1

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

5

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

19

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.

20

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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