ENGG1811 Lab 8: Using Matlab
Objectives
After completing this lab, students should be able to
 Use Matlab as a calculator working with arrays in one and two
dimensions
 Understand how to plot variables in 2 dimensions
 Perform array operations
 Use the comet function to aid in visualising function movement
Assessment
Your tutor may ask you to rerun some of Part A and B, using the command
history in your Matlab session, so don't exit Matlab until the assessment.
If you've done it at home it should only take a couple of minutes to
redo, or you could do it in a script. They will check your Part C script
fully, and will withhold some marks if the script is not properly
documented. You will also need to show your tutor an example from Part D.
The online assessment is about matrix operations, and can be done at
any time.
Video Help
Please watch the following help
videos (no audio)
before attempting the exercises.
They provide comprehensive introduction to the Matlab topics required for
the lab exercises.
If you are familiar with a topic(s),
you can increase display speed
(speed upto 2.0X , button located at the bottom right corner)
to quickly glanced through the topics covered in the videos.

Arrays (assignment, elements, etc.), [Duration 04:34]

Array Functions (size, length, zeros, ones, eye) , [Duration 08:04]

Colon (:) operator and linspace , [Duration 02:11]

SubArrays , [Duration 03:59]

Array and Matrix Operations (./ , .* , .^ , sum, etc.) , [Duration 12:00]

Plotting Graphs (plot, comet, etc.), [Duration 08:45]

Example: Rates_of_Diffusion (from Week07 Lecture), [Duration 08:05]
Organising your work
After starting up matlab, you will be working in the default folder
MATLAB, We recommend that you organise your work by creating folders under
MATLAB. Create a folder called ENGG1811. Change into ENGG1811 folder and
create a folder called lab08. Change into the directory lab08 and you can
store all your files for this week in this directory.
Part A: Arrays
There are many steps in this part, but most are very simple, almost
trivial. Don't add a semicolon to the answers, we want the results to
appear on screen.
Example: typing the Matlab command
id = [1, 0; 0 1]
is one way of creating the identity matrix of order 2 and displaying the
result.
 Create a 2x3 array m whose first row contains the values 1, 2
and 4 and whose second row contains 3, 7 and 16.
 Confirm how big m is with the size and length functions.
 Assign to a new variable ms the two rows of m swapped
around. That is, the first row of ms is the second row of m,
and the second row of ms is the first row of m. Hint:
the symbol : when used as a subscript means "all values", so m(:,2)
represents the second column of m.
 Divide m by ms, using array (not matrix) division,
assign the result to mdiv.
 Multiply the first row of mdiv by the second row of mdiv,
again using array operations. Don't assign to anything this time, as the
result should be obvious if you've followed the instructions.
 Assign the transpose of m to mt. You may use either '
and .' for transpose because the array contains only real number, but
don't forget that ' is transpose and complex conjugate, while .' is
transpose only.
 We'll do matrix multiplication more seriously later on, but type
these expressions and interpret the results in terms of the shapes of
the two operands and that of the result (if there is any):
m * mt
mt * m
m .* mt
mt .* m
 (Chapman exercise 5.38, abridged). Component failure rates are
expressed as mean time between failures (MTBF). A system that relies on
several components will have an overall MTBF that's a function of the
individual component MTBFs:
Given the following component MTBFs, calculate the system MTBF in hours:
Component: 
Antenna 
Transmitter 
Receiver 
Computer 
MTBF (hours): 
2000 
800 
3000 
5000 
You should use just two Matlab statements: one to assign the four
numbers to a vector, and one to perform the calculation using suitable
array operations on the vector.
Part B: Plotting
 Create a vector x that has values from 2 to 2, with each
value differing from the previous by 0.01. Use the colon operator.
 Repeat the exercise using linspace, assign to x2.
(Hint: You can use x to see how
many points you need.)
 The vectors are not necessarily identical (the isequal
function can compare them). The default display is to show 4 decimal
places. You can turn on maximum display precision with
format long g
which shows 14 decimal places. Compute the difference between the
secondlast element of x and x2.
You may see a small nonzero value which shows there is a small
difference between them. We will discuss numerical errors later on in
the course .
 Consider the following two functions:
Hint: you can use abs(x)
to get x.
Calculate and plot them on the same figure.
Use separate calls to the plot
function, using hold to overlay them (type "help hold"
if you don't know how to use that command).
 Plot can display multiple plots with additional pairs of
arguments. Close the figure window and try it. Add titles and labels to
each axis (don't worry about fancy formatting).
 The figure window has some useful features, including a zoom (the
magnifierwithaplus icon). Use it to examine the bottom of the cusp
function.
 The plots so far have been pure functions, with a single value of y
for each x. This time plot a circle of unit radius using the
equations
Start with 200 points for theta, from 0 to 2π and create x
and y vectors from that.
Hint: to ensure you get a circle rather than an ellipse, before
the plot command type
clf; axis equal; hold on;
Part C: Van der Waals Equation
The relationship between the volume V
of a vessel holding n moles a certain gas
and pressure P in the gas at a certain
absolute temperature T is approximated by
van der Waals equation, a modification of the ideal gas law:
where R is the universal gas constant (You
can use 8.314 J/K/mol, which is not the most accurate version, for this
lab) and the two values a
(representing attraction between particles) and b
(representing the particle volume) are characteristic of the particular
gas.
In derived SI units, for oxygen, a =
138.2 kPa (L/mol)^{2} and b =
0.0319 L/mol.
For this exercise, assume that n is 100 moles, and
T
is 1.94 times T_{c}.
The PV curve has a different shape depending on a critical temperature
and associated volume:
Below the critical temperature the substance is in a mixed gas and liquid
form.
Write Matlab commands to do the following:
 assign known values to variables (for example: a, b, n, R)
 calculate T_{c} and
V_{c}
 plot the PV curve for the range of volume V from
0.35V_{c} to 1.2V_{c}.
Hint: use linspace
to generate vector
V and calculate P from it.
 You can use the following to add labels, title and legends to your graph
(assuming variables n and T are properly defined):
xlabel('Volume (L)')
ylabel('Pressure (kPa)')
title(['Pressure volume relationship for ',num2str(n,4)])
legend(['T = ',num2str(T,3)])
Your graph may look similar to the following:
Part D: Using the comet command
You are asked to write a script to perform the operations specified
below. There are five set of parameters specified below, you only need to
use one of them. Make sure your script is properly documented.
The comet function (or command, there's no difference) is an
animated plotter that displays the trace progressively, though at a fixed,
fast speed. The newly displayed points start out in one colour (the head
of the comet), and gradually change colour as the head moves on.
A good use for comet is to trace Lissajous figures. They are described by
equations of the form
where a and b
are usually small integers and δ
represents a phase shift. They are relevant in signal processing.
For a nifty animation:
 First create a 2000element vector for the independent variable t,
ranging from 0 to 2π (you need at least that many to slow down the
display to a convenient speed!).
 Assign to variables a, b and delta one
of the combinations in the table below. Use one line so you can easily
return to the statement and change the numbers.
 Assign vectors x and y according to the formula
(remember to reexecute this when you change the other variables).
 Copy the following series of commands, which clears the display and
rescales it so that the figures don't graze the axes.
clf; axis([1.2, 1.2, 1.2, 1.2]); axis('equal'); hold on;
 Make sure the figure window is visible, then issue the actual display
command:
comet(x, y)
With the uparrow key it's easy to change the parameters, or you could
put the steps in a script and ask for a, b and delta
each time it's run.
a 
b 
δ 
Figure 
1 
1 
0 
Line 
1 
1 
π/2 
Circle (same as in part B) 
2 
1 
π/4 
2lobed parabolic 
5 
4 
π/2 
5x4 lobes 
1 
3 
π/2 
Youknowwho's logo 
Wikipedia has
plenty of info about Lissajous figures, including a reference to an
oscillating echanical signal light called a Mars
Light, which traces a Lissajous curve with a=1,
b=2 to make it more noticeable.
References: Chapman (2012), Exercises 5.38 and 4.11
(starting point only).
Part C background material:
Salzman, WR (2004). Critical Constants of the van der Waals Gas, http://www.chem.arizona.edu/~salzmanr/480a/480ants/vdwcrit/vdwcrit.html.
Last accessed 20140503.
RosePetruck, C (1998). Chem 201: FirstOrder Phase Transitions,
Brown University. http://casey.brown.edu/research/crp/Edu/Documents/00_Chem201/7_phase_equil_1/7phase_equilibria1frames.htm.
Last accessed 20140503.
Van der Waals Constants for Gas courses.chem.indiana.edu/c360/documents/vanderwaalsconstants.pdf.
Last accessed 20140503.