Preliminary Mathematics Extension 1
Assessment Task Two
Weighting: 30%
Due Date: Friday 12 June 2020
TOPIC
GRADE
Functions - Pendulums
Outcomes:
ME11-1 uses algebraic and graphical concepts in the modelling and solving of problems involving
functions and their inverses
ME11-2 manipulates algebraic expressions and graphical functions to solve problems
ME11-6 uses appropriate technology to investigate, organise and interpret information to solve
problems in a range of contexts
ME11-7 communicates making comprehensive use of mathematical language, notation, diagrams
and graphs
STUDENT NUMBER
34448027
ACADEMIC HONESTY DECLARATION
This assessment task is my own original work. No part of this work has been copied from any other source or person except where due
acknowledgement is made. I have read the Student Academic Honesty Policy and understand its implications.
SIGNATURE:
emily
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
DATE: 8/6/21
1
Nature of the task
This assignment involves the application of functions to investigate simple and complex pendulums.
The assignment comprises two parts.
Part A – Investigating Simple Pendulums explores the relationship between the length of a pendulum
and its period.
Part B – Investigating Complex Pendulums explores pendulums swinging in two directions.
Submit a hard copy during the lesson, and any digital files before 9:30 am on the day the assignment
is due.
Marking criteria
You will be assessed on how well you:



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select and use appropriate mathematical processes to investigate, organise and interpret
calculations
communicate mathematical ideas using appropriate mathematical language
use appropriate technology to investigate, organise and interpret information
provide reasoning and justification to support conclusions
demonstrate understanding of graphical relationships, inverse functions and parametric
equations
__________________________________________________________
Part A: Investigating Simple Pendulums
Introduction
Traditionally pendulums are associated with devices for keeping time such as
clocks or metronomes. They involve a mass called a bob that is suspended
from a fixed point and oscillates between two extremes.
Pendulums are used in many different ways. For example, seismometers which
measure Earth movements and earthquakes use pendulums. Pendulum
models have also informed the design and development of friction pendulum
bearings which are used in buildings in earthquake danger zones. Friction
pendulum bearings allow a building to move during an earthquake in a similar
way to a pendulum but on a microscopic scale consequently minimising
damage.
The applet ‘Single Pendulum’: https://www.myphysicslab.com/pendulum/pendulum-en.html
allows experimentation with a simulation of a pendulum swinging under the effects of gravity.
Some terminology:
 the bob is the object that hangs from a fixed point and is set swinging back and forth
 the length of a pendulum is the distance from the fixed point to the bob
 the period is the time taken for the bob to make one return trip to its original position
 an oscillation is one complete cycle from one extreme position to the other and back
again.
Aim
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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To establish a relationship between the length of a pendulum and the period.
Method
Complete the following steps in order to investigate this relationship.
(a) An experiment was conducted to explore the relationship between the length of a
pendulum, 𝑙 centimetres, and the period of the pendulum swing, 𝑃 seconds.
For each pendulum length, the time taken for the first 30 oscillations was recorded, and
hence the average period for one oscillation determined. The results are summarised
in the following table:
Length of pendulum
in centimetres
𝑙
5
10
15
20
25
30
35
40
Period in seconds
𝑃
0.455
0.633
0.779
0.925
1.005
1.101
1.191
1.269
Length of pendulum
in centimetres
𝑙
45
50
55
60
65
70
75
Period in seconds
𝑃
1.354
1.421
1.491
1.551
1.657
1.677
1.721
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-
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(i) State which variable you would classify as independent and which as
dependent in this experiment and why.
The independent variable is the length of the pendulum as it is being changed each
test
The dependent variable is the period as the time, in seconds, is being measured
(ii) State any assumptions that may have been made in the process of conducting
this experiment.
The displacement of the bob remained constant for each oscillation
Acceleration remained constant as the only force acting upon the object was gravity
and all experiments were conducted in the same spot where the gravitational field was
a constant strength
Nothing was in the path of the bob’s displacement and no external forces operated on
the system except for the initial impulse of motion
Constant variables must have been kept to allow for a valid experiment which include
the device used to measure time, the device used to measure length, the bob used
and the position where the bob was released from at each test
(iii) State anything you initially notice about the relationship between the length of
the pendulum, 𝑙, and the period, .
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As the length of the pendulum increases, the period of the bob increases
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The time increases by 0.178 seconds from 5 to 10 cm, 0.096 seconds from 25 to
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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30cm, 0.085 seconds from 40 to 45 seconds, 0.06 seconds from 55 to 60 seconds and
0.044 seconds from 70 to 75cm
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This displays how it is not a direct relationship and the graph is increasing at an
increasing rate, the gradient is getting steeper as the graph progresses
(b) Use the given data, graphing technology and your knowledge of the shapes of graphs
to explore an appropriate model for the relationship between 𝑙 and 𝑃 and find a formula
to represent that relationship.
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Graphing the data on Desmos produced a curve that resembled the shape of 𝑦 = √𝑥
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The graph of 𝑦 = √𝑥 was compared to the curve displayed by the data, however, the
vertical dilation of 𝑦 = √𝑥 was far greater than that of the data
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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This was a result of length being recorded in centimetres, the function was converted
to SI units by multiplying x by 10-2 to observe if the changes resembled the curve
produced by the data
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Then the function was vertically dilated by a scale factor of 2 as the y values of the
function appeared to be half of the y values of the data
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Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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Therefore, the relationship between T and L can be represented as, 𝑇 = 2√𝐿 where all
variables are measured in SI units but for the units used in this investigation
𝑇 = 2√𝐿 × 10−2 would be the appropriate formula
(c) Galileo is attributed with determining that the period of a pendulum was related to the
length of the pendulum and was not related to the size of the mass of the bob or the
size of the arc through which the pendulum swings.
(i) Research the formula Galileo discovered that links the period for a pendulum to
the length of the pendulum (stating your source) and provide a brief explanation
of the terms used.
𝐿
𝑔
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The formula Galileo discovered is that 𝑇 = 2𝜋√
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T is the period of pendulum in seconds
L is the pendulum’s length in metres
g is acceleration due to gravity which is a constant 9.8ms-2 when on Earth
𝜋 is a constant variable with a value of approximately 3.14
This displays how T and L are the only variables, therefore pendulums are only altered
by their period and length
School For Champions 2019, ‘Equations For a Simple Pendulum, viewed 29 May
2021, https://www.school-forchampions.com/science/pendulum_equations.htm#.YLHtHy0RpQI
(ii) Does the formula you found using the data from the experiment in part (b)
correspond to Galileo’s formula for a pendulum? Justify the correlation or
differences between your results and the general form of the formula.
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https://learn.desmos.com/regressions
The formula I found initially appears to slightly resembles Galileo’s formula, however,
the constants g and 𝜋 were included in his
Despite appearing different, when both formulas are graphed on the same set of
axes, the difference is minute
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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Both formulas correspond with the data and the difference is negligible
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After breaking down Galileo’s formula (image below), the strong correlation between
these curves is explained as Galileo’s formula multiplies the formula I suggested by a
constant of approximately 1.004 (when on Earth)
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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The strong correlation between the formulae is modelled using data from the
experiment:
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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The difference between y values (period) increases as the x value (length) increases
for these formulae, however, the difference in y values is approximately 0.004
seconds and 0.016 seconds for the length of 35cm and 75cm, respectively, which
remains negligible as it is very difficult to measure these time periods
On Desmos, neither formula perfectly aligns with the data as the time was measured
to the nearest millisecond therefore accuracy and experimental error must be
considered in the results
A difference between the general formula and the results obtained is expected as it is
very difficult to achieve this level of precision when measuring time
(d) Use graphing technology and your knowledge of inverse functions to plot the graph of
Galileo’s formula for the period of a pendulum and its inverse function on the same
axes. Explain how the inverse function might also be established using algebra. How
might the inverse function be used in a practical sense?
Using graphing technology, the inverse function of Galileo’s formula would look like:
The inverse function can be obtained using algebra by swapping the variables of time and length
𝐿2 𝑔
then restricting the domain, resulting in 𝑇 = 4𝜋2 for a domain [0 ∞)
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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In a practical sense, the inverse function can be used to determine the length of the pendulum
if the period is known or the period of the pendulum if the length is known as the inverse function
swaps these variables:
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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(e) If we assume Galileo’s formula is correct, we can use our knowledge of pendulums
and their motion to explore gravity in different locations in our solar system. For
example, on the Moon, the acceleration due to gravity is one-sixth of the acceleration
due to gravity on Earth (𝑔). What effect would this have on the period of a pendulum?
9.8
6
=
49
30
ms-2
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The acceleration would be
-
This weaker force results in a smaller constant being in the denominator of the fraction
which produces a larger quotient, this causes a larger period as the period is the
product of the square root of the quotient and 2 𝜋
There is less gravity ‘pulling down’ or a weaker force would be acting on the bob
resulting in it having less velocity than on Earth therefore the period would increase
If L is 10cm, the period would be approximately 1.55 seconds on the moon and
approximately 0.63 seconds on Earth:
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Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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(f) If an explorer visiting Mars takes a pendulum and adjusts the length to ensure an
oscillation of 4 seconds, the length of the pendulum is 1.54 m. What can they assume
the acceleration due to gravity must be on Mars from this experiment?
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Acceleration due to gravity on mars is
77𝜋2
m𝑠 −2
200
or approximately 3.8ms-2
(g) The exact measurement for gravity at the equator is 9.78 ms-2 whereas the
measurement for gravity at the North Pole is 9.83 ms-2.
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(i) If a pendulum is required to have a period of 4 seconds at the North Pole, what
length should it be?
Approximately 3.96 metres
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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(ii) How would the same pendulum need to be adjusted in order to have the same
period at the equator? Could this result be generalised for any pendulum?
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The length would need to be approximately 3.98m, therefore length would need to be
extended by 0.02m at the equator
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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For period to remain constant, length needs to increase when the force of gravity
increases
𝐿
Generalisation for this period to remain constant in different places: 𝑘 = 2𝜋√𝑔 where k
is a constant period, g is the force of gravity in different places and L is the length of
the pendulum which varies with the strength of the force of gravity
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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(h) Submit your digital file for this investigation as Simplependulum_yourname.filetype.
Part B: Investigating Complex Pendulums
Introduction
The pendulums considered so far swing in one direction only. Pendulums may also swing in
more than one direction. It can then be helpful to consider the pendulum’s motion from two
perpendicular directions in order to analyse its movement.
An example of a pendulum swinging in this way can be found at ‘Sand pendulums – Lissajous
patterns part one’: https://www.youtube.com/watch?v=uPbzhxYTioM.
In analysing a pendulum’s motion from two perpendicular directions, we consider it to have two
parametric equations, in the 𝑥-direction and in the 𝑦-direction. This is explained in the video
‘Lissajous figures in the sand’: https://www.youtube.com/watch?v=77ED3qtjKZk.
Aim
To explore the motion of a pendulum moving in two directions.
Method
(a) (i) Using graphing technology, draw the graph given by the parametric equations 𝑥 =
cos 2𝑡 and 𝑦 = sin 𝑡, 0 ≤ 𝑡 ≤ 2𝜋.
https://learn.desmos.com/parametric-equations
(iii) Choose at least 8 points on the graph, including any critical points, and explain how
their coordinates are obtained from the given parametric equations.
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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cos 2t corresponds to the x coordinate of each point while sin t determines the y value
To explain how the coordinates are obtained, take the inverse sin of the value of the y
coordinate to determine t then substitute it into cos 2t to prove the value of x
This relationship is a one-to-many with two y values sharing one x value at every point
except the vertex (1,0)
Due to the fact that sin-1 is an odd function while cos-1 is even, the same x value will be
produced when inputting t and -t while a negative value will be produced when t is
negative for sin-1
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Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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The points (-1,-1), (-1,1) and (1,0) reflect the range and domain of this graph when the
inverse function is taken of their values:
The relationship has a domain of [-1,1] as cos determines the x values and cos-1(1)=0
and cos-1(-1)= 𝜋 which is the range of cos-1 graph, therefore this graph can only exist
between these points
Similarly, the range of this relationship reflects the range of the sin-1 graph as the range
𝜋
𝜋
is [-1, 1] and sin-1(-1) = - 2 and sin-1(1) = 2
(iv) Find the Cartesian equation for the graph, explaining your reasoning and
confirming that it is correct by drawing the graph using the Cartesian equation and
graphing technology.
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Cartesian equation is 𝑦 2 = −
𝑥−1
2
for a domain of [-1, 1]
2
The left hand side remains as y as taking the square root of the right hand side would
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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result in a curve that is either above or below the x-axis which would not equate to the
parametric curve
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(b) (i) Using graphing technology, draw the graph given by the parametric equations:
𝑥 = cos 𝑡 and 𝑦 = sin 2𝑡, 0 ≤ 𝑡 ≤ 2𝜋.
(iii) Find the Cartesian equation for the graph, explaining your reasoning and
confirming it is correct by drawing the graph using the Cartesian equation and
graphing technology.
-
Cartesian equation is 𝑦 2 = −4𝑥 2 (𝑥 − 1)(𝑥 + 1)
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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(c) Curves that are generated in this manner are called Lissajous curves and take the
general form: 𝑥 = 𝐴cos(𝑎𝑡 + 𝑐), 𝑦 = 𝐵sin( 𝑏𝑡 + 𝑑), 0 ≤ 𝑡 ≤ 2𝜋.
Investigate these curves using graphing technology. Comment on the effect that
changing the values of 𝐴, 𝐵, 𝑎, 𝑏, 𝑐 and 𝑑 have on the curve.
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Whenever A=B, a=c and b=d, the graph is symmetrical about the x and y axis, no
matter the values
When A = B, a=b and c=d, the graph is a circle with a radius of A/B
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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B stretches the graph vertically (increases range) while A stretches horizontally
(increases domain) when all other values are 1, the graph produced is an oval
a and b affect put a ‘twist’ in the graph with a making the twist horizontal and b making
it vertical
every multiple of 3, the curve becomes closer to the shape of a wave as the number
increases and the sum of crests and troughs is n-1 where n is value of a or b
Every 3 integers, the shape is similar, if n equals a multiple of 3, n+1 gives a double
helix shape while n-1 gives a squashed double helix shapes with ‘twists’ occurring
further from the x-axis
The same pattern and shapes occur for a, except vertically and the other variable (out
of a or b) must be equal to 1
With a and b, the shape is defined by the ratio of these values and not the magnitude
of the values, a:b is in a 7:1 ratio and the same shape is produced:
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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c and d rotate the shape on a diagonal axis with c rotating the shape to the left until it
has a value of 5 and d rotating to the right until a value of 5
c and d follow a pattern for their shape when all other variables are 1: it rotates to the
left (c) or right (d) until 5 where it switches to the other side, then switches again at 8
and 11 then becomes a straight line at 12, switches again at 14, 17 and 20 then
becomes a circle at 23, this pattern repeats for every 22 (1 doesn’t count) values but in
the opposite direction, therefore the first ‘cycle’ is completed at 45
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This pattern only occurs if a=b where a and b can be any real number
When the other variables are not 1, c and d still control the ‘switches’ but they equate
to the graph being flipped upside down or left to right
When a=c (and b and d = 1) and the value is even, the graph produced is a single line
in a ‘wave shape’ with the sum of crests and trough equating to one less than this
value (n-1) whereas when the value is odd, an even relationship is produced with the
value of a/c indicating the number of crests on one side of the y-axis
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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When b=d, the value equates to the number of crests/troughs on one side of the x-axis
despite the value being even or odd when a and c = 1, the graph is reflected about the
x-axis
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https://support.desmos.com/hc/en-us/articles/202528919-Sliders
Justify your conclusions using your knowledge of parametric and Cartesian equations,
providing evidence of your investigation.
(d) Submit your digital file for this investigation as Complexpendulum_yourname.filetype.
End of assignment
Checklist
Part A – Problem Solving, Reasoning and Justification
(a)
(i), (ii) and (iii) are answered correctly with at least one assumption given for part (iii).
(b)
(i)
The graph is correctly drawn, and a regression model is applied with a clear justification
given that it fits the shape of the curve.
(ii) An appropriate formula is quoted that could represent the relationship between 𝑙 and 𝑃.
(c)
(i)
The correct formula is stated, and the terms used are explained, with the source stated.
(ii) The two formulae are compared, and similarities and differences are justified correctly.
Part A – Understanding, Fluency and Communication
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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Part A – Problem Solving, Reasoning and Justification
(d)
There is a correct graph of both the inverse and original function and the inverse function is
correctly stated.
(e)
The correct answer is shown, and working is clearly communicated.
(f)
The correct answer is shown, and working is clearly communicated.
(g)
(i)
The correct answer is shown, and working is clearly communicated.
(ii)
The correct answer is shown, and working is clearly communicated.
Part B – Understanding, Fluency and Communication
(a)
(b)
(i)
The correct graph is shown.
(ii)
There is a clear explanation of how the coordinates of the graph are obtained from the
parametric equations.
(iii)
The correct Cartesian equation is shown, and working is clearly communicated.
(i)
The correct Cartesian equation is shown, and working is clearly communicated.
(ii)
The correct graph is shown, with evidence it has been drawn from the Cartesian
equation.
Part B – Problem Solving, Reasoning and Justification
(c)
There is extensive evidence of investigating a large variety of values for 𝐴, 𝐵, 𝑎, 𝑏, 𝑐 and 𝑑
using graphing technology. Complex generalisations have been made and justified and
include noting any assumptions that have been made, including the noting of special cases.
Mathematics Extension 1 Year 11 Assessment Task 2: Functions
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