Post on 18-Jan-2017
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Extended Essay
How does the rotational frequency of the rotor blade affect the mass that a
helicopter can lift?”
Candidate Name: Nikita Miliakov
Candidate Number: 004437-0075
Centre Number: 004437
Subject: Physics
Supervisor’s Name: Shawn Pernasilici
Main essay word count: 3524
Abstract word count: 258
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Abstract:
The essay examines the question of “How does the rotational frequency of the rotor blade
affect the mass that a helicopter can lift?”. The hypothesis for the essay was : “The rotational
frequency of the rotor and the mass the helicopter can carry will be directly proportional…”
To answer the research question, an experimental approach was used. The experiment was
conducted using a radio controlled helicopter and the frequency of rotation of the rotor blade was
measured using a stroboscope and electronic scales. First attempt of the experiment failed due to
inability to identify correct procedure, however the mistakes allowed room for improvement and
modifications in the experiment, with an addition of the measuring cylinder on an electronic
scales and the attempt #2 was conducted. The graphical analysis helped to identify that the
relationship between mass and frequency appears to be a quadratic.
Further on, the relationship between the mass and the frequency was derived theoretically, which
lead to the following relationship:
𝒇𝟐 =𝒎𝒈
𝟖𝝅𝟑𝝆𝝀𝟐𝑹𝟒
The results of my research shows that theoretical and experimental results are in a good
argument with each other within the limitations of the experiment. Experimental approach have
shown the theory behind the experiment to be identified correctly. My calculated values of the
rotor inflow ratio (𝜆) of the real helicopter compared to the radio-controlled helicopter are very
similar, meaning that the same physical principles were used in designing both of them. The
research conducted might be of useful value to further investigation of the effect of the variable
angle of attack of rotor blades.
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Table of Content
Extended Essay ....................................................................................................................................... 1
Abstract: ................................................................................................................................................. 2
Introduction: ........................................................................................................................................... 4
Background Information: .................................................................................................................... 4
Research Question: ............................................................................................................................. 4
Hypothesis: ......................................................................................................................................... 4
Information about the Helicopter: ............................................................................................................ 4
Data about helicopter: ..................................................................................................................... 5
Attempt #1 .............................................................................................................................................. 6
Materials and Apparatus .................................................................................................................. 6
Equipment Set-up: ........................................................................................................................... 7
Procedure: ....................................................................................................................................... 7
Results: ........................................................................................................................................... 7
Graphical analysis of the data: ......................................................................................................... 8
Attempt #2 .............................................................................................................................................. 9
Equipment Set-up: ........................................................................................................................... 9
Procedure: ....................................................................................................................................... 9
Results: ......................................................................................................................................... 10
Analysis of the Data: ............................................................................................................................. 10
Data Processing: ................................................................................................................................... 11
Formula derivation for theoretical proof : .......................................................................................... 12
Relating to the real life: ......................................................................................................................... 15
Conclusion:........................................................................................................................................... 18
Evaluation ............................................................................................................................................ 18
Limitations ........................................................................................................................................ 18
Bibliography: .................................................................................................................................... 20
Appendices: .......................................................................................................................................... 21
Raw Data: ......................................................................................................................................... 21
Appendix Table 1 .......................................................................................................................... 21
Appendix table 2: .......................................................................................................................... 22
Justification of the appendix tables: ............................................................................................... 23
Uncertainties proof: ....................................................................................................................... 23
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Introduction:
Background Information:
Radio controlled helicopters are very interesting toys as they follow many physical laws. Many
toy makers are trying to create their designs of remote control helicopters, and because of that,
there are plenty to choose from. However, when it comes to the flying capabilities, some
helicopters are better than others. The helicopters vary in their masses, sizes and rotational
frequency- the frequency with which the main rotor, which provides thrust, rotates. There are
theories that the rotational frequency of the rotor blades is directly proportional to the mass it can
carry1. This is justified by Newtons 3rd law of motion, as well as the momentum law, as both talk
about the relationship of two forces affecting each other. However, it doesn’t fully justify the
theory, as even though there is a presence of Newtons 3rd law, the helicopter manages to lift off,
hence there must be something else to it.
Research Question:
The objective of this essay is to investigate the following question:
“How does the rotational frequency of the rotor blades affect the mass that a helicopter can
lift?”
Hypothesis:
The rotational frequency of the rotor blades and the mass the helicopter can lift will be directly
proportional, and it will be easy to see it through the graphical analysis program2 by plotting a
graph of mass vs. frequency. In addition, the rotor inflow ratio of the real helicopter will be equal
or very similar to the toy helicopter, which will be examined theoretically, as both should follow
the same laws.
Information about the Helicopter:
The helicopter that was used in the experiment – is a typical helicopter that is possible to find in
the toy shops3. The helicopter consists of two rotors- first one is the main rotor, that is providing
lift to the helicopter, and second rotor is for stabilizing the helicopter while it is on air, providing
anti-torque4.
1 See bibliography 3 2 Graphical Analysis Program: Logger Pro 3.5.5 3 See bibliography 4 4 See bibliography 10
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Data about helicopter:
Length: 17.80 ±0.05 cm
Width: 5.50±0.05 cm
Height:7.00±0.05 cm
Main Rotor Diameter:13.00±0.05 cm
Rear rotor diameter:3.00±0.05 cm
Note: the uncertainty of the measurement was taken as half of the smallest increment of the
measuring equipment.
The Experiment
To answer the question, experimental approach was chosen as it is the best way to identify the
relationship between rotational frequency and the mass. Image 4 and 5 display the helicopters,
similar to the one that is being investigated. Pictures show the functioning of the helicopters’
main rotor. Image 5 is showing that the air is being pushed down by the helicopters’ rotor blades,
creating lift to raise the helicopter. This shows that helicopters’ design is based on Newton’s
third law: “For every action force, there is equal and opposite reaction force”. Image 4 shows
the air flow in the rotor of the helicopter.
Image 1: Tested helicopter and remote control used
Image 2: Helicopter angled view
Image 3:
Helicopter
Side View
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Attempt #1
Materials and Apparatus 5
For the first attempt, several materials were required. Stroboscope was used to determine
the frequency of rotation of blades. For stroboscope to function properly, the experiment was
conducted in the dark room. A radio-controlled helicopter described above was used as a testable
apparatus for measuring the rotational frequency and mass. To measure the mass of the
helicopter, electronic scales were used. In addition, the scales were used to measure the change
in weight as the frequency of rotation of blades of the helicopter varies. To hold the helicopter in
one place duct tape was used. In addition, to make it easier, a padding for stroboscope, in terms
of couple of books, was used in order to equate the level between the helicopter and the
5 See bibliography 9
Generated Lift
Image 43: Air flows in the helicopter's rotor
Image 5: drawing, displaying the wing directions (arrows downwards) that creates lift (arrows upwards)
Air is being pushed
down in the middle
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stroboscope. Standardized equipment such as table and electric sockets were used in order to
conduct the experiment.
Equipment Set-up:
All the equipment was collected
and set up as of diagram 1 in order for
the experiment to perform well. The
electronic scales and stroboscope were
plugged in into the electric sockets. The
helicopter was put down on the electric scales and taped down with duct tape.
Procedure:
After the experiment set up, the helicopter and the remote control were turned on. The
helicopter was first launched at the lowest possible power output that count be varied through the
remote control, and this caused the rotational frequency to be lowest as well. At the same time,
the stroboscope frequency was adjusted in order for the blades to look still in the air, which will
mean that the frequency with which the stroboscope flashing is equal to the rotational frequency
of the helicopter. The data for mass was recorded five times consecutively in order to find the
average later and have near ideal result. After that the power output, hence the frequency of
rotation, was increased. The mass data was recorded five times consecutively again. These steps
were repeated four more times in order to receive a variety of data for graphical analysis.
Results:
During the collection of the results, the data did not show what was expected, as the helicopter
was pushing on the scales, rather than pulling them up. In addition, the uncertainty shown on
stroboscope was equal to 0.01Hz, however due to the fact that the data was recorded with a
human eye, the uncertainty was assumed to be 1Hz.
Table 1: Raw data from attempt #1 of the experiment6
Trials Frequency (Hz) ±1 𝐻𝑧 Average Mass lifted (kg)
1 90 0.0008±0.0003
2 94 0.0011±0.0001
3 100 0.0011±0.0001
4 109 0.0013±0.0002
5 111 0.0014±0.0003
6 112 0.0015±0.0002
6 For data collected, see appendix table 1. For the justification of uncertainty calculation see justification of the table
Diagram 1: Example of Equipment set-up
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Graphical analysis of the data:
The data collected show some relationship between mass and frequency, however the closest
curve that was fitted in was a quadratic, hence the results must be further analyzed. However, the
uncertainties are too large, as well as the curve fitted is dipping towards the end, which suggests
that the data was not collected properly, and together with that fact that the helicopters’ mass was
increasing but not decreasing suggests that the experiment failed. Looking at the theory behind
helicopter flights, Newtons Third Law takes place as one of the major obstacles in the
experiment, because as soon as the helicopter reaches the required rotational frequency, the air
flow is being pushed down on the scales, neutralizing and even overcoming the upward force,
due to the Newton’s Third law of motion. In order to solve the problem, the solution was found-
to put the helicopter much higher, as the air flow will dissipate and not push on the scales.
However, it was necessary to find something to put it on, and it could not be too thin, which
would have been the best idea, as it would have not been able to hold the helicopter properly.
With some research done from school resources, I decided to use a measuring cylinder as it was
the most relevant equipment available. Therefore, the equipment was modified, adding a
measuring cylinder:
Measuring cylinder- size :
Length: 30.00±0.05𝑐𝑚
Width of the platform: 8.20±0.05𝑐𝑚
Graph 1: Mass vs. Rotational Frequency (Attempt #1)
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Attempt #2
During attempt #2, the measuring cylinder was added. In addition, the amount of trials has
increased for more relevant data up to 12 different frequencies. Other than that, the equipment
set up stayed exactly the same.
Equipment Set-up:
At first, all the materials and apparatus
were collected. Then electronic scales and
stroboscope were plugged in into the power
sockets. After, the measuring cylinder, the
addition for this method was fixed on the top of
the scales upside down, as the bottom of the
cylinder is flat. The cylinder then was taped down using duct tape. Then, the helicopter was
placed on top of the cylinder and fixed with duct tape in order for it not to fly off during the
experiment and to lift weight. In the end, electric scales were zeroed so that any change in the
mass of the helicopter could be clearly seen, as the helicopter would in theory lift the whole
system. Diagram 2 clearly showing an updated equipment set-up
Procedure:
The procedure of attempt #2 is very similar to the previous attempt. The helicopter and
the remote control were turned on, and the helicopter was launched on the lowest power output,
hence lowest frequency. At the same time, the stroboscope frequency was adjusted in order for
the blades to look still in the air, which will mean that the frequency with which the stroboscope
flashing is equal to the rotational frequency of the helicopter. Then five consecutive records of
mass were taken at that certain frequency. After, the power output was increased, hence the
frequency of rotation increased as well. The data for mass was recorded five times consecutively,
and the data for frequency was recorded as well. The procedure was repeated for ten more
frequencies of the rotation of the helicopter blades. Similarly to the first attempt, the uncertainty
in frequency was assumed as 1Hz as the data for frequency was collected with a human eye.
In order to receive the graphical analysis with correct shape of the curve, absolute value of
average value of mass were used.
Diagram 2: Example of updated equipment
setup for attempt #2.
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Results:
Table 2: Raw data from attempt #2 of the experiment7
Trials Frequency Hz ±1𝐻𝑧 Average Mass (kg)
1 61 -0.0031±0.0001
2 64 -0.0035±0.0001
3 68 -0.0038±0.0001
4 88 -0.0066±0.0001
5 93 -0.0071±0.0001
6 97 -0.0078±0.0001
7 99 -0.0090±0.0001
8 100 -0.0096±0.0002
9 104 -0.0096±0.0001
10 104 -0.0096±0.0001
11 109 -0.0101±0.0001
12 112 -0.0107±0.0001
Analysis of the Data:
7 For the way of calculating uncertainty for mass, see Appendix table 2 and justification to it.
Graph 2: Mass Vs. Rotational Frequency graph (attempt #2)
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Data Processing:
Similarly to the attempt #1, the graphed data have shown relationship of a quadratic formula
between mass and rotational frequency, however this time, the incline of the slope is positive, the
uncertainties are quite small, and the procedure was proving the theory correct. All of these
suggests that the data of the attempt #2 is correct and the relationship between the mass the
helicopter can lift and the rotational frequency of its blades had been found. To further
investigate the relationship, another graph was formed of mass versus frequency squared. For
these purposes, all frequency values must be squared.
8 Values in the table are squared values from table 2. For uncertainties see “proof of uncertainties” in appendix 9 See table 3
Table 4: The values used for graphical analysis
Rotational Frequency Squared( Hz2)8 Absolute Average mass values9
3700±100 0,0031±0,0001
4100±100 0,0035±0,0001
4700±100 0,0038±0,0001
7800±200 0,0066±0,0001
8600±200 0,0071±0,0001
9500±200 0,0078±0, 0001
9800±300 0,0090±0, 0001
10100±200 0,0096±0, 0002
10800±300 0,0096±0,0001
10900±200 0,0096±0,0001
11800±300 0,0101±0, 0001
12500±200 0,0107±0, 0001
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The graph above10 proves that the relationship between mass and frequency squared is true, as it
has given a best fit line as a straight line. The uncertainties of frequency are so tiny that it is
difficult to see them, and uncertainties of mass are also small. Some values are off from best fit
line which can be explained as a human error, due to the fact that it is almost impossible to
record with high accuracy and precision exact values of mass and frequency with a human eye.
To further justify the relationship between the mass and frequency squared, it is possible to
derive a formula for the relationship.
Formula derivation for theoretical proof 11:
Assuming that a smooth air stream, pointing in the direction of the rotor blade is being passed
through the rotor blades area, we can find out the formula to find the relationship between
frequency squared and mass. The mass flow through the disk is given by the equation of:
𝑑𝑚
𝑑𝑡= 𝜌𝐴𝑣
10 Graph 3. 11 See bibliography 1,3,5 and 6.
Equation 1
Graph 3: Mass vs. Rotational Frequency Squared graph (attempt #2)
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Where 𝑑𝑚
𝑑𝑡 is the rate of mass passing through the rotor disc area, 𝜌 is the density of air, A is the
area swept out by the rotor blade, and v is the speed of the airflow downwards. The force
exerted by the rotor is equal to the rate of change of momentum:
𝐹 =𝑑𝑝
𝑑𝑡
The air above the helicopter is at rest, however, the air beneath it is not, hence the speed of the
air can be assigned as 𝑧. Assuming that the air stream cannot be compressed and not viscid, then
the force exerted by the rotor is:
𝐹 = (𝑑𝑚
𝑑𝑡) 𝑧
The conservation of energy states that the rate of work done by the rotor must be equal to kinetic
energy, hence:
𝐹 =1
2(𝑑𝑚
𝑑𝑡)𝑧2
Dividing equation 4 by equation 3 leads to the result that 𝑧 = 2𝑣. Combining all information
together, the force that the helicopter exerts is:
𝐹 = 2 (𝑑𝑚
𝑑𝑡) 𝑣 = 2𝜌𝐴𝑣2
The speed of the tip of the rotor blade is 𝜛𝑅, where 𝜛 is angular frequency of the rotor blade
and R is the radius of the blade. General relationship between the speed of the air v through the
plane of the rotor blade and the speed of the tip of the rotor 𝜛𝑅 is complicated. The rotor of the
helicopter has a blade twist, called an angel of attack, which is twisted to keep it as constant as
possible in order to generate enough lift. If the angle is too small- there will be wasted energy.
The angle of attack 𝛼 is measured away from the vector sum of v and 𝜛𝑅 (image 6). The speed
Near Rotor Hub
Near Rotor Tip
Image 6: Angle of attack explanation
Equation 2
Equation 3
Equation 4
Equation 5
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of the air drawn downwards past the rotor blade is relatively consistent; however, the tangent
velocity of the rotor blade increases as the radius increases. Near the hub, the aerofoil has to be
at the large angle to the horizontal plane defined by the trail of the rotor. The aerofoil at the tip of
the rotor has to be at a smaller angle to the horizontal plane because the directions of the vectors
sum of v and 𝜛𝑅 has changed. By limiting the investigation to the hovering of the helicopter, we
can define the parameter 𝜆- rotor inflow ration. Rotor inflow ration- is the ratio between the
speed of the air through the rotor plane to the speed of the rotor tip:
𝜆 =𝑣
𝜛𝑅
Using equation 6 to replace 𝑣 in the equation 5, the new equation appears:
𝐹 = 2𝜌𝐴𝜆2𝜛2𝑅2
Due to the fact that 𝜛 is angular frequency that is found by 𝜛 = 2𝜋𝑓, and 𝐴 = 𝜋𝑅2, it is
possible to relate them the force required to make the helicopter hover to the frequency of the
rotor blade:
𝑚𝑔 = 8𝜋3𝜌𝜆2𝑅4𝑓2
The result is the equation of the form:
𝒇𝟐 =𝒎𝑔
8𝜋3𝜌𝜆2𝑅4
Where 𝑓2 is frequency squared, m- is mass of the helicopter, g- is an acceleration due to a free
fall, 𝜌 is air density, 𝜆2 is a property called the rotor inflow ratio squared, and 𝑅4is the radius of
the rotor blade to the power of four.
Graph 3 shows the ratio between frequency squared and mass the helicopter can carry. To find
out the value of the rotor inflow ratio, the value of mass must be modified to correspond with the
formula. For the need of it, acceleration due to a free fall was used as 9.81ms-1, density of air at
the room temperature of 25 degrees Celsius and atmospheric pressure 102 or more kilopascals,
the density of air, 𝜌 is equal to approximately 1.2041kgm-3.The radius of the rotor blade is half
Equation 6
Equation 7
Equation 9
Equation 8
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of its diameter, which was mentioned before, hence radius is equal to 13.00 𝑐𝑚
2= 7.5 𝑐𝑚.
Therefore, it is possible to rearrange the formula to obtain the value of 𝜆:
𝜆 = √𝑚𝑔
8𝜋3𝜌𝑅4𝑓2
Using the values listed above the formula and plugging them into the formula it is possible to
calculate that the rotor inflow ration is equals to 0.030. Since rotor inflow ratio is the ratio of
inflow air speed to rotor tip speed, it is possible to calculate the speed with which the air passes
through the plane of the rotor at certain frequency. At the frequency of 68.23Hz:
𝜛𝑅 = 2𝜋𝑓𝑅 = 2𝜋 ∗ 68.23 ∗ 0.075𝑚 ≈ 32.15 𝑚𝑠−1
By multiplying this value by the 𝜆, it is possible to get the air speed passing through the rotor
plane:
32.15 ∗ 0.030 = 0.96𝑚𝑠−1
This is a very reasonable result, including the fact of neglecting other forces.
Relating to the real life:
The theory behind radio controlled helicopter is that by experimenting with it, it is possible to
compare the data collected to the data from real life helicopters
In order to do so, it is necessary to compare data
of real helicopter to the data obtained. As it is the
maximum lifting capacity that is being
investigated, it will require comparing the airflow
speed passing through the rotor plane at the
maximum rotational frequency of the rotor
blades. In order to compare the remote control
helicopter to real helicopters, it is necessary to
keep in mind that the large the helicopter, the less rotational frequency has to be, as the angular
frequency formula states that ϖR=2πfR, 10 where R is the radius of the rotor and f is frequency
of rotation of the rotor, hence, it is necessary to know both the rotational frequency of the real
Equation 10
Image 7: Eurocopter AS365 N3+
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helicopter and the radius of its rotor. At first, using formula 10, it is possible to find rotor inflow
ratio. For this cause, a well-known helicopter was being researched, Eurocopter AS365 N3+
(Image 7). All information about the helicopter was obtained from its official specification12.. It
is known that the maximum RPM (Revolution per Minute) of the main rotor is 365. Knowing
this, it is possible to identify that rotational frequency is equal to 6.1Hz. Maximum loaded mass
of the helicopter is 4300kg. For more equal comparison between real helicopter and toy
helicopter, the values of acceleration due to a free fall and density of air were used the same as
before: g=9.81ms-1 and 𝜌 = 1.2041𝑘𝑔𝑚−3. The diameter of the rotor, as declared on the
website, is 11.94m, hence the radius of the rotor is 5.97m.
𝜆 = √4300 ∗ 9.81
8𝜋3 ∗ 1.2041 ∗ 5.974 ∗ 6.12= 0.056
Using this, we can calculate the speed with which the air passes through the plane of the rotor at
certain frequency. At the frequency of 6.1Hz:
𝜛𝑅 = 2𝜋𝑓𝑅 = 2𝜋 ∗ 6.1 ∗ 5.97 ≈ 36.41 𝑚𝑠−1
By multiplying this value by the 𝜆, it is possible to get the air speed passing through the rotor
plane:
36.41 ∗ 0.056 = 2.04𝑚𝑠−1
There is a large difference in values. This uncertainty is difficult to explain, but looking at the
pictures of both helicopters, it is clear that toy helicopter has got two propellers, compared to the
Eurocopters’ four. What this may mean is that there is less lift created by the toy helicopter, as
the total wing area is less, meaning less air is pushing down. Having in consideration Newton’s
Third law, it means that the air speed will be less too, even though the frequency of rotation
partially decreased the problem, however, the radius of the rotor is much smaller, meaning that
the frequency can only partially decrease the difference, and hence the. This leads to the
conclusion that if the toy helicopter would have two more propellers, the value of the air speed
passing through the rotor plane could be very similar. To test so, it is possible to double the value
of helicopters’ rotor inflow ratio:
2𝜆 = 0.059
12 See bibliography 2 and 8.
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Furthermore, with this value we can calculate the air speed passing through the rotor plane, using
existing value of 𝜛𝑅 from before:
𝜛𝑅 ∗ 𝜆 = 32.15 ∗ 0.059 = 1.91𝑚𝑠−1
The value obtained is very similar to the value of the real helicopter. Knowing both values, it is
possible to find percentage discrepancy of the updated value of sped passing through the rotor
plane of the toy helicopter to the real helicopter:
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑑𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦 =𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
𝐴𝑐𝑡𝑢𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 𝑥100 =
2.04−1.91
2.04𝑥100 = 6.4%
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Conclusion: In the end, the experimental and theoretical proof suggests that the relationship between the mass
the helicopter can carry and the rotational frequency of the blade does exist. However, it was
proven that it is not a linear relationship but a quadratic. This in its turn justifies that the
rotational frequency squared is directly proportional to the mass the helicopter can lift. In
addition, the comparison between the real helicopter and toy helicopter had shown very similar
values of the air speed passing through the rotor plane. The percentage discrepancy of the
calculated airflow through the rotor plane is 6.4%, meaning that the hypothesis set in the
beginning of the experiment was partially correct. To conclude, this means that the physics
behind the toy helicopters and real helicopters are exactly the same, or very similar.
Evaluation
The experiment conducted was rather successful. It required two attempts to complete it. First
attempt failed as there was little understanding of the problem, and the problem that arose was
not expected. After modifying the experiments’ equipment list and procedure, the data was
collected properly and it followed the hypothesis created beforehand. The most challenging part
of the essay was a formula derivation, as it required external research and excellent
understanding of the topic, and sometimes thinking outside the box. The data collected is
sufficient, however, with the experimental and theoretical results collected and analyzed, the
question of whether or not the angle of attack really affecting the rotor inflow ratio appears,
which can be analyzed with the data collected in this essay.
Limitations
The essay has got some limitations that may have affected the results and the conclusion of the
essay. First, the way the data was recorded was not the most ideal. When the data was collected,
there was a great change of human error, as values of mass kept changing, and for the humans it
is impossible to keep up with a change, and hence five trials of mass were taken to find an
average which would partially decrease human error. Even though the limitation was addressed,
there were still some human error, as it can be seen from graph 2 and 3, as plotted points are not
exactly on the line of best fit. Some uncertainty could also be caused by the fact that the
measuring cylinder was used in trial two, which could cause some air still pushing down on the
scales, as it is not fully guaranteed that all air flow will dissipate before scales. Second, one of
the greatest limitations of the research was the discharging battery of the helicopter. The battery
in the helicopter was of a very small volume, because of that it caused to abort some values
being recorded, and including the fact that there is a small gap between the recording of mass
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values and frequency values, it could be large enough to cause small uncertainty in values of
mass of frequency. Another large limitation of the research is the fact that many external forces
were neglected and only relationship between frequency of the rotor blade and the mass was
investigated, which mean that the helicopter was investigated in “ideal conditions”, meaning that
it is only possible to achieve them in the lab, and information collected cannot be used outside of
it. Even though there are many limitations for the project, the data and the results collected can
further be used for external use.
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<http://seniorphysics.com/physics/helicopterEEI.pdf>.
8) "Operational Evaluation Board Report." EUROPEAN AVIATION SAFETY AGENCY. EUROPEAN
AVIATION SAFETY AGENCY, n.d. Web. 15 July 2013.
<http://www.easa.europa.eu/certification/experts/docs/oeb-reports/eurocopter/EASA-OEB-Final-
Report-Eurocopter_AS365_EC_155%20B--B1-02-08022012.pdf>.
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<http://upload.wikimedia.org/wikipedia/commons/c/cb/Vortex_ring_helicopter.jpg>.
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n.d. Web. 21 May 2013.
<http://science.howstuffworks.com/transport/flight/modern/helicopter.htm>.
21
Appendices:
Raw Data:
Appendix Table 1
Frequency
±𝟎. 𝟎𝟏 𝑯𝒛
Trial # Mass ±𝟎. 𝟎𝟏 (𝒈) Mean Mass
±𝟎. 𝟎𝟏 (𝒈)
Uncertainty of
Mass
90.20 Trial 1 17.82 17.86 0.32
Trial 2 17.65
Trial 3 17.93
Trial 4 17.91
Trial 5 17.97
93.56 Trial 1 18.10 18.11 0.08
Trial 2 18.11
Trial 3 18.15
Trial 4 18.07
Trial 5 18.13
100.23 Trial 1 18.13 18.18 0.09
Trial 2 18.20
Trial 3 18.22
Trial 4 18.18
Trial 5 18.15
108.70 Trial 1 18.36 18.31 0.18
Trial 2 18.18
Trial 3 18.26
Trial 4 18.31
Trial 5 18.22
110.51 Trial 1 18.42 18.40 0.28
Trial 2 18.47
Trial 3 18.19
Trial 4 18.39
Trial 5 18.42
111.88 Trial 1 18.41 18.51 0.18
Trial 2 18.46
Trial 3 18.53
Trial 4 18.57
Trial 5 18.59
22
Appendix table 2:
Frequency
±𝟎. 𝟎𝟏 𝑯𝒛
Trial # Mass ±𝟎. 𝟎𝟏 (𝒈) Mean Mass
±𝟎. 𝟎𝟏 (𝒈)
Uncertainty of
Mass
61.28 Trial 1 -3,10 -3.12 0.06
Trial 2 -3,18
Trial 3 -3,14
Trial 4 -3,11
Trial 5 -3,06
63.86 Trial 1 -3,52 -3.49 0.04
Trial 2 -3,51
Trial 3 -3,48
Trial 4 -3,49
Trial 5 -3,45
86.23 Trial 1 -3,85 -3.83 0.04
Trial 2 -3,79
Trial 3 -3,81
Trial 4 -3,84
Trial 5 -3,87
97.19 Trial 1 -7,90 -7.82 0.06
Trial 2 -7,84
Trial 3 -7,85
Trial 4 -7,73
Trial 5 -7,80
98.83 Trial 1 -8,09 -8.03 0.07
Trial 2 -8,05
Trial 3 -8,02
Trial 4 -7,96
Trial 5 -8,03
88.43 Trial 1 -6,72 -6.63 0.10
Trial 2 -6,52
Trial 3 -6,69
Trial 4 -6,65
Trial 5 -6,58
92.67 Trial 1 -7,28 -7.12 0.12
Trial 2 -7,14
Trial 3 -7,08
Trial 4 -7,05
Trial 5 -7,07
100.38 Trial 1 -9,68 -9.59 0.18
Trial 2 -9,77
Trial 3 -9,57
Trial 4 -9,42
Trial 5 -9,49
23
103.75 Trial 1 -9,45 -9.60 0.10
Trial 2 -9,65
Trial 3 -9,62
Trial 4 -9,65
Trial 5 -9,64
104.37 Trial 1 -9,59 -9.64 0.14
Trial 2 -9,59
Trial 3 -9,81
Trial 4 -9,53
Trial 5 -9,69
108.66 Trial 1 -10,22 -10.14 0.09
Trial 2 -10,18
Trial 3 -10,05
Trial 4 -10,06
Trial 5 -10,17
111.94 Trial 1 -10,68 -10.71 0.11
Trial 2 -10,75
Trial 3 -10,80
Trial 4 -10,58
Trial 5 -10,75
Justification of the appendix tables:
After data collection, the data for mass was divided by a thousand to equate grams to kilograms.
Further on, for the appendix table 1, the initial mass of the helicopter was subtracted from the values to
see the change in weight. Uncertainty values were found by taking away the lowest value of mass from
the try from the largest trial of mass.
Uncertainties proof:
In the experiment, there were many uncertainties recorded. For the frequency uncertainty, the smallest
increment of the stroboscope was used, and hence it is 0.01Hz. For mass, the uncertainty was found by
subtracting from the highest value recorded in trials the lowest value recorded in trials. When it came to
calculating uncertainty for frequency squared, the formula for absolute and relative uncertainties were
used, hence:
Δ𝑓2
𝑓2=
Δ𝑓
𝑓+
Δ𝑓
𝑓
For these purposes, the example for first uncertainty will be calculated:
Δ𝑓2
𝑓2=
1
61+
1
61= 0.0328 − 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 𝑖𝑛 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑠𝑞𝑢𝑎𝑟𝑒𝑑
To find an absolute uncertainty, it is necessary to multiply the relative uncertainty with the value of
frequency squared received from the table 4:
24
Δ𝑓2 = 𝑓2 ∗ 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 = 𝑓2 ∗ 0.0328 = 122𝐻𝑧 ≈ 100𝐻𝑧
Hence the frequency squared value with uncertainty will be :
3700 ± 100𝐻𝑧
The following table summarise all uncertainties for frequency squared:
Values of Frequency
squared (𝑯𝒛𝟐) Uncertainty for 𝒇𝟐 (𝑯𝒛𝟐) Rounded uncertainty (𝑯𝒛𝟐)
3700 123 100
4100 127 100
4700 145 100
7800 178 200
8600 233 200
9500 195 200
9800 259 300
10100 202 200
10800 280 300
10900 209 200
11800 302 300
12500 224 200