APS106 – Lab #2

APS106 – Lab #2
Preamble
This week you will practice writing functions. In the first part you will write two functions
– magnitude(x,y) and phase(x,y)- that convert a 2-dimensional vector into a
magnitude and phase angle, respectively. In the second part, you will write a function to
approximate the vertical position of a charged particle in an electrostatic precipitator1
.
By completing this lab, you will develop, utilize, and practice the following skills:
• setup, debug, and run a program
• interpret real world parameters as variables
• utilize math functions
• utilize built-in functions
• develop effective test cases
• read Python documentation
Use appropriate variable names and place comments throughout your program.
Deliverables
For this lab, you must submit all functions within a single file named ‘lab2.py’ to MarkUs
by the posted deadline.
Five test cases are provided on MarkUs to help you prepare your solution. Passing all
these test cases does not guarantee your code is correct. You will need to develop
your own test cases to verify your solution works correctly (see part 1.1 for more
details). Your programs will be graded using ten secret test cases. These test cases will
be released after the assignment deadline.
IMPORTANT:
• Do not change the file name or function names
• Do not use input() inside your program
1 Don’t worry if you haven’t heard of an electrostatic precipitator before. In these labs we will use example of
engineering technologies to motivate problems, but we will always provide you all information and equations you
need to write the code.
Part 1 - Problem and Motivation
For many problems in engineering and sciences2
, analysis can be simplified by
converting between different coordinate systems. A common transformation is to
convert 2-dimensional cartesian coordinates (x and y components) to polar coordinates
(magnitude and phase angle). Figure 1 below illustrates how a vector can be defined
using either coordinate system.
Figure 1. Example of vector defined by both cartesian and polar coordinate systems where r
represents the magnitude and φ represents the phase angle.
We can calculate the magnitude, r, and phase angle, φ, with the following equations:
𝑟 = √𝑥
2 + 𝑦
2 (1)
𝜑 = tan−1
(
𝑦
𝑥
) (2)
where x and y represent the two components of a vector in cartesian coordinates.
Part 1.1 Calculate Vector Magnitude
Download the file lab2.py from quercus.
2 Examples include analysis of AC circuits, mechatronic control systems, ionic flow in fluids, and quantum
mechanics
For the first part of this lab, you will complete the function magnitude(x,y) which
accepts x and y as floating point input parameters and returns the magnitude as
calculated using equation 1.
Sample Test Cases
Note: each week we will provide some example test cases that you can use to design
and evaluate your programs. Passing these sample test cases does not guarantee
your code is correct. You will need to consider your own additional test cases to
ensure your program works correctly. An important part of programming is designing
test cases to ensure your program functions correctly with different input combinations.
The example test cases will be given with brief descriptions to give you insight into the
scenarios being tested and how you might design further test cases.
Inputs Expected Output (to six
decimal places)
Description
x y magnitude
0 0 0.0 Zero vector
10.2 63.2 64.017810 Quadrant 1: x and
y both positive
0 45.0 45.0 Border between
quadrant 1 and 2
-11.3 -3.9 11.954079 Quadrant 3: x and
y both negative
Hint: Notice that not all quadrants are tested in the above examples.
Part 1.2 – Calculate Vector Phase Angle
For the second part of this lab, you will complete the function phase which accepts two
inputs, x and y, as floating-point input parameters and returns the phase angle in
radians as calculated using equation 2.
To do this, you will need to use an inverse tangent function from Python’s math module.
The available trigonometric functions can be found here3
. Note that there are two
different inverse tangent or arctangent functions. Read the descriptions and determine
which you believe should be used in your function. If you are not sure, your TA will be
happy to discuss with you during your PRA session.

3 https://docs.python.org/3.7/library/math.html#trigonometric-functions
Sample Test Cases
Inputs Expected Output (to six
decimal places)
Description
x y phase
0 0 0.0 Zero vector
10.2 63.2 1.410784 Quadrant 1: x and
y both positive
0 45.0 1.570796 Border between
quadrant 1 and 2
-11.3 -3.9 -2.809260 Quadrant 3: x and
y both negative
Part 2 – Problem and Motivation
In this part of the lab, you will write a function to calculate the vertical position of a
charged particle in a simple electrostatic precipitator.
An electrostatic precipitator is a device that is typically used to capture and remove
harmful particles from the exhaust of industrial and/or chemical processes. The device
works in two stages:
1. Particles in the exhaust are charged (either positively or negatively)
2. The particles pass through an air channel where high voltage electrodes attract
the charged particles (see figure 2). The electrodes create an electric field that
exerts a coulomb force on charged particles.
If we assume this force is constant, the vertical height of the particle can be
approximated using the following equation:
ℎ(𝑡) = −
1
20000
𝑞𝐸
𝑚
𝑡
2 + ℎ𝑖𝑛𝑖𝑡
Variables:
• q is the particle charge in nanocoulombs (can be positive or negative)
• E is the electric field strength between the electrodes in kilonewtons per coulomb
• m is the mass of the particle in nanograms
• t is the time since the particle entered the air channel in microseconds
• hinit is the initial height of the particle in centimetres
• h(t) is the height of the particle in centimetres at time t
Figure 2. Example of charged particle movement between high voltage electrodes. Particles
enter through the opening on the left and are attracted to the oppositely charged electrode. In
this example, a positively charged particle enters in the middle of the two electrodes and is
accelerated towards the negatively charged electrode by the coulomb force, Fc. When the
particle reaches the electrode, it sticks to the surface and stop moving.
Your task is to complete the function particle_height which will calculate and
return the vertical height of a particle within the electrostatic precipitator. The
function accepts five input parameters:
1. q – The charge of the particle in nanocoulombs
2. E – The electric field strength in kilonewtons per coulomb
3. m – The mass of the particle in nanograms
4. L – The distance between the two electrodes in centimetres
5. t – The time since the particle entered the air channel in microseconds
For simplicity, we will assume the following:
1. The bottom electrode is at a height of 0 cm and the top electrode is at a height of
L cm, L will be greater than zero
2. The top electrode will be positively charged and the bottom will be negatively
charged
3. Particles enter the channel halfway between the two plates (i.e., hinit = L/2 cm)
4. Particles have zero vertical velocity when entering the air channel
5. Particles immediately stop moving when contacting one of the electrodes
6. All other forces (gravity, drag, etc.) are negligible

Sample Test Cases
Inputs
Expected
Output (to six
decimal
places)
Description
q (nC) E (kN/C) m (ng) t (μs) L (cm) Height (cm)
0 150 9.2 3.6 5 2.5 Zero force
scenario
2.3 150 9.2 26.8 5 1.1533
Positive
Charge; has
not yet
contacted
electrode
-2.3 160 9.2 36.8 5 5.0
Negative
charge;
attached to top
electrode
Hint: This problem is designed to be completed without the use of conditional
statements (i.e., if statements) as those will be introduced later in the course. For this
lab, you may find the built-in Python functions max and min helpful. Use:
help(min)
help(max)
in the python interpreter to find out more about these functions.