ± ................. Part 2: Determining the rotational inertia of cradle and disc • Use the peg on the steel disc to center it on the cradle, adding a small piece of paper under the disc to prevent it from slipping. As before, determine the rotational inertia It of cradle-plus-disc. m (kg) 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 a1 (m/s2 ) a2 (m/s2 ) hai (m/s2 ) Table 3.2: Acceleration data for cradle assembly with disc It = .............................. σ(It ) = .............................. .............................. .............................. .............................. .............................. It = ................. ± ................. • Subtract Ic from It to obtain Id , the rotational inertia of the disc alone. Id = .............................. σ(Id ) = .............................. .............................. .............................. .............................. .............................. Id = ................. ± ................. 27 • Use the equation from the review question to calculate the theoretical rotational inertia for the disc that you used, as well an estimate of the error. Id(theoretical) = .............................. σ(Id ) = .............................. .............................. .............................. .............................. .............................. Id(theoretical) = ................. ± ................. IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Discussion Begin by presenting a tabulated summary of your results. Address the following issues as part of your discussion; as always, you should expand this list with your own observations, points and ideas. Do your theoretical end experimental results for Id agree with one another? Explain. Use the largest value obtained for the acceleration a to estimate the maximum error introduced in the rotational inertia I by the approximation used in Equation 3.6. Are the results for m0 for the cradle and the cradle-plus-disc systems consistent with your observations? Describe how the two systems behaved when m = 0, then analyse the results from your fits to reach your conclusions. If you spun the cradle using the smallest pulley on the drum (r = 1.539 cm), should the rotational inertia IC of the system change? How, if at all, should the acceleration a change as a function of the pulley radius r? Explain your reasoning by referring to the relevant equation(s). Perform a trial using the smallest pulley with pull mass m = 30 g on the unloaded cradle. Compare this a with the previous result for a obtained using the second smallest pulley and same pull mass. Do these two a values obtained by varying the pulley radius r agree with the results predicted by the theory? Show your data, equations, calculations and results below. ...................................................................... ...................................................................... ...................................................................... ...................................................................... ...................................................................... ...................................................................... 28 EXPERIMENT 3. ANGULAR MOTION first name (print) last name (print) Brock ID (ab13cd) TA initials grade Experiment 4 Collisions and conservation laws The velocity ~v of a body of mass m is determined by its speed, a scalar quantity, and its direction of motion, represented by a unit vector. If we consider a collision of two such masses m1 and m2 with velocities ~v1b and ~v2b before the collision, and velocities ~v1a and ~v2a after the collision, we note that it is generally difficult, if not impossible, to predict these resulting velocities ~v1a and ~v2a . To accomplish this, one would need to have a complete knowledge of the physical characteristics of the objects (size, shape, et cetera) and of the geometry of the interaction. The linear momentum P~ of a body is a vector equal to the product of its mass m (a scalar) and its velocity ~v (a vector). The law of conservation of linear momentum states that: If there are no net external forces acting on the masses, then the total momentum P~b before the collision is equal to the total momentum P~a after the collision. A mathematical formulation of this law for a collision between two masses m1 and m2 is a vector equation and can be expressed as follows: P~1b + P~2b = P~1a + P~2a m1~v1b + m2~v2b = m1~v1a + m2~v2a . (4.1) Rearranging Equation 4.1 in terms of the change in the momentum ∆P~1 of m1 and ∆P~2 of m2 reveals that the net change will be zero and that the vectors will be oriented anti-parallel to one another: P~1b − P~1a + P~2b − P~2a = 0 → (P~1b − P~1a ) = −(P~2b − P~2a ) → ∆P~1 = −∆P~2 Similarly, the change in the velocity for the two masses will result in two anti-parallel vectors: m1 (~v1b − ~v1a ) = −m2 (~v2b − ~v2a ) → m1 ∆~v1 = −m2 ∆~v2 (4.2) Since the velocity vectors are collinear, the vector equation can be simplified to a scalar equation and expressed in terms of the magnitudes of the velocities |~v2a − ~v2b | and |~v1b − ~v1a |. Rearranging Equation 4.2 as a ratio of masses m1 and m2 : m1 |∆~v2 | |~v2b − ~v2a | |~v2a − ~v2b | =− =− = . m2 |∆~v1 | |~v1b − ~v1a | |~v1b − ~v1a | (4.3) The mass ratio equation predicts that for equal masses m1 and m2 , the change in the velocity of the two masses should be the same. It also predicts that the two resulting velocity vectors will point in the same direction since the the components of vector 1 have been reversed. It is also of interest to know whether kinetic energy K = mv 2 /2 is conserved during a collision. The parameter Q, defined as the ratio of the total kinetic energy Ka after the collision to the total kinetic 29 30 EXPERIMENT 4. COLLISIONS AND CONSERVATION LAWS energy Kb before, is used as a measure of the energy lost during a collision. Note that Q depends on the square of the velocity and hence will be very sensitive to variations in v. Q= Ka = Kb 1 2 2 m1 v1a 1 2 2 m1 v1b 2 2 + m v2 + 12 m2 v2a m1 v1a 2 2a = 1 2 2 2 m1 v1b + m2 v2b + 2 m2 v2b (4.4) The kinetic energy, and therefore Q, are scalar quantities. Q can range in value from 0 to 1. If Q = 1, the kinetic energy of the system is conserved and the collision is said to be elastic. A collision is said to be inelastic when Q < 1. This is the expected result of our experiment since some of the kinetic energy is changed to heat and sound energy during the collision, and there will be frictional forces acting on the masses throughout the interaction. On the other hand, rotational kinetic energy may be imparted onto the pucks as they are pushed. In this experiment the mass ratio of two colliding pucks will be calculated to determine whether or not linear momentum was conserved during the collision. The ratio Q will be used to estimate the kinetic energy lost during the interaction. Review questions • Sketch the collision shown in Figure 4.2 then add the vectors ~v1 = ~v1b −~v1a and ~v2 = ~v2a −~v2b . What can you conclude about the magnitude and direction of these two vectors? Should the momentum be conserved in the collision? And the velocity? When is the velocity conserved? Explain. ................................................... ................................................... ................................................... • Show below that, if a certain condition is met, Equation 4.3 can be rewritten in terms of distance vectors rather than velocity vectors, to analyse your collision record as in Figure 4.2. Then you can then measure the magnitude of the velocity vectors which have units of distance/time with a ruler that measures distance. What is this necessary condition? ...................................................................... ...................................................................... ...................................................................... • Derive the error σ(m1 /m2 ) in the theoretical mass ratio m1 /m2 using the appropriate error propagation rules. The answer is given by Equation 4.5 in the analysis section of this lab. Do not simply copy Equation 4.5 as your answer. Show a complete step by step solution (see Appendix). CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 31 Procedure and analysis The equipment consists of a flat glass plate in a metal frame which can be levelled by varying the height of four adjustable legs. The glass plate is covered by a conductive black pad. A sheet of regular white paper is placed over this conducting layer. The sheet must be flat and free of any kinks or other deposits. The collision components are two heavy metal pucks. Each puck is tethered to a plastic hose that provides the puck with compressed air and carries within it an electrode wire. The hose should hang freely, without any twists or interference from the rest of the equipment. When turned on, the air exits from a hole at the bottom of the puck, creating a high pressure layer between the puck and the paper and causing the puck to levitate and move with negligible friction. When the frame is properly levelled, the puck will float in place, without any lateral movement. A spark timer is used to provide the high voltage pulses required in the experiment. One terminal of the timer is connected through the electrode wire in the hose to an insulated needle at the bottom of the puck. The other terminal of the spark timer is connected to the conductive sheet, completing the electric circuit. The firing rate of the timer can be adjusted with a control knob on the unit. To turn on the timer, depress and hold the pushbutton switch at the front of the spark timer unit. Sparks are generated from the needle through the white paper to the conducting paper beneath. These pulses produce black spots on the bottom of the white paper, marking the positions of the pucks at equal time Figure 4.1: Trails left by moving pucks intervals. Consequently, it is not necessary to know the actual time interval between firings of the spark timer. In your calculations refer to this unknown time unit as ”u” (unknown time unit). Puck speeds would then be expressed in units of cm/u or mm/u. • Place a new sheet of white paper over the conducting layer, making sure that the paper is flat and clean. Very slowly turn on the air until the pucks begin to float. Excessive air pressure will burst the air hose. Level the frame by adjusting the height of the four legs until the pucks float in place. • To record the collision, have an assistant depress the switch of the spark timer just before you release the pucks, and release the switch just after the pucks rebound from the edges of the frame. Every member of the team needs to make their own collision record to analyse. • Place the pucks against the launchers in adjacent corners of the frame, and release them toward the centre of the paper, where they will collide. Do several trial runs. While any collision is theoretically valid, for ease of analysis your collision should be fairly symmetric as in Figure 4.1 and take about 3 seconds to complete. Small initial velocities will cause the pucks to slow down due to friction and yield incorrect results. The pucks should not rotate as this would give them rotational kinetic energy. We are concerned only with translational kinetic energy. • The mass of each puck has been measured and is displayed on the puck. Assign this mass value to the corresponding trace for future reference, keeping in mind that the traces appear on the bottom of the paper. Be sure to match each mass with the corresponding trace, otherwise your Q value will not make sense. • A good collision record will show for each of the for trails a series of 4-6 collinear dots, equidistant and longer than 100 mm in total. If your collision did not turn out, flip the sheet of paper paper and try again. 32 EXPERIMENT 4. COLLISIONS AND CONSERVATION LAWS • Evaluate the quality of your collision record. Measure the distance between the adjacent dots of a trail. Did the distance between the dots in each trail remain the same and did these dots form a straight line? Is the trail a valid representation of a vector? Note your observations below: ................................................................................ ................................................................................ ................................................................................ It can be seen from Equation 4.3 that the generation of the vectors ~v1b , ~v2b , ~v1a , and ~v2a is required. For example, the vector ~v1b can be obtained from the collision record by joining a series of dots along the Puck 1 trail (before the collision) spanning n time intervals. • Draw four proper vectors on your sheet, as shown in Figure 4.2. It is important to use this same number n of time intervals for all the vectors so that you are, in effect, applying the same time scale n ∗ u to each vector. The vectors ~v2a − ~v2b and ~v1b − ~v1a depend on the magnitude and direction of the component vectors, hence a graphical vector addition must be performed. A reasonably long vector or line segment with well defined endpoints can be transposed very accurately by visually estimating the final placement of this vector as follows: Figure 4.2: Join dots to obtain vectors • Using a long ruler, extend the line segment AB beyond the region of point D. • Place the ruler so that the edge rests on point C and is parallel to the previously drawn line. • Draw a line through C beyond points A and D. The measurement error for all vectors is σ(v) = ± .............. • Measure near A and D the perpendicular distance between the two lines to verify that they are indeed parallel. Determine the length AB with a ruler or the span of a compass and draw a line segment of length AB from C to D to define the vector CD. Figure 4.3: Transposition of a vector This method of extending the length of a line segment can also be used on the resultant vectors to see if they are parallel. By the law of conservation of momentum, the change in momentum of one puck is equal and opposite to that of the other puck. The two resultant vectors should then be nearly equal in magnitude since m1 ≈ m2 and parallel since the a − b components in the vectors are reversed. 33 m1 |v1b | |v1a | |~v1b − ~v1a | Time units m2 |v2b | |v2a | |~v2a − ~v2b | (g) (mm) (mm) (mm) (u) (g) (mm) (mm) (mm) Table 4.1: Experimental data: masses and vector magnitudes • Compare the magnitudes of directions of your resulting vectors. To quantify the test for parallelism, calculate the angle θ between the two extended line segments. It is given by θ = arctan(dy/dx), where dx is the length of the line and dy is the difference in the distance of the two lines at their opposite ends. Show your calculations below. ...................................................................... ...................................................................... ...................................................................... • Determine a theoretical value (m1 /m2 ) and error σ(m1 /m2 ) for the mass ratio from the given values of m1 and m2 . The measurement error for these masses is ±0.1 g. Show your calculatios below m1 = m2 ............................. m1 σ m2 = .................................. ............................. (Theoretical) ................................. m1 = ............... ± ............... m2 • Use Equation 4.3 to calculate a value and error for the experimental mass ratio of the pucks. m1 = m2 σ ............................. ............................. (Experimental) m1 m2 = .................................. .................................. m1 = ............... ± ............... m2 34 EXPERIMENT 4. COLLISIONS AND CONSERVATION LAWS • Use Equation 4.4 and the theoretical m1 and m2 values determine Q and σ(Q). You can approximate σ(Q) by letting Q = (A + B)/(C + D). Show the complete solution on a separate sheet of paper. Hint: solving and calculating a result for the σ(Q) error equation can be a tedious error-prone venture. To simplify the task, note the symmetry in the equation, apply a divide-and-conquer strategy and avoid repeating calculations. For example, there are four identical mv 2 terms, each of which will appear several times in the Q and σ(Q) equations. Evaluate these four terms only once, record the result and then compare the four values. You would expect them to be similar since the m and v values are also similar. Then you can confidently evaluate Q and apply the values to σ(Q). Likewise, the error equation for each term will be identical in form and these error terms should also evaluate to similar results. Q= σ(Q) = ............................. .................................. ............................. .................................. Q = ............... ± ............... IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Discussion Include your collision record, your worksheets and a tabulated summary of your results. Discuss the results in terms of momentum changes. Was momentum conserved? How do you establish this? What was the quality of your vector addition? Did the experimental and theoretical mass ratios agree within their respective margin of error? Explain. Discuss your result for the parameter Q. Was energy conserved? Was there a significant loss of energy during the interaction? Was the collision elastic or inelastic? Was your Q value greater than unity? Can Q be greater than unity? Consider how someone performing this experiment could arrive at a value of Q > 1. Review the test you performed on the two resultant vectors and use your findings to support your conclusions regarding the quality of the data obtained from the analysis of the collision record. The conservation of energy requires that no energy may be lost during an interaction. However, it can be changed to other forms, which may not be monitored during the interaction. In this collision experiment, what other forms of energy would the initial kinetic energy be transformed into? Consider how the apparatus used in this experiment may have introduced random and systematic errors into the experimental procedure. Which of these errors do you think would have a significant impact on the experiment? Which are likely to be insignificant? first name (print) last name (print) Brock ID (ab13cd) TA initials grade Experiment 5 The ballistic pendulum In this experiment we explore the transfer and conservation of energy and momentum in a collision of two objects. One of these objects is a small projectile of mass m that is given a certain velocity v by a launcher. The second object is a stationary pendulum. As the projectile hits the pendulum, a re-distribution of energy and momentum takes place. In a certain class of collisions, the projectile is captured by the pendulum. For such inelastic collisions, there is only one combined object that is moving at the end, and carries all of the kinetic energy K and momentum P . If the mass of the pendulum is M then the total mass of the combined object at the end of the collision is MT = M + m. If the bob is stationary when the projectile hits, it contributes nothing to the total kinetic energy and momentum of the system before the collision. Thus the Law of Conservation of Momentum in this case yields: Figure 5.1: Ballistic Pendulum Pbef ore = mv = Paf ter = (M + m)vT = MT vT (5.1) where vT is now the velocity of the combined object immediately after the collision. Knowing v and MT , we can use Equation 5.1 to determine vT . As the pendulum begins to swing after the collision, another physical process takes place; a conversion of the kinetic energy of the moving object into into gravitational potential energy as it swings up, losing kinetic energy and gaining potential energy. At the bottom of the swing, the pendulum of mass MT has all of its energy in the form of kinetic energy. At the top of the swing, all of the pendulum’s energy is converted into gravitational potential energy, and as MT momentarily pauses and reverses its motion, the kinetic energy falls to zero. Thus we can write: 1 Ebottom = K = MT vT2 = Etop = MT gh. 2 (5.2) Combining Equations 5.1 and 5.2 to eliminate vT gives us an expression that relates the initial velocity v of the projectile to the final elevation h of the combined object. v= MT p 2gh, m 35 (5.3) 36 EXPERIMENT 5. THE BALLISTIC PENDULUM It is more convenient to express h in terms of the angle θ of the swing v= MT q 2gRcm (1 − cos θ) m (5.4) where Rcm is the distance from the pivot point to the centre of mass of the combined rod, block, and block contents. There is another energy conversion taking place, even before the collision. In the experiment, the launcher gives the projectile its initial kinetic energy and momentum by releasing the potential energy and momentum of a compressed spring an converting it into the kinetic energy of the moving projectile. Conservation of energy in this case yields 1 1 Einitial = P E = kx2 = Ef inal = K = mv 2 2 2 (5.5) where x is the spring displacement from its equilibrium (relaxed) length and k is the spring constant. As the projectile is released from rest, the projectile has no initial kinetic energy. As the projectile begins to accelerate, a conversion of energy takes place where the potential energy stored in the spring is converted into the kinetic energy of the projectile. At the point where the projectile releases from the spring, all of the spring’s stored potential energy has been converted into the projectile’s kinetic energy. We can then determine a value for the spring constant k of the launcher by equating the initial potential energy of the spring to the final kinetic energy of the projectile: k= mv 2 x2 (5.6) Review question • Derive an equation that expresses the error σ(v) in the velocity v given by Equation 5.4, expressed below as a product of terms. Perform a change of variables and apply the appropriate error propagation rules (Appendix B) to each term to arrive at a final answer. Show a complete step by step solution. 1/2 v = MT ∗ m−1 ∗ 21/2 ∗ g1/2 ∗ Rcm ∗ (1 − cos θ)1/2 ...................................................................... ...................................................................... ...................................................................... ...................................................................... ...................................................................... σ(v) = ................................................................ CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 37 Procedure and analysis CAUTION: Always wear safety glasses while using the launcher. The launcher component of the ballistic pendulum consists of a precision spring encased in an aluminum barrel. One end of the spring is secured to the closed end of the barrel. The other end is attached to a piston that slides along the inside of the barrel. The projectile rests on the face of the piston. A trigger mechanism allows the piston to be locked in one of three force settings: short, medium or long range. Note: When the launcher is in the discharged position, the spring is subjected to a small compression, or preload, so that it will not rattle when released. This preload is in the order of 1 mm. For this experiment, we will assume that when the launcher is discharged the potential energy stored in the spring is approximately zero. • Load the ballistic pendulum by placing the projectile in the barrel and pushing it with the plunger rod until the trigger locks. This is the short range setting. Further compression selects the medium range and finally, the long range setting. The work done to compress the spring is now stored as the spring’s potential energy V = kx2 /2, where k is the spring constant and x is the compression distance. The launcher mechanism is designed so that the elastic limit of the spring is not exceeded during compression. • Once the launcher is loaded, slowly withdraw the plunger, making sure that the trigger is indeed locked and that the projectile has not rolled away from the face of the piston, as this would cause an innacurate firing of the launcher. • To fire the launcher, gently pull the string upward to release the trigger. The projectile of mass m will be accelerated by the expansion of the spring to achieve a final velocity v at the moment that it finally loses contact with the face of the piston. At this point, the potential energy V stored in the spring will have been totally converted into the kinetic energy K = mv 2 /2 of the projectile. The pendulum consists of a rod and bob of combined mass M attached to a pivot point. When an impact takes place, the pendulum catches the impacting mass m, changing the total mass of the pendulum to MT = M + m, and is caused to swing about the pivot point to a maximum angle of deviation θ, relative to the initial vertical position of θ = 0◦ . The pendulum drags with it a pointer that stops at the limit of the swing and identifies the value of θ on a degree scale concentric with the pivot. The friction of the pointer is negligible. All of the kinetic energy of the projectile is transferred to the pendulum which, being initially at rest, gains a kinetic energy K = MT v 2 /2. After rising against the force of gravity g a maximum height h from the vertical position, the pendulum stops as all of the pendulum’s kinetic energy has been converted to gravitational potential energy V = MT gh. This type of collision is classfied as totally inelastic since the bodies involved stick together, moving as one after the collision. • Check that two brass weights are attached to the pendulum and that the bob is properly oriented in order to catch the projectile. • Swing the pendulum to 90◦ and lock it in position with a slight push. Load the steel ball projectile into the launcher, and latch the trigger at the short range setting. Gently lower the pendulum to the vertical position, and move the angle indicator to the 0◦ mark. If the indicator does not reach zero, you will need to subtract the offset from your angle reading. 38 EXPERIMENT 5. THE BALLISTIC PENDULUM range θ1 θ2 θ3 θ4 θ5 hθi σ(θ) short medium long Table 5.1: Experimental angle values at three force settings • Perform five short range launches, recording the angle θi reached in trial i = 1 . . . 5 in the appropriate spaces of Table 5.1. These values should be within ± 0.5◦ of one another, otherwise redo the set of measurements. What is the resolution of the protractor scale? .................. • Repeat the five above trials using the medium and then the long range settings. • Calculate an average value hθi for the five angles θi obtained in each of the three sets of trials and enter these in Tables 5.1 and 5.2. To avoid some lengthy standard deviation calculations, let the error in the angle θi be the measurement error of the angle scale, σ(θ) = ± 0.25◦ . Discuss the validity of this assumption in your conclusions. • With the digital scale, measure the masses m and MT . m = .................. ± .................. kg MT = .................. ± .................. kg • Remove the pendulum arm and determine the centre of mass point of the pendulum/ball combination by balancing the unit on the edge of the measuring apparatus, as shown in Figure 5. Make sure that the arm of the pendulum is parallel with the length of the scale. The centre-of-mass distance Rcm is the distance from the edge to the centre of the pivot hole on the arm of the pendulum of mass MT . Be sure to replace the pendulum arm when you are done. Rcm = .................... ± .................. m Figure 5.2: Experimental setup for determining the centre-of-mass of the pendulum 39 • With the launcher discharged and the ball removed, use the scale on the plunger to determine the offset depth of the face of the piston from the front end of the barrel. You will need to subtract this offset from all of the following depth measurements. offset = .................... ± .................. m • With the ball removed, compress the piston until it latches at the short range setting. Measure the depth from the face of the piston to the front end of the barrel. Subtract from this length the offset depth of the piston and record the result as x in Table 5.2. • Repeat the above step for the medium and long range settings and complete Table 5.2. • Use Equation 5.4 and the error equation derived in the review section of this lab to calculate the initial velocity v and the error σ(v) of the steel ball projectile at the three force settings. Enter the results in Table 5.2. Show your calculations below for the short range force setting and include all other calculations as part of your Discussion. v= MT q 2gRcm (1 − cos θ) m σ(v) = |v| σ(MT ) σ(m) σ(Rcm ) | cos[θ + σ(θ)] − cos[θ − σ(θ)] | + + + |MT | |m| 2|Rcm | |4(1 − cos θ)| ................... ............................................................ ................... ............................................................ v = .................... ± .................. m/s • Calculate the maximum kinetic energy of the steel ball at the moment that it lost contact with the piston of the launcher and enter the value in Table 5.2. Show a complete calculation for the short range setting and include all other calculations as part of your Discussion. K= 1 mv 2 2 σ(K) = ............................ .............................................. ............................ .............................................. K = .................... ± .................. J If no energy is lost (or gained) during the interaction, this kinetic energy K is equal to the potential energy V = kx2 /2 of the spring before the ball was discharged, K = kx2 /2. This equation is that of a straight line Y = M X with Y = K, X = x2 and slope M = k/2. By plotting K as a function of x2 we can extract from the slope a value for the spring constant k of the launcher spring. To do this, you will measure x relative to a fixed reference point, the end of the barrel, at the three force settings. 40 EXPERIMENT 5. THE BALLISTIC PENDULUM v (m/s) x (m) x2 (m2 ) K (J) ± ± ± ± ± medium ± ± ± ± ± long ± ± ± ± ± range hθi short ◦ Table 5.2: Parameters for the calculation of the kinetic energy K and the force constant k • Shift focus to the Physicalab software and enter in the data window the three data pairs and corresponding errors in the format (x2 , K, σ(K), σ(x2 )). Select scatter plot. Click Draw to generate a graph of your data. Your graphed points should well approximate a straight line. Select fit to: y= and enter A*x+B in the fitting equation box. Click Draw to perform a linear fit of the data. Label the axes and include a descriptive title. Click Print to generate a hard copy of your graph. Every student requires their own graph. • Summarize the values for the slope, the Y-intercept Y(X=0) and their associated errors, then calculate a value for k and σ(k): slope = .................. ± .................. J/m2 Y(X=0) = .................. ± .................. J k = .................. ± .................. J/m2 IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Discussion Tabulate your final data and attach your computer printout and all calculation worksheets. Discuss whether your results show a linear relationship between K and x2 and if three data pairs are sufficient to define such a relationship. Does the spring force constant k change as the spring compression x is increased? Explain. Consider the value for the Y-intercept from the Least Squares Fit of your data. What does this value represent? Is your result consistent with your expectations? Explain. Refer to your three sets of angle data. Discuss the validity of the assumption made regarding the magnitude of the error σ(θ) in the angle θ. You calculated the force constant k of the launcher spring by observing the interaction of the projectile with the ballistic pendulum apparatus. What needs to be true for this method to be valid? Can we use this experimental setup to determine the actual energy lost by the projectile between the time that it leaves the piston and the time that it collides with the pendulum bob? Explain. first name (print) last name (print) Brock ID (ab13cd) TA initials grade Experiment 6 Harmonic motion A simple harmonic oscillator (SHO) is a model system that is widely used, and not just in the elementary physics exercises.The reason for this is profound: the same fundamental equations describe the motion of a mass-on-a-spring, and also of the interatomic forces that hold all matter together. Literally, everything we touch, at the atomic level, is held together by springs connecting pairs of atoms. The details of these interactions are studied in Quantum Mechanics and Solid-State Physics, but two masses connected by a coiled elastic wire represent an excellent model system,from which much can be learned. With no external forces applied to the material, the interatomic springs are at their equilibrium length, neither stretched nor compressed. The application of an external stretching force to the material will cause these springs to extend, thereby increasing the bulk length of the material. When the applied external force is removed, the springs return to their equilibrium lengths, restoring the material to its original dimensions. This restoring force may be overcome by A large enough applied external force will cause the object to deform permanently or to break. The maximum force applicable without permanent distortion is called the elastic limit of the material. Hooke’s Law states that the amount of stretch or compression x exhibited by a material is directly proportional to the applied force F . The proportionality constant is called the spring constant k. The spring constant, k, has the units of Newtons per metre (N/m) and is a measure of the stiffness of the material being considered. This proportionality between force and elongation has been found to hold true for any body as long as the elastic limit of the material is not exceeded. Perhaps the most convenient way of expressing a relationship of the Hooke’s Law is to write; F = −kx, (6.1) The minus sign expresses the fact that the force exerted by the spring, Fs , is always opposing the deformation. It is often described as a restoring force If Fs is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency, which does not depend on the amplitude. Harmonic motion is equivalent to an object moving around the circumference of a circle at constant speed. In the absence of friction, the oscillations will conrinue forever.Friction robs the oscillator of it’s energy, the oscillations decay, and eventually stop altogether. Frequently, this friction term is proportional to the velocity of the moving mass (think of the air resistance), Fd = Rv. If a frictional force Fd = −Rv, proportional and opposed to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the strength of the friction coefficient R, the system can either oscillate with a frequency slightly smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator); or decay exponentially to the equilibrium position, without oscillations (overdamped oscillator). 41 42 EXPERIMENT 6. HARMONIC MOTION Mass on a vertical spring When a mass m is attached to a hanging spring, the spring is subjected to a force Fg = mg and will be stretched to a new equilibrium position y0 where Fg = −Fs ⇒ mg = ky0 ⇒ m= k y0 . g (6.2) If the mass is displaced from y0 and released, it will begin to oscillate about y0 according to y = A0 cos(ω0 t + φ), ω0 = q k/m = 2πf0 = 2π T0 (6.3) where A0 and φ are the initial amplitude and phase angle of the oscillation, T0 is the period in seconds, and ω0 is the angular speed in radians/second. If a damping force Fd is present, the oscillation decays exponentially at a rate determined by the damping coefficient γ: q R (6.4) , ωd = ω02 − γ 2 2m Note two interesting details; first, the damped frequency of oscillations, ωd , is made smaller than ω0 by the subtraction of γ 2 under the square root. However, if R is not very large, this reduction is not all that noticeable, even though the decrease in the amplitude due to the e−γt term may be readily observed. Second, in the case of the air resistance, Fd = −Rv is an approximate relation, valid only for slow speeds. As the speed increases, the turbulent air flow offers resistance that is proportional to the square of the velocity, Fd = −Cv 2 . Here C is the so-called drag coefficient. Unfortunately, this expression cannot be solved analytically and would have to use numerical methods (computer) to evaluate. Note also that the hanging spring is stretched by its own weight and may exhibit twisting as well as lateral oscillations when stretching. One other simplifying assumption: this experiment assumes an ideal massless spring connected to a point mass m. Even with all these approximations, Equation 6.4 lends itself very nicely to an experimental investigation. y = A0 cos(ωd t + φ)e(−γt) , γ= Review questions Two springs with spring constants k1 and k2 are connected together as shown in figure 6.1. With mass m1 attached, the springs have lengths d1 and d2 . Increasing the pull mass to m2 stretches the springs a distance x1 and x2 for a total stretch of x. Derive an equation for the effective force constant ke of the two spring in series. (Hint: The stretching force F is the same for both springs.) What is ke when k1 = k2 ? Does the k of a spring depend on the length of the coiled spring? ................................................. ................................................. Figure 6.1: Effective spring constant ................................................. ................................................. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 43 Procedure and analysis The experimental setup consists of a vertical stand from which hangs a spring. A platform hanging from the bottom of the spring accepts various mass loads. The spring mass system is free to move in the vertical direction. A collinear rangefinder measures the distance to the bottom of the platform by sending a stream of ultrasonic pulses and measuring the time for each echo to return. Part 1: Static properties of springs In the following exercises you will determine the force constants of two different springs, labelled #1 and #2, and verify that their combined force constant satisfies the equation derived in the review section of the experiment. Note: Use positive k values in all the following steps. Equation 6.2 represents a straight line with slope k/g = ∆m/∆y so that k = |∆m/∆y| ∗ g. By varying m, measuring y0 and plotting the resulting data, the force constant k for the spring can be determined. Note that the absolute distance y0 does not matter, however the change in distance ∆y with mass ∆m is critical. • Suspend spring #1 from the holder and load it with the 50 g platform. Use a ruler to carefully measure the distance y from the top of the table to the bottom of the platform. Record your result in Table 6.1. Determine the mass required to bring the platform near the top of the table and record this new distance, then select several intermediate masses and repeat the measurements. mass (kg) y (m) Table 6.1: Data for determining the spring force constant k1 of spring #1 • Use the Physicalab software to enter the data pairs (y, m) in the data window. Select scatter plot. Click Draw to generate a graph of your data. Select fit to: y= and enter A+B*x in the fitting equation box. Click Draw to perform a linear fit on your data. Label the axes and enter your name and a description of the data as part of the graph title. Click Print to generate a hard copy of your graph. Show below the steps used to evaluate k1 and the associated error σ(k1 ). (Recall that the error is always rounded to one significant digit while the value is rounded to the same decimal place as the error. Appendix B includes a review of error propagation rules). B = ............. ± ............. A = ............. ± ............. k1 = ............................... σ(k1 ) = ............................... ................................ ................................ ................................ ................................ k1 = ............... ± ............... 44 EXPERIMENT 6. HARMONIC MOTION • Repeat the above procedure using spring #2, recording your results in Table 6.2, then determine the spring constant k2 for the spring. Show all calculations on a separate sheet and include these as part of your lab report. mass (kg) y (m) Table 6.2: Data for determining the spring force constant k2 of spring #2 B = ............. ± ............. k2 = ............... A = ............. ± ............. ± ............... • Connect together springs #1 and #2, and repeat the procedure, entering your data in Table 6.3, to determine the experimental effective spring constant ke of the two springs: mass (kg) y (m) Table 6.3: Data for determining the effective spring force constant ke of springs #1 and #2 B = ............. ± ............. ke = ............... A = ............. ± ............. ± ............... • Calculate the theoretical effective spring constant ke and σ(ke ) for the two springs: ke = k1 k2 k1 + k2 σ(ke ) = ............................... ................................ ................................ ................................ ................................ ke = ............... ± ............... 45 Part 2: Damped harmonic oscillator You will now explore the behaviour in time of an oscillating mass. By accumulating a series of coordinate points (ω, m) and fitting your data to Equation 6.4, you can use several methods to determine the spring constant for the oscillating mass system. Begin by selecting one of the two springs to use: • Starting with a 100 g, raise the platform a small distance, then release it to start the mass swinging vertically. Wait until the spring/mass system no longer exhibits any erratic oscillations. • Shift focus to the Physicalab software. Check the Dig1 box and choose to collect 200 points at 0.05 s/point. Click Get data to acquire a data set. • Select scatter plot, then Click Draw . Your points should display a smooth slowly decaying sine wave, without peaks, stray points, or flat spots. If any of these are noted, adjust the position of the rangefinder and acquire a new data set. • Select fit to: y= and enter A*cos(B*x+C)*exp(-D*x)+E in the fitting equation box. Click Draw . If you get an error message the initial guesses for the fitting parameters may be too distant from the required values for the fitting program to properly converge. Look at your graph and enter some reasonable initial values for the amplitude A of the wave and the average (equilibrium) distance E of the wave from the detector. C corresponds to the initial phase angle of the sine wave at x=0. • The angular speed B (in radians/s) is given by B= 2π/T . Estimate the time x= T between two adjacent minima of the sine wave then estimate and enter an initial guess for B. • The damping coefficient D determines the exponential decay rate of the wave amplitude to the equilibrium distance E. When D = 1/x, the envelope will have decreased from A0 to A0 e−1 = A0 /e = A0 /2.718 ≈ A0 /3. Make an initial guess for D by estimating the time t=x required for the envelope to decrease by 2/3. • Check that the fitted waveform overlaps well your data points, then label the axes and include as part of the title the value of mass m used. Click Print button to generate a printout of your graph. Every student should print a copy of ONE sample graph. • Record the results of the fit in Table 6.4, then complete the table as before, being careful to not have any of the hanging masses fall on the rangefinder: mass (kg) y0 (m) ω0 (rad/s) γ (s−1 ) Table 6.4: Experimental results for damped harmonic oscillator • The spring constant k can be determined as before from the slope of a line fitted through the (y0 , m) data in Table 6.4. Generate and print the graph, then record the result below: k(y0 ) = ............... ± ............... 46 EXPERIMENT 6. HARMONIC MOTION • The spring constant k can also be determined fron the period of oscillation of the mass. Rearranging the terms in Equation 6.3 yields m = k/ω02 . Enter the coordinate pairs from Table 6.4 in the form (ω0 , m) and view the scatter plot of your data. Select fit to: y= and enter A+B/x**2 in the fitting equation box. This is more convenient than squaring all the ω0 values and fitting to A+B/x. Click Draw to perform a quadratic fit on your data. Print the graph, then enter below the value for the spring constant: k(ω0 ) = ............... ± ............... • The damped harmonic oscillator equation 6.4 predicts that the frequency of oscillation depends on the damping coefficient γ so that your experiment actually yields values for ωd rather than ω0 . Calculate ω0 from ωd using an average value for γ. Which ωd value should be used? Why? Is the difference between ωd and ω0 significant? ................................................................................ ................................................................................ IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Discussion Begin by tabulating your spring constant results. In the linear fit of your (y, m) data, what are the units and meaning of the fit parameter A? Do the experimental and theoretical results for ke agree within experimental error? Explain. Do the results for k for the spring used in parts 1 and 2 agree within experimental error? Explain. It is assumed that the rangefinder is properly calibrated, so that it measures length correctly. Based on your results, is this assumption valid? Explain your conclusion. How would you determine if the rangefinder is not properly calibrated? Check the calibration of your rangefinder against the ruler. Show your results and calculations below. Why would you want to select two calibration points to be a maximum distance apart. Are two calibration points sufficient? Why? ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ first name (print) last name (print) Brock ID (ab13cd) TA initials grade Appendix A Review of math basics Fractions a b ad + bc + = ; c d cd a b = , c d If then ad = cb and ad = 1. bc Quadratic equations Squaring a binomial: Difference of squares: (a + b)2 = a2 + 2ab + b2 a2 − b2 = (a + b)(a − b) 2 The two roots of a quadratic equation ax + bx + c = 0 are given by x = −b ± √ b2 − 4ac . 2a Exponentiation (ax )(ay ) = a(x+y) , ax = ax−y , ay a1/x = √ x a, a−x = 1 , ax (ax )y = a(xy) Logarithms Given that ax = N , then the logarithm to the base a of a number N is given by loga N = x. For the decimal number system where the base of 10 applies, log10 N ≡ log N and log 1 = 0 (100 = 1) log 10 = 1 (101 = 10) log 1000 = 3 (103 = 1000) Addition and subtraction of logarithms Given a and b where a, b > 0: The log of the product of two numbers is equal to the sum of the individual logarithms, and the log of the quotient of two numbers is equal to the difference between the individual logarithms . log(ab) = log a + log b a log = log a − log b b The following relation holds true for all logarithms: log an = n log a 47 48 APPENDIX A. REVIEW OF MATH BASICS Natural logarithms It is not necessary to use a whole number for the logarithmic base. A system based on “e” is often used. Logarithms using this base loge are written as “ln”, pronounced “lawn”, and are referred to as natural logarithms. This particular base is used because many natural processes are readily expressed as functions of natural logarithms, i.e. as powers of e. The number e is the sum of the infinite series (with 0! ≡ 1): e= ∞ X 1 = n! n=0 1 1 1 1 + + + + · · · = 2.71828 . . . 0! 1! 2! 3! Trigonometry Pythagoras’ Theorem states that for a right-angled triangle c2 = a2 + b2 . Defining a trigonometric identity as the ratio of two sides of the triangle, there will be six possible combinations: b c c csc θ = b sin θ = a c c sec θ = a cos θ = sin(θ ± φ) = sin θ cos φ ± cos θ sin φ cos(θ ± φ) = cos θ cos φ ∓ sin θ sin φ tan θ ± tan φ tan(θ ± φ) = 1 ∓ tan θ tan φ b sin θ = a cos θ a cos θ cot θ = = b sin θ tan θ = sin 2θ = 2 sin θ cos θ cos 2θ = 1 − 2 sin2 θ 2 tan θ tan 2θ = 1 − tan2 θ 180◦ = π radians = 3.15159 . . . 1 radian = 57.296 . . . ◦ sin2 θ + cos2 θ = 1 To determine what angle a ratio of sides represents, calculate the inverse of the trig identity: if b sin θ = , c then θ = arcsin b c For any triangle with angles A, B, C respectively opposite the sides a, b, c: a b c = = , (sine law) sin A sin B sin C The sine waveform If we increase θ at a constant rate from 0 to 2π radians and plot the magnitude of the line segment b = c sin θ as a function of θ, a sine wave of amplitude c and period of 2π radians is generated. Relative to some arbitrary coordinate system, the origin of this sine wave is located at a offset distance y0 from the horizontal axis and at a phase angle of θ0 from the vertical axis. The sine wave referenced from this (θ, y) coordinate system is given by the equation y = y0 + c sin(θ + θ0 ) c2 = a2 + b2 − 2ac cos C. (cosine law) first name (print) last name (print) Brock ID (ab13cd) TA initials grade Appendix B Error propagation rules • The Absolute Error of a quantity Z is given by σ(Z), always ≥ 0. • The Relative Error of a quantity Z is given by σ(Z) |Z| , always ≥ 0. • To determine the error in a quantity Z that is the sum of other quantities, you add the absolute errors of those quantities (Rules 2,3 below). To determine the error in a quantity Z that is the product of other quantities, you add the relative errors of those quantities (Rules 4,5 below). Relation Error 1. Z = cA σ(Z) = |c|σ(A) 2. Z = A + B + C + ··· σ(Z) = σ(A) + σ(B) + σ(C) + · · · 3. Z = A − B − C − ··· σ(Z) = σ(A) + σ(B) + σ(C) + · · · (Error terms are always added.) 4. Z = A × B × C × ··· (Use only if A is a single term, i.e. Z = 3x.) 5. Z= 6. Z = Ab σ(Z) |Z| σ(Z) |Z| σ(Z) |Z| 7. Z = sin A σ(Z) = σ(sin A) = 8. Z = log A σ(Z) = σ(log A) = A B = = = σ(A) σ(B) |A| + |B| σ(A) σ(B) |A| + |B| |b| σ(A) |A| + σ(C) |C| + ··· (Note the absolute value of the power.) | sin[A+σ(A)]−sin[A−σ(A)] | 2 | log[A+σ(A)]−log[A−σ(A)] | 2 (Similar for cos A) • a, b, c, . . . , z represent constants • A, B, C, . . . , Z represent measured or calculated quantities • σ(A), σ(B), σ(C), . . . , σ(Z) represent the errors in A, B, C, . . . , Z, respectively. How to derive an error equation Let’s use the change of variable method to determine the error equation for the following expression: y= Mq 0.5 kx (1 − sin θ) m • Begin by rewriting Equation B.1 as a product of terms: 49 (B.1) 50 APPENDIX B. ERROR PROPAGATION RULES y = M ∗ m−1 ∗ [ 0.5 ∗ k ∗ x ∗ (1 − sin θ)] = M ∗ m −1 1/2 ∗ 0.5 ∗ k 1/2 ∗ x 1/2 1/2 (B.2) 1/2 ∗ (1 − sin θ) (B.3) • Assign to each term in Equation B.3 a new variable name A, B, C, . . . , then express v in terms of these new variables, y=A ∗ B ∗ C ∗ D ∗ E ∗ F (B.4) • With σ(y) representing the error or uncertainty in the magnitude of y, the error expression for y is easily obtained by applying Rule 4 to the product of terms Equation B.4: σ(y) σ(A) σ(B) σ(C) σ(D) σ(E) σ(F ) = + + + + + |y| |A| |B| |C| |D| |E| |F | (B.5) • Select from the table of error rules an appropriate error expression for each of these new variables as shown below. Note that F requires further simplification since there are two terms under the square root, so we equate these to a variable G: A = M, B= m−1 , C = 0.51/2 , D = k1/2 , E = x1/2 , F = G1/2 , G = 1 − sin θ, σ(A) = σ(M ) Rule 1 σ(B) |B| σ(C) |C| σ(D) |D| σ(E) |E| σ(F ) |F | Rule 6 σ(m) = | − 1| σ(m) |m| = |m| = 21 σ(0.5) =0 |0.5| = 21 σ(k) = σ(k) 2|k| |k| 1 σ(x) σ(x) = 2 |x| = 2|x| σ(G) = 21 σ(G) |G| = 2|G| σ(G) = σ(1) + σ(sin θ) = 0 + since σ(0.5) = 0 Rule 6 Rule 6 Rule 6 | sin[θ+σ(θ)]−sin[θ−σ(θ)] | 2 Rules 3,6 • Finally, replace the error terms into the original error Equation B.5, simplify and solve for σ(y) by multiplying both sides of the equation with y: σ(y) = |y| σ(M ) σ(m) σ(k) σ(x) | sin[θ + σ(θ)] − sin[θ − σ(θ)] | + + + + |M | |m| 2|k| 2|x| |4(1 − sin θ)| (B.6) first name (print) last name (print) Brock ID (ab13cd) TA initials grade Appendix C Graphing techniques A mathematical function y = f (x) describes the one to one relationship between the value of an independent variable x and a dependent variable y. During an experiment, we analyse some relationship between two quantities by performing a series of measurements. To perform a measurement, we set some quantity x to a chosen value and measure the corresponding value of the quantity y. A measurement is thus represented by a coordinate pair of values (x, y) that defines a point on a two dimensional grid. The technique of graphing provides a very effective method of visually displaying the relationship between two variables. By convention, the independent variable x is plotted along the horizontal axis (x-axis) and the dependent variable y is plotted along the vertical axis (y-axis) of the graph. The graph axes should be scaled so that the coordinate points (x, y) are well distributed across the graph, taking advantage of the maximum display area available. This point is especially important when results are to be extracted directly form the data presented in the graph. The graph axes do not have to start at zero. Scale each axis with numbers that represent the range of values being plotted. Label each axis with the name and unit of the variable being plotted. Include a title above the graphing area that clearly describes the contents of the graph being plotted. Refer to Figure C.1 and Figure C.2. Figure C.1: Proper scaling of axes Figure C.2: Improper scaling of axes The line of best fit Suppose there is a linear relationship between x and y, so that y = f (x) is the equation of a straight line y = mx + b where m is the slope of the line and b is the value of y at x = 0. Having plotted the set of coordinate points (x, y) on the graph, we can now extract a value for m and b from the data presented in the graph. Draw a line of ’best fit’ through the data points. This line should approximate as well as possible the trend in your data. If there is a data point that does not fit in with the trend in the rest of the data, you should ignore it. 51 52 APPENDIX C. GRAPHING TECHNIQUES The slope of a straight line The slope m of a straight-line graph is determined by choosing two points, P1 = (x1 , y1 ) and P2 = (x2 , y2 ), on the line of best fit, not from the original data, and evaluating Equation C.1. Note that these two points should be as far apart as possible. m = m = rise run ∆y y2 − y1 = ∆x x2 − x1 (C.1) Figure C.3: Slope of a line Error bars All experimental values are uncertain to some degree due to the limited precision in the scales of the instruments used to set the value of x and to measure the resulting value of y. This uncertainty σ of a measurement is generally determined from the physical characteristics of the measuring instrument, i.e. the graduations of a scale. When plotting a point (x, y) on a graph, these uncertainties σ(x) and σ(y) in the values of x and y are indicated using error bars. Figure C.4: Error Bars for Point (x, y) For any experimental point (x ± σ(x), y ± σ(y)), the error bars will consist of a pair of line segments of length 2 σ(x) and 2 σ(y), parallel to the x and y axes respectively and centered on the point (x, y). The true value lies within the rectangle formed by using the error bars as sides. The rectangle is indicated by the dotted lines in Figure C.3. Note that only the error bars, and not the rectangle are drawn on the graph. The uncertainty in the slope Figure C.5 shows a set of data points for a linear relationship. The slope is that of line 2, the line of best fit through these points. The uncertainty in this slope is taken to be one half the difference between the line of maximum slope line 1 and the line of miinimum slope, line 3: σ(slope) = slopemax − slopemin 2 (C.2) The lines of maximum and minimum slope should go through the diagonally opposed vertices of the rectangles defined by the error bars of the two endpoints of the graph, as in Figure C.5. Figure C.5: Determining slope error 53 Logarithmic graphs In science courses you will encounter a great number of functions and relationships, both linear and nonlinear. Linear functions are distinguished by a proportional change in the value of the function with a change in value of one of the variables, and can be analyzed by plotting a graph of y versus x to obtain the slope m and vertical intercept b. Non-linear functions do not exhibit this behaviour, but can be analyzed in a similar manner with some modification. For example, a commonly occuring function is the exponential function, y = aebx , (C.3) where e = 2.71828 . . ., and a and b are constants. Plotted on linear (i.e. regular) graph paper, the function y = aebx appears as in Figure C.6. Taking the natural logarithm of both sides of equation (C.3) gives ln y = ln aebx ln y = ln a + ln ebx ln y = ln a + bx ln e ln y = ln a + bx (since ln e = 1) Figure C.6: The exponential function y = aebx . Equation (C.4) is the equation of a straight line for a graph of ln y versus x, with ln a the vertical intercept, and b the slope. Plotting a graph of ln y versus x (semilogarithmic, i.e. logarithmic on the vertical axis only) should result in a straight line, which can be analyzed. There are two ways to plot semilogarithmic data for analysis: 1. Calculate the natural logarithms of all the y values, and plot ln y versus x on linear scales. The slope and vertical intercept can then be determined after plotting the line of best fit. 2. Use semilogarithmic graph paper. On this type of paper, the divisions on the horizontal axis are proportional to the number plotted (linear), and the divisions on the vertical axis are proportional to the logarithm of the number plotted (logarithmic). This method is preferable since only the natural logarithms of the vertical coordinates used to determine the slope of the lines best fit, minimum and maximum slope need to be calculated. Semilogarithmic graph paper The horizontal axis is linear and the vertical axis is logarithmic. The vertical axis is divided into a series of bands called decades or cycles. • Each decade spans one order of magnitude, and is labelled with numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1. • the second “1” represents 10× what the first “1” does, the third 10× the second, et cetera, and • there is no zero on the logarithmic axis since the logarithm of zero does not exist. A logarithmic axis often has more than one decade, each representing higher powers of 10. In Figure C.7, the axis has 3 decades representing three consecutive orders of magnitude. For instance, if the data to be plotted covered the range 1 → 1000, the lowest decade would represent 1 → 10 (divisions 1, 2, 3, . . . , 9), 54 APPENDIX C. GRAPHING TECHNIQUES Figure C.7: A 3-Decade Logarithmic Scale. the second decade 10 → 100 (divisions 10, 20, 30, . . . , 90) and the third decade 100 → 1000 (divisions 100, 200, 300, . . . 900, 1000). Another advantage of using a logarithmic scale is that it allows large ranges of data to be plotted. For instance, plotting 1 → 1000 on a linear scale would result in the data in the lower range (e.g. 1 → 100) being compressed into a very small space, possibly to the point of being unreadable. On a logarithmic scale this does not occur. Calculating the slope on semilogarithmic paper The slope of a semilogarithmic graph is calculated in the usual manner: m = slope rise = run ∆(vertical) = . ∆(horizontal) For ∆(vertical) it is necessary to calculate the change in the logarithm of the coordinates, not the change in the coordinates themselves. Using points (x1 , y1 ) and (x2 , y2 ) from a line on a semilogarithmic graph of y versus x and Equation C.4, the slope of the line is obtained. m = m = m = rise run ln y2 − ln y1 x2 − x1 ln (y2 /y1 ) x2 − x1 (C.4) Note that the units for m will be (units of x)−1 since ln y results in a pure number. Analytical determination of slope There are analytical methods of determining the slope m and intercept b of a straight line. The advantage of using an analytical method is that the analysis of the same data by anyone using the same analytical method will always yield the same results. Linear Regression determines the equation of a line of best fit by minimizing the total distance between the data points and the line of best fit. To perform “Linear Regression” (LR), one can use the preprogrammed function of a scientific calculator or program a simple routine using a spreadsheet program. Based on the x and y coordinates given to it, a LR routine will return the slope m and vertical intercept b of the line of best fit as well as the uncertainties σ(m) and σ(b) in these values. Be aware that performing a LR analysis on non-linear data will produce meaningless results. You should first plot the data points and determine visually if a LR analysis is indeed valid.