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Physics for Science and Engineering III
Physics 112
Department of PhysicsIllinois State UniversityNormal, IL 617904560
Fall 2010Edited by B. K. Clark and R. F. Martin, Jr.
PART I Computational Labs
Lab Number
Title
CL1 Introduction to the Unix Environment 4
CL2 The FORTRAN Compiler 8
CL3 Pendulum Motion: The Euler Method for Solving FirstOrder Differential Equations
11
CL4 Damped Oscillator: FeynmanNewton Method for Solving FirstOrder Differential Equations
14
CL5 Monte Carlo Simulation of Light Scattering 17
PART II Experimental Labs
Lab Number
Title
EL1 Vibrating Strings (BC4) 21
EL2 A Study of Thin Lenses (BC3) 24
EL3 The Michelson Interferometer (BC1) 28
EL4 Measure the Wavelength of Light with a Ruler (BC5) 31
EL5 The Photoelectric Effect: Measurement of h/e (BC2) 32
Appendix
A Brief Guide to Writing Lab Reports 36
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PART I
Computational Labs
3
Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Computational Physics Laboratory IIntroduction to the UNIX Environment
Introduction
Historically, theory and experiment have been the main tools for advances in science and engineering, and these two methodologies are well represented in this course. Progress in computer technology, both software and hardware, has generated a new methodology computational science and engineering for learning and doing physics. Presently, computational physicists and engineers are as prevalent as theoreticians and experimentalists. These Units will introduce you to the basic tools of computational science.
Purpose
This Unit provides an introduction to the Physics Department computer system. In particular, you will be introduced to the following areas:
1. Basic UNIX commands for manipulating files and directories.2. Files and the hierarchical directory structure of UNIX3. Code editing4. Basic networking
Unix was developed in the 1970’s and its structure still defines how modern operating systems work. Both Windows and MacOS use Unixlike file structure (e.g. files and hierarchical folders), Linux is a Unix derivative, and there are scientific and engineering workstations that run versions of Unix. A brief guide to Unix commands is given below and in Chapter 2 of the ISU Physics Department Computer Manual available online at http://www.phy.ilstu.edu/~rfm/phy112lab.
Unit 1 Assignment and Requirements (Due at the end of the laboratory period)
1. Read these lab instructions and relevant parts of the online manual.
2. Take the short quiz.
3. Send an email containing information about your physics, math, and computer experience: what courses you have taken in these areas and any outside experience you’ve had. Save a copy of this file in your home directory with permissions set so that a faculty member can read it (see the chmod command). You should then send me an email message telling me that the file is available and where it is saved.
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Appendix 1. Unix and Networking Commands (Courtesy of Prof. Bogue)
Unix Commands
Command Brief Description
cp file1 file2 Copy file1 to a new file named file2
rm file1 Delete the file named file1Note: file1 is gone forever
mv file1 file2 Rename the file named file1 to file2Note: file1 is gone, but file2 exists with the contents of file1
mkdir dir1 Create a new directory named dir1
rmdir dir1 delete the directory named dir1Note: dir1 must be empty
ls List the files in current directory
ls dir1 List the files in the directory named dir1
cd dir1 Change the current directory to dir1. Note: cd without argument changes to home
directory
more file1 List contents of file1 to the screen, one screen at a time. Note: spacebar advances to next screen and return advances one line
passwd Change password
pwd Displays current directory
logout Close interactive session
man command Displays the manual page for command
Networking Commands
ssh hostname Open a connection to the computer whose network name is hostname
sftp hostname Open a connection to hostname for purpose of file transfer. Then use the following
commands:
get file1 Retrieve file1 from hostnamesend file1 Send file1 to hostname
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ls List files in current directory of hostnamecd dir1 Change directory to dir1 on hostname
Appendix 2. Jove (emacs) Commands (Courtesy of Prof. Bogue)
The following list summarizes some basic editing commands for the Jove version of the emacs text editor, arranged by function. Historically, emacs was one of the most sophisticated screenoriented text editors (compare it to the clunky vi editor sometime...), although such beasts have largely been replaced by code editing applications that do more than just edit text. We include it here to give you some historical knowledge who knows? You may actually like it.
Some of the commands have mnemonic names which are given with the abbreviation in boldface. Learning these will allow you to perform most editing tasks needed for programming. However, emacs is really much more sophisticated than these commands indicate, allowing you to edit multiple files simultaneously, do text formatting, run Unix shell processes, customize emacs commands, and more. For further information consult the web.
General
^ = control keyESC = escape key^U = multiply the following command by 4
Moving by characters, lines, words
^B 1 space LEFT (Back)^F 1 space RIGHT (Forward)^U^F 4 spaces RIGHT^U^U^F 16 spaces RIGHT
ESC B 1 word LEFTESC F 1 word RIGHT
^P 1 line UP (Previous)^N 1 line DOWN (Next)^A to BEGINNING of current line^E to END of current line (End)
Moving by screens and more
^V FORWARD 1 screenESC V BACKWARD 1 screen
ESC < BEGINNING of fileESC > END of fileESC G go to specific line (prompts for line number)
ESC , FIRST LINE on screen (note “,” is unshifted “<“)ESC . LAST LINE on screen (note “.” is unshifted “>“)
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Deleting and Changing
^H delete character LEFT^D delete CURRENT character (Delete)^K delete to END of line (Kill)ESC D delete word RIGHTESC K delete to end of SENTENCE (i.e. to a period or blank line)
Searching and Replacing
^\ search FORWARD (prompts for search string)^R search REVERSE (prompts for search string)ESC R replace all FORWARD (prompts for search and replace strings)ESC Q replace with Query FORWARD (prompts for search and replace strings):
– spacebar replaces the string
– delete key skips the replace and goes to next occurrence
– return key stops further replacements
Cutting, Copying, and Pasting Text
^spacebar set a mark (invisible) at location of cursor^WCUT text between the cursor and the mark (moves it to a buffer) (Wipe)ESC W COPY text between the cursor and the mark into a buffer^Y PASTE the last text that was cut or saved at the cursor position (Yank)
Inserting Files
^X^I Insert the file (filename prompted for) at the cursor position
Quitting, Saving
^X^\ Save the file being edited (overwrites if file already exists)^X^W Write a new file (prompts for filename)^X^C Exit from the file being edited.
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Computational Physics Laboratory 2Introduction to the FORTRAN Compiler
Introduction
The first unit introduced the Unix operating system, the BASH shell and simple shell commands, ssh and other networking commands, and the use of text editors. These are essential tools for using any Unixbased computer, including meitner, an Intel quadcore cpu. Two additional tools are required to use the computer effectively for learning and doing physics: a highlevel programming language and a graphics package. The programming language used in this course is Fortran 95, a widely used high level programming languages in the sciences and engineering. For visualizing your results, you will use the KaleidaGraph data analysis and graphics software (note: you can also use GNUplot, an open source graphics program, if you prefer). With a familiarity with KaleidaGraph, you should be able to adapt to other packages you may come across in the future.
Purpose
In this project, you will be introduced to the following:
1. Basic FORTRAN commands2. Compiling, linking, and running FORTRAN programs3. Plotting with KaleidaGraph
Information on these topics can be found in the physics department computer manual. You are also encouraged to use the original software manuals and the web.
Unit Assignment and Requirements
Do each of the following:
1. Read the Fortran chapter and the sections on KaleidaGraph in the Analysis Software chapter.
2. Write a FORTRAN program to compute the trajectory of the simple pendulum assuming the solution is of the form
= A cos(t + ),
where A is the amplitude, is the frequency, t is the time, and is the phase offset. Choose initial angles of 1° and 10° and a pendulum length of 1.00 m. You must output the data to an ascii file. Each of the files should consist of time, position, velocity, and acceleration. Make plots of each position, velocity, and
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acceleration as functions of time. What are the phase relationships between position, velocity, and time? Repeat your calculations and analyisis for 0 = 2° and d/dt = 0.200 rad/sec.
3. Transfer a plot image into a text editor document and confirm that it is of sufficiently high quality with appropriately sized fonts to be readable.
4. Email the instructor a graph (try a jpeg or png) of one of your graphs at publication quality resolution with a short description.
5. Turn in a full lab report by the instructor specified due date and time.
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Appendix: Sample Program
program pendulum
program pendulum!! Declare variables!real*8 :: period, phi, t, amp, pi, g, lreal*8 :: thet, thetd, thetdd, thet0, thet0d, winteger :: i!! (describe the following section of code)!open(30, FILE = 'pend.txt') !output file for datawrite(30,'(A)') ' time angle ang. vel. ang. accel.'pi = 4.0d0*atan(1.0d0)g = 9.8d0 ! acceleration due to gravity in m/s2l = 1 ! length of pendulum in mw = sqrt(g/l) ! (describe variable)period = 2.0d0*pi/w ! (describe variable)thet0 = 1.0d0*(Pi/180.0d0) ! (describe variable)thet0d = 0.0d0 ! (describe variable)amp = sqrt(thet0**2+(thet0d/w)**2) ! (describe variable)phi = atan(thet0d/(w*thet0)) ! (describe variable)print*, 'initial angle = ',thet0,' initial ang. vel. = ',thet0dprint*, 'frequency = ',wprint*, 'amplitude = ',amp,' phase = ',phi!! (describe the following section of code)!do i = 1,5000 t = period *dble(i)/1000.d0 thet = amp * dcos(w*t + phi) thetd = w*amp * dsin(w*t + phi) thetdd = w**2 *amp * dcos(w*t + phi)!! (describe the following section of code)! if (mod(i,10) == 0) then write(30,'(4(e12.5,3x))') t, thet, thetd, thetdd endifenddo!close(30)end
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Computational Physics Laboratory 3: Pendulum MotionThe Euler Method for Solving FirstOrder Differential Equations
Introduction
In this unit you will apply your knowledge of FORTRAN and graphics to solve Newton's Second Law for the motion of a pendulum with and without the small angle approximation. In the last computational assignment, you solved the pendulum problem in the small angle approximation. Within that approximation, there is an analytic solution. However, no such solution exists in the absence of the small angle approximation.
Newton's Second Law,
F t =ma t =md2 s
dt 2 , (1)
for an object of mass m, is a secondorder differential equation. Eq. (1) can be expressed as two firstorder differential equations,
dvdt
=F t dt
(2)
and
dsdt
=v t . (3)
Eqs. (2) and (3) are simultaneous equations which can be solved, in principle, for the velocity v(t) and position s(t) of the object. For simple forces F(t) the solution can often be found mathematically, as you will do in the differential equations course, but for complicated forces such an analytic solution may be difficult or impossible to find. However, you can prove that a solution always exists (assuming F(t) is a wellbehaved function), so how do you find it? If you can’t find it mathematically, you can always try a computational method.
The simplest numerical technique for solving a first order differential equation of the form
df t dt
=q t (4)
is to convert it to a finite difference equation. The Euler algorithm does just that, by approximating the derivative by a finite difference,
df t dt
=Δf t
Δt≈
[ f t+h −f t ]h
, (5)
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where h = t is the finite timestep. Solving (4) and (5) simultaneously gives
f t+h ≈f t +hq t . (6)
The initial value f(t = 0) must be known to start the iterative solution, which simply involves application of Eq. (6) in a loop over t to get f(t) at all times desired.
In this case, a small translation must be made in moving from an equation in s, v, and a to , d/dt, and d2/dt2. We begin with
which can be separated into the two equations
ddt
θ̇ =−gl
sinθ,
and
The above equations can be implemented via code similar to
do i = 1,10000 a = g*sin(xold)/l v = vold + a*dt x = xold + vold*dt t = told + dt write(10,*) t,x,v,a vold = v xold = x told = tenddo.
This code is an example of the Euler algorithm. In the EulerCromer algorithm, vold is replaced with v in the x equation.
Like in digital sampling (the previous lab), there are two types of errors that can enter such a numerical solution: (i) roundoff error is produced by the finite accuracy of storing numbers in computer memory; (ii) truncation error is produced by the approximate solution algorithm, in this case the finite value for the time step h in Eq. (7). You don’t have much control over roundoff error: it is determined by the computer hardware you are using, although highlevel compilers such as Fortran95 have software algorithms (such as double precision) that can reduce roundoff error at the expense of processing speed. You do have some control over truncation error: for example, it can be reduced by reducing the magnitude of h, again at the expense of processing speed, or by using a more accurate algorithm (as you will in the next computational project).
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dθdt
=θ̇ .
d2 θdt 2
=−gl
sin θ,
Sometimes trucation error can be so severe that the numerical solution is unstable (e.g. it never settles down or it blows up to infinity), in which case the numerical solution fails and you have to go to a more accurate algorithm.
Objectives
In this unit you will do the following:
1. Use the Euler and EulerCromer algorithm to solve Newton's Second Law for the motion of a pendulum acted upon by gravity.
2. Produce plots of your numerical solutions.
3. Compare these results to the analytic solution obtained with the small angle approximation.
Assignment
1. Modify your code from Computational Physics Laboratory 2 to implement the Euler and EulerCromer algorithms. I recommend saving a backup copy before you start editing.
2. Choose initial angles of 1° and 10° and assume the pendulum begins at rest to compare the two algorithms. I would begin with the time step that you used in the last computer lab. Determine if one, both, or either of the algorithms provide stable solutions. Is there any observable effect if you increase or decrease the time step by a factor of 10 for the Euler method?
3. Compare your results with the results obtained with the analytic solution in the small angle approximation. Include plots of the differences in pendulum positions for the three methods. What information can be discerned from these plots.
4. Turn in a full lab report by the instructor specified due date and time.
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Computational Physics Laboratory 4: Damped OscillatorFeynmanNewton Method for Solving FirstOrder Differential Equations
Introduction
In this unit you will apply your knowledge of FORTRAN to solve Newton's Second Law for the motion of an oscillator acted upon by a velocitydependent drag or damping force. The drag is a force Fdamp which is directed opposite from the direction of the motion and which has a magnitude
Fdamp = cv, (1)
where c is the damping coefficient and v is the velocity of the oscillating mass. Newton's Second Law for an oscillator of mass m acted upon by a linear restoring force with springconstant k is
F t =ma t =md2 xdt 2 =−kx t −cv t , (2)
and is a secondorder differential equation. As in Unit 3, Eq. (2) can be expressed as two firstorder differential equations
dvdt
=F t m
(3)
and
dxdt
=v t . (4)
You can try to use the Euler method to solve this problem, but you will find that it is unreliable at best. This problem is solved better by using an improvement to the Euler method known as the FeynmanNewton Method or the halfstep method. In this method the solution to a firstorder differential equation
dxdt
=q t (5)
is obtained by using the value of the forcing function q(t) halfway between t and t + h, i.e. iterating the expression,
f t+h ≈f t +hq t+h /2 . (6)
This algorithm is known as the FeynmanNewton (FN) method or halfstep method for obvious reasons, and is more accurate than the Euler method. For a small timestep h, the Euler method has a truncation error proportional to h, while the FN method’s truncation error is proportional to h2. Thus, for a timestep of 0.1 the FN error is about onetenth of the Euler error. You can learn more about the error of algorithms by taking PHY 318.
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Objectives
In this unit you will do the following:
1. Use the FeynmanNewton algorithm to solve Newton's Second Law for the motion of an oscillator with a velocitydependent drag or damping force.
2. Compare the result to the Euler method and to the analytic solution.
3. Plot both x(t) and v(t) vs. t and v(x) vs. x (phase space plot).
Assignment
1. Solve the problem for the damped oscillator by two methods:
a.Euler Method: Solve the damped harmonic oscillator with the Euler method. Use h = 0.25 s and h = 1.0 s, and integrate from t = 0 to t = 25 s. Take k = 1 N/m, c = 0.1 kg/s and m = 1 kg, and use initial conditions x(0) = 1 m and v(0) = 0 m/s. Hand in the source code.
Make a single plot showing x(t) as a function of t for both values of h. How do the curves compare? Which seems to be more accurate? Why?
b. FeynmanNewton Method: Modify your program to use the FeynmanNewton method and solve the same problem. Use h = 0.25 s and h = 1.0 s, and integrate from t = 0 to t = 25 s. Hand in the source code.
Make a single plot showing x(t) as a function of t for both values of h. How do they compare? Which seems to be more accurate? Why?
Make a second plot of v(t) and a(t) vs. t for h = 0.25 s. Describe how v and a behave. Can you explain why?
2. Compare with analytic solution: The exact solution,
x(t) = 1.00125 xo e 0.05t cos(0.0500209 0.998749t),
can be obtained analytically if you know some differential equations. This analytic solution is for x(0) = xo and v(0) = 0 m/s and the above parameters. Compare both the Euler and FN numerical solutions for x(t) (using h = 0.25 s) to the analytic solution at various times. Which numerical method best approximates the analytic solution? Is the result what you expected? Explain.
3. Phase space plot: Make a plot of your data from part 1b above with velocity v(t) on the ordinate (yaxis) and the position x(t) on the abscissa (xaxis). The result is a picture of the trajectory of the mass in “phase space”. From the plot, determine where the mass is when the v = 0 m/s. What is its speed when x = 0 m? Where will the mass be and what will its speed be as t → ∞? (This is called a “fixed point”). What happens to the initial energy of the mass (what is it?) as time passes? Where does it go?
Appendix
The following is a sample loop which uses the FeynmanNewton algorithm. You should be able to integrate this into your program from lab CL3.
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!!....Initialize acceleration, position, and velocity.! Get velocity after half timestep (FN method)!accel = (c1*xstart + c2*vstart)xold = xstartvold = vstart + h*accel/2.!!....Loop for stepping in time! Note VOLD is half step ahead of XOLD!do 10 j = 1,numpts xnew = xold + h*vold accel = (c1*xnew + c2*vold) vnew = vold + h*accel timex = time + h timev = timex + h/2. write(10,'(5(e12.5,3x))') timex,xnew,timev,vnew,accel xold = xnew vold = vnewenddo!!.... Note that this code can be converted into the ! Euler method by eliminating the H*ACCEL/2 in the! expression for VOLD, moving the new calculation ! of ACCEL to the line after VNEW is calculated, and! replacing VOLD in the ACCEL equation with VNEW.
Note that only the velocity is computed right at the halftimestep, i.e. the FN method is being used to solve Eq. (4). Eq. (5) uses a “hybrid” acceleration computed from v at the halfstep and x after a full step. Thus, in the loop, the new position is computed first, then the acceleration, then the new velocity. If you try to compute ACCEL using XOLD instead of XNEW you might find that everything blows up! This is an example of a numerical instability and is an example of how careful you must be in using numerical methods to solve physics problems.
Strictly speaking, the position and velocity are calculated at different times, so I have accounted for that in the sample above by saving times for each variable. Thus, the phase space plots are only approximate (since you should be plotting x(t) and v(t) at the same t). However, for small h the difference is small.
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Computational Physics Laboratory 5Modeling Light Scattering Using Random Numbers
Introduction
In this unit, you will model the scattering of light by a dielectric slab using the computer and a (pseudorandom) random number generator. The result of your simulation will be an estimate for the halfdistance x1/2 of the slab. If we treat a beam of light incident on a dielectric slab as a beam of quanta (commonly called photons), we can see that the photons must scatter out of the incident beam as the beam travels through the slab. If there was no scattering, then the sky would have no color, and we could not see the path a laser takes through a plastic slab. We could make the measurement on a series of dielectric slabs using a series of filters called neutral density filters. Each filter blocks a specified fraction of the light (between 5 and 99 %). If we use a filter designed to scatter 50 % of the light, then we have 50 % of the light after 1 filter, 25 % of the light after 2 filters, etc. The use of these filters is essential to measurements in nonlinear optics. Of course we can do the calculation analytically for comparison, but a great deal can be learned from a computer simulation of the process.
If the scattering of a photon is an independent, random event, then the number of photons N(x) remaining in a light sample at location x is
N(x) = No exp(–kx), (1)
where No is the number of photons at x = 0. The decay rate coefficient k is related to the halfdistance by k = ln(2)/x1/2, where x1/2 is the distance where N(x) = N0/2
The scattering process can be simulated by generating a set of random numbers r where 0 ≤ r ≤ 1. We assign each photon that has not been scattered a random number from the set. We assume that the probability of a photon scattering in the unit of distance is p where 0 ≤ p ≤ 1, and the probability that it does not scatter is 1– p. Then a photon assigned random number r will scatter if 0 ≤ r ≤ p; otherwise the photon does not scatter. If the photon scatters, the number N of photons that remain in the beam is reduced by 1. After all photons are tested, this completes one appropriate interval of distance. This process is repeated for successive distance intervals, where only the set of photons N that have not previously scattered is considered at each step. The process ends when N = 0, or after a reasonably large number of distance intervals (loops). The use of random numbers to simulate a physical process is sometimes referred to as the Monte Carlo method, after that famous city where randomness can determine whether you win or lose your money.
The random number generator on any computer only approximates a series of truly random numbers since the computer uses an algebraic algorithm that is completely deterministic. However, UNIX and most FORTRAN compilers include very good random number generators, so you will use the builtin generator available on meitner.
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Objectives
1. Write a program which models light scattering using the Monte Carlo method.
2. Explore the effect of varying No and p on the scattering process.
3. Find the probability p which best simulates the 1 mm thick NG5 filter, which transmits 53 % of the light and scatters 47 % of the light. Technically, surface scattering (reflection) and bulk scattering are not the same, but we will assume all scattering is bulk scattering for simplicity.
Assignment
1. Set up a FORTRAN code to implement the Monte Carlo simulation of light scattering:
a.Vary N: Fix p = 0.05 and vary No = 103, 104, and 105. It does not really matter what values you choose for N0, so long as your range covers at least a factor of 100. Similarly, any small value for p will work here. Make one plot (of N vs. x) with curves for all three No’s on it. Describe the shape of the curves and how the different No’s affect it. Of course, you must also choose a step size for x. If p is in units of 1/2 mm, then x is in units of 2 mm.
Make another plot with a logarithmic yaxis (a “semilog” plot). Now describe the shape of the curves and the effect of varying No.
b. Vary p: Fix No = 104 and vary p = 0.005, 0.05, 0.1, 0.5. Make one plot (of N vs. x) with curves for all four p’s on it. Describe the shape of the curves and how the different p’s affect it. Here, the important point is the factor of 1000 change in p.
Make another plot with a logarithmic yaxis (a “semilog” plot). Now describe the shape of the curves and the effect of varying p.
2. Simulation for the NG5 filter. The NG5 neutral density filter is 1 mm thick and transmits approximately 53 % of the incident light in the visible spectrum. Choose an appropriate step size for x, and determine which value of the probability p corresponds to NG5 filter. Should x be 1 mm or many times less? Use the method for finding k in the halfdistance equation discussed above and its relationship to p to provide a starting point. Explain your reasoning and show work as appropriate. Make a plot of log(N) vs. x for this value of p (Using No = 104, or any other reasonable value).
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Appendix
A sample program that seeds and then calls a random number generator using the gfortran compiler is shown below. You need to properly merge the relevant parts into your code. There are a number of commands that you have not seen before. If you want to know what they do (and you should), hunt for them using the web.
program randomgen integer, dimension(:), allocatable :: seed integer n, clock double precision x Call random_seed(size=n) allocate(seed(n)) Call system_clock(count=clock) seed = clock+37*(/(i1,i=1,n)/) Call random_seed(put=seed) deallocate(seed) do i = 1, 10 call random_number(x) print *, i, x enddoend
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PART II
Experimental Labs
20
Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Experimental Physics Laboratory 1Vibrating Strings (BC4)
Introduction
Standing waves on a string of length L are described in Chapter 18 of Physics for Scientists and Engineers by Serway and Jewett. The basic concept is that a wave will interfere with its own reflection. When this happens, the wave and its reflection are generally out of phase and will destructively interfere, leading to partial cancellation. However, when the phase relationship is just right, constructive interference is maximized and standing waves are observed. Standing waves are characterized by a set of nodes and antinodes, where the amplitudes of the antinodes are maximized at resonance. Fig. 1 shows the standing wave patterns for the fundamental frequency and the second and third harmonics. It is clear that each end of the string is fixed and there are n-1 nodes and n antinodes, where n is the number of the harmonic (the fundamental is the first harmonic). The points of greatest constructive interference are called antinodes, and have the greatest amplitudes. The points of complete destructive interference are called nodes, and remain stationary. The endpoints are ignored.
Fig. 1. The fundamental (first), second and third harmonics for a string with fixed ends.
Standing waves can be observed in the laboratory using a set-up similar to the one shown in Fig. 2. The experimental set-up consists of a string that is attached to a post at one end of an optical bench, and it is draped over a pulley at the other end of the bench. A vibrator is placed in between the post and pulley. When the vibrator is active, standing waves can be created between the vibrator and pulley by moving the vibrator back and forth with respect to the pulley until the source wave and reflected wave are in phase with each other.
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Figure 2. Experimental Set-up.
The distance between successive nodes of a standing wave is half the wavelength . Standing waves only occur when
L=nλ2
, (1)
where n can be any positive integer. As we found in class, the wavelength, frequency, f, and velocity, v, of the wave are related as f = v, which can be substituted into Eq. (1) to yield
f=nv2L
. (2)
In the lecture part of the course, we showed that the velocity of the wave on the string is given by
v=Tμ
, (3)
where T is the tension and is the mass density of the string. Eq. (2 ) and (3) can now be combined to yield
f= n
2Lμ1/2 T 1/2 . (4)
Eq. (4) shows us that the oscillation frequency is linearly dependent on the harmonic number and inversely proportional to the square root of the mass density. A rearrangement of this equation might be more useful depending on the measurements that we wish to complete.
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Table
Vibrator
Mg
We can precisely determine the mass density by carrying out a linear least squares fit of several pairs of n, f values for the same values of , T, and L. In fact this is generally a very precise way to determine the mass density. Most commonly available mass scales to not provide sufficient precision to be useful for very low mass strings. We should also consider that the mass density might change as the tension is increased. How might Eq. (4) be used to determine the mass density as a function of tension? Consider the fact that the mass density decreases as tension is increased, consequently, the mass density in Eq. (4) is not a constant when frequency is observed as a function of tension.
Objective
This exercise provides an opportunity to verify many of the equations describing wave motion on strings. We will experimentally verify the equations describing standing waves and determine the mass density of a string. Additionally, we will verify the amplitude and frequency dependence of the power for a wave on a string. Perhaps more importantly, this experiment gives you a chance to gain practical familiarity with the behavior of a vibrating string.
Procedure
Attach the components to the optical bench as shown in Fig. 2. The strings are guitar strings. Be sure to pick two strings of different mass density. Starting with a vibrator frequency of 100 Hz and a mass of 150 g, or any other combination of values that yield clear standing waves, determine the optimal vibrator position for observing two antinodes. Feel free to vary any value to gain an improved understanding of standing waves. Determine the mass density of the string and observe as many standing wave harmonics as possible. Make sure you identify standing wave frequencies to the nearest Hz. Can you observe a standing wave at the fundamental frequency? You can check your determination of the mass density of the string by doubling the string or choosing a different tension and repeating your measurements. Do the new measurements correspond to the new mass density? Now try a set of measurements with a different tension. Does a node always occur at the vibrator? You should end up with four different sets of data.
Tinker with the set-up to make sure that you completely understand it. Before leaving the room, make sure that your data is self-consistent. If it is not, then identify the problem and correct it? If you place a finger on the vibrating string, what happens as you move your finger back and forth along the string? Do you observe any higher harmonics at the same time you observe some of the lower harmonics?
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Experimental Physics Laboratory 2Thin Lenses (BC3)
Introduction
A converging lens refracts a parallel beam of light, incident from the, left, to a real focus, to the right of the lens. A diverging lens refracts a parallel beam of light, incident from the left, so that it appears to come from a virtual focus to the left of the lens. It is easy to distinguish a converging lens from a diverging lens; the converging lens is thicker in the middle than at the edges, while the diverging lens is thicker at the edges than at the middle. Sample ray diagrams are shown for each type of lens in Fig. 1.
converging lens
ff
siso
diverging lens
-f
sosi
-f
Figure 1. Ray diagrams for diverging and converging lenses. Object and image distances as well as focal lengths are defined in the diagrams.
The thin lens equation relating the object distance so, image distance si, and focal length f is
1s o
1s i
=1f , (1)
and is valid for both real and virtual images. Objects are real and positive when they are on the side of the lens from which light is coming. Images are real and positive when they are on the side of the lens to which light is
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going. Otherwise, objects and images are negative. Converging lenses have positive focal lengths, and diverging lenses have negative focal lengths.
Objective
The objective of this experiment is to gain a practical familiarity with thin lenses. To do so, you will determine the focal lengths of two converging lenses by several techniques, determine the focal length of a diverging lens, and study some of the general properties of the formation of real images using a converging lens.
Procedure
Unless otherwise indicated by your instructor you will be given three converging lenses of approximate focal lengths 5 cm, 10 cm and 20 cm. The thickest lens’ focal length is approximately 5 cm. You will also be given a diverging lens with a focal length of approximately – 10 cm.
Part I. Focal length of converging lenses.
A. Parallel rays: Place the 10 cm nominal focal length lens in a lens holder on the optical bench some distance from a screen located at the 100 cm position at the extreme right on the bench. Project the image of a distant object (such as a building or tree) on the screen by orienting the bench such that the lens is between the distant object and the screen and then adjust the distance between the lens and the screen until the image is as sharp as possible. Since the object distance is very large (approximately infinity) the screen will lie in the focal plane of the lens and the distance between the lens and the screen will be the focal length. Record the position of the lens that produced the sharpest image on the screen. Repeat this with the approximately 20 cm converging lens.
B. Thin lens formula: Place the illuminated object near the 100 cm end of the optical bench, the screen near the other and the nominally 10 cm converging lens on the bench between them. Adjust to the same height. Then move any or all of them around until an image, that is smaller than the object, is formed on the screen. Record the object and image distances, and calculate the focal length of the lens. Now measure the length of the arrow Yo, on the illuminated object and the length of the image of the arrow, Yi. Calculate the magnification from the measured distances and lengths. How well do they agree? Repeat using the nominally 20 cm focal length lens.
C. Coincidence: Mount a plane mirror at the 100 cm position, place the approximately 10 cm lens up close to it and adjust the position of the illuminated object until a sharp image is formed on the screen held to one side of the object and in the same plane as the object. The light is returning nearly along its original path. For this to be true the rays must strike the mirror at near normal incidence, which means that they are parallel and the image will be the same size as the object. Thus the distance between the lens and the screen (and also
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the illuminated object) is the focal length. Record the focal length of the lens and repeat your measurements for the approximately 20 cm converging lens.
D. Conjugate foci: Place the illuminated object at the 0 cm position and the screen at the 100 cm position. Determine the object and image positions for the nominally 10 cm focal length converging lens when it is located between 0 cm and 50 cm from the object. Determine a second pair of object and image position with the lens in the range 50 cm to 100 cm. The first object (image) position should equal the second image (object) position. Mathematically, show that
f=so si
so +si
, (2)
from eq. (1), and determine the experimental focal length of the approximately 10 cm focal length converging lens. Repeat the measurement for the approximately 20 cm focal length lens.
Part II. Focal length of diverging lens: Place the illuminated object at the zero point and the screen at the 70 cm point on the optical bench. Place the approximately 5 cm focal length lens and the diverging lens in lens holders on the bench with the converging lens nearer to the object. Adjust their heights to be the same as the illuminated object. Bring these two lenses as close together as their holders will permit and move them back and forth as a unit until a sharp image is focused on the screen. Clamp the converging lens in place and record the position of the diverging lens. Remove the diverging lens and move the screen toward the converging lens until a sharp image of the illuminated object again appears on the screen. This image was the virtual object for the diverging lens. Record this final position of the screen. Draw a ray diagram for the approximately 5 cm focal length converging lens. Then draw a second figure with the diverging lens included. Use these figures to obtain quantities so2 and si2; remember so2 is a negative quantity and should be recorded as such.
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Part III. General properties of converging lenses: Use the longer focal length converging lens and complete Table 1. Express the image distance in terms of the focal length of the lens i.e., less than f, twice f, etc.
Table 1. General properties of image formation using a convergent lens.
Object Image
General object distance
Real or virtual
Erect or inverted
Enlarged or reduced
Location (in terms of f)
Over twice the focal lengthAt twice the focal length
Between f and 2f
At f
Less than f
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Experimental Physics Laboratory 3The Michelson Interferometer (BC1)
Introduction
Light from any source will interfere with itself if it is divided using a beamsplitter and recombined, or split with one or more slits and allowed to recombine. The Michelson interferometer is an instrument that is based on a beamsplitter. The Michelson interferometer can be used to measure the wavelengths of different atomic emissions or the spacing between two very close atomic emissions like the yellow sodium doublet. Other applications of the Michelson interferometer include measuring the index of refraction of a gas, the coherence length of a light beam, and the thickness of films. The Michelson interferometer is a very versatile device.
In this experiment you will use a very simple Michelson interferometer. First, you will calibrate the interferometer using a helium-neon laser ( = 632.8 nm). Second, you will use the calibrated interferometer to measure the wavelength of the emission from a green laser.
Fig. 1. Michelson Interferometer.
The interferometer (Fig. 1) consists of a light source, a beamsplitter, one fixed mirror (mf), one mirror on a translation stage (mt), and a projection screen or an observer’s eye. When light from the laser is incident on the beam splitter, 50% of the light is reflected to mf and 50% of the light is passed on to mt. On the return path from mf, 50% of the light is reflected back to the laser and 50% of the light is transmitted to a projection screen. On the return path from mt, 50% is transmitted back to the laser and 50% is reflected to the projection screen. Thus a total of 50% of the initial light is sent to the projection screen and 50% is directed back to the laser.
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mt
mf
projection screen
light source
When the interferometer is properly aligned, the two beams that are incident on the projection screen will interfere to produce a series of bright and dark fringes. Actually if the distance from each mirror to the beamsplitter is nearly identical, a pattern of bright and dark regions that depend on the flatness of the mirrors are observed. As mt is moved the pattern becomes conical. Why? The bright fringes occur when the paths are a whole number of wavelengths out of phase, and the dark fringes occur when the paths are a whole number plus an additional half number of fringes out of phase. As mt is translated, the fringe pattern alternates between bright and dark fringes at the center of the pattern. The fringes appear to move toward or away from the center. Note that the path length between the beamsplitter and each mirror is actually double the distance because the light beam makes a round trip. The difference in round trip distance determines whether two light beams are in phase or not. The distance to the screen is irrelevant. This distance for the beam travelling the beamsplitter- mt path is
d 1 = 2λ x1 , (1)
where is the wavelength of light, and x1 is the number of wavelengths between the mirror and the beamsplitter. At another position for mt we similarly obtain
d2=2λ x2 (2)
If both of these positions correspond to a bright fringe at the center of the pattern on the screen, we have
d2−d1=2λ x2−x1 . (3)
This relates the distance mt was translated to the number of whole fringe shifts observed during the translation. The difference in distance is proportional to the number of turns, n, during which a number of fringes, mne, appear at the center of the screen. So,
dne=2λne mne , (4)
where dne = d2 - d1. Of course dne has the units of length, and it is directly proportional to the number of turns of the dial that it took to produce dne. The unknown proportionality constant, α , depends on the detailed construction and drive mechanism of the interferometer. If a second light source is examined, it will produce a similar equation to Eq. (6),
du=2λu mu , (5)where the subscript u refers to the unknown wavelength of the new light source. Eqs. (4) and (5) can be combined to yield
λu=λne du
dne
mne
mu
. (6)
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If the same number of fringes are observed for both the calibration source and the unknown source, then the number of observed fringes cancel. Finally, the distances dne and du are proportional to the number of turns of the dial, so the ratio of the number of dial turns Nne and Nu can be substituted into eq. (6) to yield
λu=λne N u
Nne
. (7)
Objective
In this experiment, the behavior of the Michelson interferometer is explored. The interferometer is then used to measure the wavelength of emitted light from a green laser.
Procedure
Use the He-Ne laser to calibrate the interferometer. The He-Ne laser should be aligned in the interferometer when you come into the lab. The instructor will provide any required guidance on alignment. You should be able to see fringes projected onto a screen or wall. DO NOT LOOK DIRECTLY INTO THE LIGHT BEAM. Once the interferometer is aligned, count about 150 fringe shifts and record the total dial movement. Your lab partner may have to count the total number of revolutions of the dial, and you can read the partial number of revolutions from the scale. You do not have to count all the fringes in one pass. I suggest counting fringes in sets of 10-15. The number of dial turns should be very nearly the same for each set of equal numbers of fringes. This provides a check on the quality of the results. Calculate the distance mt moves per turn of the dial.
Now remove the He-Ne laser from its clamp and place the green laser so that its light passes through the beamsplitter. Repeat the measurements with the new light source. From your calibration in the first part of the experiment, calculate the wavelength of the green laser emission.
Discuss the uncertainties introduced in each part of the experiment. What is the total uncertainty in each calculated value?
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Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Experimental Physics Laboratory 4Measuring the Wavelength of Light with a Ruler (BC5)
The instructions and theory for this laboratory exercise are given in class. You will need to take the theory as presented in class and test it by measuring the wavelength of two light sources with a metal ruler. The measured wavelengths can in turn be used to measure the diameter of a hair.
31
Illinois State University Physics 112Department of Physics
Physics for Scientists and Engineers III
Experimental Physics Laboratory 5The Photoelectric Effect: Measurement of h/e (BC2)
Introduction
Most metals emit electrons under the influence of electromagnetic radiation at optical frequencies. This phenomenon was observed as early at 1887, and was termed photoelectric emission. The photoelectric emission process depends strongly on the frequency of the light, and for each metal there exists a critical frequency such that light of lower frequency is absolutely unable to liberate electrons while light of higher frequency always does. For a given surface, the energy of the emitted electrons increases linearly with frequency. Electron emission occurs within a very short time interval after the arrival of the radiation, and the number of electrons emitted per unit time is strictly proportional to the intensity of the radiation.
The experimental facts given above are among the strongest evidence for a quantized electromagnetic field. Photoelectric emission cannot be explained in terms of a continuous energy distribution in the radiation field, instead, it must be assumed that the field consists of “quanta” of energy. An individual quantum has an amount of energy,
E=hν , (1)
where is the frequency of the radiation and h is Planck’s constant. These quanta are called photons.
Further, it is assumed that the electrons are bound inside the metal surface with an energy e, where is the work function and e is the magnitude of the charge on the electron. All such electrons have equal probability of absorbing a photon. It will be possible to eject photoelectrons if the frequency of light is such that
hν>eφ . (2)
Photoelectrons cannot be ejected if
hν<eφ , (3)
because the probability that an electron will absorb two photons simultaneously is very small. In the former case, the excess energy of the quantum appears as kinetic energy of the electron, so that
hν=1/2 mu2+eϕ , (4)
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which is the famous photoelectric equation proposed by Einstein in 1905. Here, m is the mass of the electron, and u is the velocity of the electron. In writing the equation in this form we express the fact that the energy is not shared just between the electron and the photon. A third body is required to balance the momentum, which in this case is the crystal lattice that recoils with negligible energy. Thus in the photoelectric effect we observe the transfer of the total energy of a photon to an electron bound in a lattice.
Eq. (4) has been verified for many materials and over a broad range of frequencies. What is experimentally measured is the kinetic energy of the emitted photoelectrons as a function of photon frequency, either with a magnetic field or in a simpler way by a retarding potential technique, as is done in this experiment. Since the work function, , is usually not known beforehand, the kinetic energy of the photoelectrons, 1 /2 mu2 , is obtained as a function of so that the slope of the straight line
E=hv−eφ (5)
yields h, and the intercept at the extrapolated point = 0, can give e. When a retarding potential V is used to measure E, we have E = eV, so that the ratio h/e is determined from the slope,
V=h /e v−φ . (6)
incident light
e-
cathode anode
amplifier circuit
voltmeter
Fig. 1. Experimental Apparatus
The arrangement that you will use to measure the value of h/e consists of a clean metal surface acting as a cathode, and an anode facing the cathode. Both are sealed in a vacuum. When radiation is incident on the cathode, electrons are emitted which reach the anode giving rise to a detectable current if the circuit between anode and cathode is completed through a sensitive current meter (or instrumentation amplifier) as shown in Fig. 1. The negative charges will condense on the anode if the impedance of the meter is sufficiently high. The maximum negative potential reached by the anode is some potential V, measured relative to the cathode. When the potential reaches V, no additional electrons arrive at the anode, because
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none will have a kinetic energy greater than eV . Alternatively, the retarding potential, V multiplied by e is equal to the energy of the fastest electrons emitted. In practice all electrons are not emitted with the same energy, and therefore the threshold at V is not very sharp. Space charge effects further reduce the definition.
Objective
In this exercise, you will examine Einstein’s theory of the photoelectric effect and determine the value of h/e.
Procedure
The phototube you will be using is a commercial vacuum type 1P39 that uses a mixture of cesium and antimony for the cathode. It is housed in a metal box that permits light to enter through a vertical slit. Connection to the instrumentation amplifier is made inside the box. A voltmeter is connected to the amplifier through banana jacks on the main unit. The primary sources of light are a mercury vapor lamp and helium lamp. The purpose of the filters is to suppress wavelengths of light shorter than the one selected. Why aren’t the longer wavelengths important? These filters are designed with an alpha-numeric code such as OG 570, which means it is a filter that transmits wavelengths of light longer than 570 nm.
Table 1 suggests filters appropriate for the mercury and helium wavelengths indicated.
Table 1. Recommended FiltersColor Wavelength (nm) Filter
Mercury
Yellow 577.0 & 579.0 OG 570Green 546.1 OG 530Blue-Green 491.6 GG 475Blue 435.8 GG 420Violet 404.7 GG395Ultra-Violet 365.0 None
Helium
Yellow 587.6 OG 570Blue-Green 492.1 GG 475Blue 447.1 GG 420Ultra-Violet 388.9 None
Begin with either lamp. Align the lamp with the phototube housing so that the slit is illuminated. You should measure signal strength of about 2 volts after a short time. There is a short time required for the electrons to collect on the anode.
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Choose a wavelength and insert the appropriate filter. Measure the stopping potential by zeroing the amplifier, which has the effect of removing any electrons from the anode. Now let the anode collect electrons and record the voltage displayed on the voltmeter when the reading stabilized. Repeat this for the other wavelengths in the spectrum of your light source. Switch to the other lamp when you are ready.
Plot a graph of stopping potential versus incident light frequency. Determine the slope and intercept along with their respective uncertainties. Obtain Planck’s constant (h/e) from the slope along with its uncertainty.
The incident intensity of the light should not affect the stopping potential. To verify this pick any convenient spectral line from either lamp. Measure the stopping potential as a function of light intensity with the neutral density filter. The neutral density filter has five different regions of varying effective optical density designed to block from 0 to 80 % of the incident light. What is the result?
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A Brief Guide to Writing Laboratory Reports for PHY112
General Comments: The purpose of writing lab reports is to learn to document and convey what you have done in a professional manner. Professionally written laboratory reports have several components and follow some general rules. As you read about each of these components and rules be aware that the numbers in parentheses represent the assessed value of each rule or component of the report.
Professionally written reports for this course employ grammatically correct English. You are to check your report for spelling (2) and grammar (2). Italicize variables and parameters (2). Consider using an equation editor for complicated equations. Always italicize equations (2). Timeliness of the report is assessed at 1 point per day. This means that you should not put off writing a report for too long of a time.
Each report has four major sections, the abstract, introduction, results and discussion, and conclusion. The next four sections describe the material that belongs in each section. The value of each section and subsection is included. Note that the totals of each subsection may be more than the total of each section. Generally, the points that each section is worth is the limit on how many point you can earn or loose in that section of a full report.
Abstract (5): What did you do (2)? Why did you do it (2)? What are your most important numerical results (2)? At a minimum, these questions should be addressed. Do not try to add too much beyond the minimum in the abstract. The purpose of an abstract is to provide a very short statement that informs the reader about what is in the report. It is not a mini-report. It is not an introduction to your report.
Introduction (10): What is the context of the report with respect to the class? Describe the situation that you are examining either experimentally or computationally. Start with a figure (2). Describe the details of the figure and add material as needed to aid in the reader’s grasp of the system (3). Describe the theoretical basis for the experiment or simulation (2). Develop the appropriate equations (2). Do not simply quote the equations. Important equations should have an equation number. Equations should always be part of a grammatically correct sentence. If you are doing a simulation, describe the computer code and relate the important lines in the code to the theory (3). Pay particular attention to changes in variable names as you move from theory to code implementation. This is also a good section to tell the reader what actual measurements you obtain in the laboratory exercise.
Results and Discussion (10): Provide a detailed discussion of what you actually did and what the results are. If data can be included as a table, it is probably a good idea to do so. Include properly labeled graphs (quantity, scale, units) as needed. A good way to start writing this section is to produce the graphs that you need to explain the importance of the work and write a paragraph about each graph. If you have many similar graphs, describe the first graph or set of graphs in detail, then, tell what is similar,
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different, and important about later graphs with respect to the first graph or set of graphs.
An uncertainty or error analysis is usually included in this section of the report. The uncertainty analysis is not a list of all of the mistakes that were made in the laboratory. If mistakes were made, then the proper procedure is to go back and do the laboratory over. Rather, the uncertainty analysis is where you provide quantitative arguments about the quality and validity of your results. For example, if you measured the mass of an object to within 1gram, how would a 1 gram error in that measurement affect your final results? Is it a 1 % effect or a 10 % effect? A very instructive way to do the uncertainty analysis is to consider how certain you are of each individual measurement that is used in determining final results. State the basis for each of the individual uncertainties. Never use human error. Mistakes are not uncertainties. Then calculate how the uncertainty in each individual measurement impacts the uncertainty in the final results for the laboratory. There are other, often statistical, methods for doing an uncertainty analysis. While other methods might be more traditional in the professional literature, they also assume the author has a detailed understanding of the statistics. The method that is suggested here will help you better understand the statistical methods when you encounter them.
Conclusion (5): A conclusion should never include comments about whether you liked or disliked the experiment. It is not a summary of the report as a whole. It should include your important numerical results (compare this to most important numerical results in the abstract). It may also include qualitative results.
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