OCR GCE Physics B G496 Coursework (experiment)

Experiment-based report for the qualification, OCR GCE Physics B (Advancing Physics)

November 2014

HOW DO DIFFERENT FACTORS

AFFECT THE FUNDAMENTAL

FREQUENCY OF A WOUND NICKEL

WIRE?

By

Maurice Yap

Peter Symonds College

How do different factors affect the fundamental frequency of a wound nickel wire?

GCE Physics – G496 Researching Physics | Maurice Yap 6946 ii

TABLE OF CONTENTS 1. Introduction ......................................................................................................... 1

1.1. Aims of the investigation .................................................................................................................................................. 1

2. Plan ...................................................................................................................... 1

2.1. Method .............................................................................................................................................................................. 1

2.2. Variables ........................................................................................................................................................................... 1

2.3. Risk assesment and precautions .................................................................................................................................... 2

2.3.1. Electrical hazard ....................................................................................................................................................... 2

2.3.2. Heavy masses ........................................................................................................................................................... 2

2.3.3. Stretched wire .......................................................................................................................................................... 3

3. Preliminary test ................................................................................................... 3

3.1. Modifications to method due to preliminary experiment ............................................................................................ 4

3.1.1. Amendments to safety precautions ......................................................................................................................... 4

3.2. Findings of preliminary test ........................................................................................................................................... 4

4. Carrying out the experiment ................................................................................ 5

4.1. Uncertainties .................................................................................................................................................................... 5

4.2. Experiment 1 ................................................................................................................................................................... 5

4.3. Experiment 2 ....................................................................................................................................................................7

4.4. Experiment 3 ................................................................................................................................................................... 9

4.5. Experiment 4 .................................................................................................................................................................. 11

5. Overall conclusion and analysis .......................................................................... 13

5.1. Comparison with Mersenne’s laws ................................................................................................................................ 14

5.2. Errors and anomalies ..................................................................................................................................................... 15

5.3. Limitations of equipment .............................................................................................................................................. 15

6. Reference List ..................................................................................................... 15

7. Appendices ......................................................................................................... 15

TABLE OF FIGURES Figure 1: Diagram of the setup of my initial method idea ........................................................................................................ 1

Figure 2: Force diagram of mass and pulley system ................................................................................................................ 2

Figure 3: Moments diagram for lever on sonometer ............................................................................................................... 3

Figure 4: Diagram of the setup of my initial method idea ...................................................................................................... 4

Figure 5: Scatter graph plotting frequency2 against tension for experiment 1 ...................................................................... 6

Figure 6: Scatter graph plotting frequency2 against tension for experiment 2 ..................................................................... 8

Figure 7: Scatter graph plotting frequency2 against tension for experiment 3.................................................................... 10

Figure 8: Scatter graph plotting frequency-1 against length for experiment 4 ..................................................................... 12

Figure 9: Maximum and minimum gradients of graph for experiment 1. ............................................................................ 16

Figure 10: Maximum and minimum gradients of graph for experiment 2. .......................................................................... 16




GCE Physics – G496 Researching Physics | Maurice Yap 6946 1

1.INTRODUCTION 1.1.AIMS OF THE INVESTIGATION

The objective is to explore how the factors of the tension and length of a wound nickel wire change the

fundamental frequency of it. It will be investigated through an experiment where a string will be plucked

and the frequency it produces will be recorded. The nature of the relationship between the frequency and

the two independent variables will then deduced from the results of the experiment.

2.PLAN 2.1.METHOD

I will attach the wire between two wooden blocks using a G-clamp and the desk. The wire will be used to

hang masses on hooks (for ease of adjustability) off the edge of the desk through a smooth pulley. The wire

will then be plucked. Using a microphone, held in place either by a microphone stand or using a boss and

clamps, that is connected to a laptop, the frequency of the sound produced will be recorded. Repeats of this

result will be taken and then the mass will be changed. A diagram of this setup is shown in figure 1.

FIGURE 1: DIAGRAM OF THE SETUP OF MY INITIAL METHOD IDEA

2.2.VARIABLES

I have decided to change the two independent variables of tension on the wire and the length of the wire. I

will conduct up to five experiments, each using a different length of vibrating wire, where the tension is

Laptop

Laptop

power

supply

cable USB cable

Universal

audio

interface

Microphone stand XLR (sound) cable

Microphone

Wound nickel wire

Wooden blocks

Smooth

pulley

G-clamp Desk

Masses

on a

hook

Padded deep

plastic tray



changed by changing the mass hanging from the wire (the force diagram for this is shown in figure 2). As

well as this, I will conduct another where the tension on the wire remains constant, but the length that is

allowed to vibrate, changes. The dependent variable that will be recorded will be the frequency produced by

the pluck.

FIGURE 2: FORCE DIAGRAM OF MASS AND PULLEY SYSTEM

Control variables will be the thickness of the wire and the elasticity of the wire (I will keep these two the

same by using the same wire and by not stretching it beyond its elastic limit). There will also be trivial

control variables like the temperature of the room. Provided that the force is a light one, the plucking force

applied to the wire does not need to be measured or controlled because it does not affect the frequency of

the sound produced.

2.3.RISK ASSESMENT AND PRECAUTIONS

2.3.1.ELECTRICAL HAZARD The laptop charger will remain plugged in during the experiment to ensure that my laptop does not run out

of battery, costing me time. I will therefore perform a visual check of its power supply cable for any visible

damage. If possible, I will plug it directly into a wall power socket as opposed to an extension lead to

prevent overload. I will ensure that the PAT test for my power supply is in date. I will make sure my hands

and the work surface are dry during the experiment and keep water bottles off the desk where I am

conduction the experiment, away from the laptop and charger. Cables could pose a tripping hazard, or could

break if accidentally forcefully pulled out, possibly leading to an electrical fire. To prevent this, I will warn

other around me of this danger and ensure that there is sufficient slack on the cables. I will also keep my

cables as tidy as possible.

2.3.2.HEAVY MASSES The use of heavy masses dictates that I must wear suitable footwear in the classroom, as must others

around me (i.e. no open toes). I will use a cushion pad in a deep tray under the masses in case they fall, in

order to catch it. This protects the equipment as well as feet. I will check that the clamp is surely tightened

T1N

m kg

mgN

T1N

T1N 𝑁2𝐿 (↑): 𝑇1 − 𝑚𝑔 = 0

∴ 𝑇1 = 𝑚𝑔



and correctly used to ensure stability. I will check condition of the mass hangers and ensure that they are

securely attached to the wire.

2.3.3.STRETCHED WIRE The wire might snap if for some reason, it structurally deforms or an excessive force is applied accidentally.

A metal bridge over the wire will be placed to stop whipping action of free ends if it does snap violently. I

will keep my face well away to ensure that I do not suffer a facial injury.

3.PRELIMINARY TEST A preliminary experiment to undertaken to evaluate the viability and suitability of my planned experiment.

I set up the apparatus as outlined in my plan. It was found that my pulley system is in fact not suitable for

my experiment. Kinetic energy from the vibrating string was very rapidly dissipated through the smooth

pulley, giving an audibly obvious fluctuating frequency. The amplitude of the sound was also very low,

inconvenient for measuring its frequency. Also, my idea of simply hanging a mass off the edge of the desk

was found to be unsuitable, because of the very heavy masses needed to generate a tension large enough to

produce a sufficiently high enough frequency to measure. The problem was emphasised when, using a 7.5

kg mass, the wire’s loop that was used for hooking masses, completely unwound; the mass fell into the

padded tray.

With the kind and extensive help of one of my college’s technicians, I was able to acquire a sonometer, a

specialist piece of equipment used for the same purpose as my experiment. This included two moveable

uprights to act as frets on a guitar, as well as a lever system that acted as a force multiplier, using moments,

shown in my diagram in figure 3. I used the equations I formed below this diagram to use different

combinations of masses and hooks to exert different tensions on the wire (force ‘T’). This was much more

convenient than my previous set up, as it eradicated the need for much of the equipment, including wooden

blocks, the clamp, the pulley and the wire (one is built into the sonometer).

FIGURE 3: MOMENTS DIAGRAM FOR LEVER ON SONOMETER

𝑀𝑜𝑚𝑒𝑛𝑡𝑠 𝑎𝑏𝑜𝑢𝑡 𝑝𝑖𝑣𝑜𝑡 (↷): 𝑀𝑎𝑔 × 𝐷 + 𝑀𝑏𝑔 × 2𝐷 + 𝑀𝑐𝑔 × 3𝐷 + 𝑀𝑑𝑔 × 4𝐷 + 𝑀𝑒𝑔 × 5𝐷 − 𝑇 × 𝐷 = 0

∴ 𝐹𝑎𝑔𝐷 + 𝐹𝑏𝑔2𝐷 + 𝐹𝑐𝑔3𝐷 + 𝐹𝑑𝑔4𝐷 + 𝐹𝑒𝑔5𝐷 = 𝑇𝐷

∴ 𝑇 = 𝑔(𝐹𝑎 + 2𝐹𝑏 + 3𝐹𝑐 + 4𝐹𝑑 + 5𝐹𝑒)

I also tried using an application on my mobile phone, originally designed for tuning musical instruments, to

measure the frequency of the sound produced by the string. It produced identical results to my microphone

and laptop setup, with the same degree of precision.

Mag Mbg

Mcg

Mdg

Meg

D

(arbitrary

length)

D D D D D

T

Pivot

Wire



3.1.MODIFICATIONS TO METHOD DUE TO

PRELIMINARY EXPERIMENT

For convenience, I have opted to use my mobile phone as the instrument to measure my dependent variable

of frequency produced, as opposed to the rather more complex and bulky microphone and laptop setup. The

sonometer will also replace my other equipment, as shown in figure 4.

FIGURE 4: DIAGRAM OF THE SETUP OF MY INITIAL METHOD IDEA

3.1.1.AMENDMENTS TO SAFETY PRECAUTIONS A warning label on the sonometer advises that the masses be hung close to the floor. Because this

isn’t practical with the equipment and furniture I have, I will instead put the padded deep tray on a

chair to reduce the distance the masses would fall and reduce their impact speed.

The hazard caused by the presence of electricity is greatly reduced because of the absence of the use

of mains electricity and the eradication of cables. My mobile phone is still nonetheless an electrical

device. I will therefore continue to make sure that my hands and my desk are dry and that water

bottles are not placed in the vicinity of the working space. This is to prevent electrocution of myself.

3.2.FINDINGS OF PRELIMINARY TEST

The ranges of suitable forces and lengths to measure were found to be as follows:

The tension exerted on the string will be between 10N and 55N.

o The lower bound is such that the sound by the plucked wire produced has a high enough

frequency to give a large enough amplitude to be detected by my mobile phone’s

microphone. It was also found that at these low frequencies, taking the frequency produced

at the wire’s second harmonic was effective (to increase to frequency and audibility of the

sound) and also accurate (dividing by two gave the same frequency as the first harmonic,

within the limits of tolerance and experimental error).

o The higher bound for tensile force of 55N is much lower than the recommended maximum

(85.75N, calculated from the 1.75kg marked on the safety sticker, multiplied by five to

account for the force multiplying lever) but was the limit recommended by the technicians.

Desk

Chair

Padded deep plastic tray

Sonometer

Wire built into

sonometer Mobile phone Force multiplier system

Masses

on a

hook



Frequency, and therefore audibility to my mobile phone’s microphone, increases with an increase in

tension. The experiment where the effect of wire length will therefore be conducted with the tension

controlled at a higher value. I have arbitrarily elected for this to be 49N.

At 49N of tensile force, the shortest length of wire that produce a sound with a sufficiently large

enough amplitude was around 15cm. At about 65-70cm, roughly the physical limit of the sonometer,

the sound produced was still of a sufficient audibility.

4.CARRYING OUT THE EXPERIMENT Four experiments were carried out over a two week period – three where the length of the wire (‘l’) was kept

constant and tension exerted on the wire (‘T’) changed and one where tension was kept constant but length

changed. The frequency of the sound produced by the vibrating wire (‘f’) was measured in all of them.

4.1.UNCERTAINTIES

The variable, ‘l’, was measured using a guide built in to the sonometer. This had increments of 1mm,

therefore, the uncertainty tolerance for ‘l’ is, by my judgement, ±1mm.

I used masses of 100g to exert tension on the wire. Measuring the mass of a large sample (in excess of 30) of

these found little variation, with a difference from 100g of 3g at the very most, though this was rare. I

judged to mean of variations to be around 1.5g either side. The uncertainty for ‘t’ is therefore ±1.5% of the

stated tension (1.5g / 100g = 0.015 = 1.5%, where mass is directly proportional to the force of the weight it

exerts), though statistically, this may be pessimistic as variations either side of the stated (“correct”) mass

will cancel each other out over the total of a large number of masses used.

‘f’ was measured using an application that has increments of 0.1Hz. Using multiple tuning forks to test the

device’s calibration found no variation between given result and the expected result. The uncertainty is

therefore ±0.05Hz.

Experimental uncertainties for the measured frequency were calculated using the following formulas:

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒

2

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦2 =(𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)2 − (𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)2

2

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦−1 =(𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)−1 − (𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)−1

2

4.2.EXPERIMENT 1

This was measuring f, where l was fixed at 0.6m and T was changed.

I carried out this experiment by setting the length of the vibrating wire to 0.6m using the sonometer’s

uprights, then exerting the required tension on the string for each test using the force multiplying pivot. I

then plucked the wire lightly and recorded the frequency shown by my mobile phone’s tuner application. I

repeated each test five times excluding anomalies in order to calculate the average frequency and the

uncertainty in the value.



FIGURE 5: SCATTER GRAPH PLOTTING FREQUENCY2 AGAINST TENSION FOR EXPERIMENT 1

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60

Fre

qu

ency

2(k

Hz2

)

Tension (N)



Tension (N) ± 1.5%

Frequency (Hz)

Frequency2 (kHz2)

Tension (N) ± 1.5%

Frequency (Hz)

Frequency2 (kHz2)

11.76 37.8±0.2* 1.43±0.02* 33.32 62.6±0.3 3.92±0.03

12.74 39.2±0.2* 1.54±0.02* 34.30 63.6±0.5 4.05±0.06

13.72 40.6±0.1* 1.65±0.008* 35.28 64.2±0.3 4.12±0.03

14.70 42.0±0.1* 1.76±0.008* 36.26 64.9±0.3 4.21±0.03

15.68 43.2±0.1* 1.86±0.009* 37.24 65.7±0.3 4.31±0.03

16.66 44.7±0.2* 1.99±0.02* 38.22 66.3±0.3 4.40±0.04

17.64 46.2±0.5* 2.13±0.04* 39.20 67.0±0.3 4.49±0.04

18.62 46.9±0.2* 2.20±0.02* 40.18 68.3±0.5 4.66±0.07

19.60 48.4±0.1* 2.35±0.01* 41.16 68.8±0.1 4.74±0.007

20.58 49.1±0.1* 2.41±0.01* 42.14 69.7±0.3 4.86±0.03

21.56 50.3±0.3* 2.53±0.03* 43.12 70.4±0.1 4.96±0.007

22.54 51.5±0.2* 2.66±0.02* 44.10 71.2±0.3 5.06±0.04

23.52 52.6±0.2* 2.77±0.02* 45.08 72.0±0.3 5.18±0.04

24.50 53.6±0.1* 2.87±0.01* 46.06 72.7±0.3 5.29±0.04

25.48 54.4±0.1* 2.96±0.01* 47.04 73.6±0.3 5.41±0.04

26.46 55.2±0.2* 3.04±0.02* 48.02 74.4±0.5 5.54±0.07

27.44 56.1±0.4 3.15±0.04 49.00 74.9±0.3 5.61±0.04

28.42 57.4±0.2 3.29±0.02 49.98 75.7±0.3 5.73±0.04

29.40 58.4±0.3 3.41±0.04 50.96 76.2±0.4 5.81±0.05

30.38 59.3±0.2 3.51±0.02 51.94 77.3±0.2 5.98±0.03

31.36 60.4±0.3 3.65±0.04 52.92 77.6±0.2 6.02±0.03

32.34 61.3±0.4 3.75±0.04 53.90 78.8±0.2 6.21±0.03

*These results were derived from readings taken at the first harmonic.

Although all values deduced from recorded results are quoted to three significant figures in the table above,

no rounding took place during calculation. A full table of raw results with calculations is in the appendix.

I have chosen to plot f2 against T on the scatter graph of figure 5 because this was the best manipulation of

the two variables of T and f that produced an apparent linear relationship. This has also been done in

figures 6 and 7. The mathematical manipulations compared to this were other polynomial relationships (y =

xa), an exponential relationship (y = abx), a linear relationship (y = a+bx), a logarithmic relationship (y =

a+blnx) and a power relationship (y = axb).

Figure 5 shows a straight line of best fit between the plotted variables. It also appears to almost go through

the origin, although an alternative line of best fit that intersects the origin could be formed using the error

bars for experimental uncertainty and equipment tolerance. Furthermore, it is a logical assumption that a

completely slack wire (zero tension, T = 0) would theoretically produce a frequency of 0 Hz. It can therefore

be deduced from these results that tension is directly proportional to frequency2.

The constant gradient of this line of best fit is 0.113±0.002kHz2N-1 to three significant figures. The

uncertainty in this value was, derived using the maximum and minimum gradients of the line of best fit,

shown in the appendix (figure 9).

4.3.EXPERIMENT 2


I set the length of the vibrating wire to 0.5m using the sonometer’s uprights, then preceded in the same way

as the previous experiment.




0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60

Fre

qu

ency

2(k

Hz2

)

Tension (N)



Tension (N) ± 1.5%

Frequency (Hz)

Frequency2 (kHz2)

Tension (N) ± 1.5%

Frequency (Hz)

Frequency2 (kHz2)

11.76 45.6±0.1* 2.08±0.009* 33.32 73.7±0.3 5.43±0.04

12.74 47.6±0.1* 2.26±0.01* 34.30 74.7±0.2 5.58±0.03

13.72 49.0±0.1* 2.40±0.005* 35.28 76.0±0.3 5.78±0.04

14.70 50.6±0.1* 2.56±0.02* 36.26 77.0±0.3 5.92±0.04

15.68 52.3±0.4* 2.74±0.04* 37.24 78.7±0.2 6.20±0.02

16.66 53.4±0.1* 2.86±0.01* 38.22 79.5±0.2 6.33±0.03

17.64 54.9±0.3* 3.02±0.03* 39.20 80.8±0.3 6.54±0.05

18.62 56.6±0.1* 3.21±0.01* 40.18 81.7±0.2 6.68±0.03

19.60 58.4±0.3* 3.41±0.03* 41.16 82.5±0.3 6.81±0.05

20.58 59.1±0.1* 3.49±0.01* 42.14 83.2±0.2 6.93±0.03

21.56 60.0±0.2* 3.60±0.02* 43.12 84.1±0.2 7.07±0.03

22.54 61.2±0.4* 3.74±0.05* 44.10 85.2±0.2 7.26±0.03

23.52 62.6±0.4* 3.92±0.04* 45.08 86.2±0.3 7.42±0.04

24.50 64.2±0.4 4.13±0.06 46.06 86.9±0.3 7.56±0.04

25.48 65.2±0.1 4.25±0.01 47.04 87.6±0.3 7.68±0.06

26.46 66.1±0.2 4.36±0.03 48.02 88.8±0.4 7.89±0.07

27.44 67.1±0.2 4.51±0.03 49.00 90.5±0.1 8.19±0.02

28.42 67.8±0.2 4.60±0.03 49.98 91.2±0.2 8.32±0.04

29.40 68.9±0.1 4.75±0.02 50.96 92.1±0.4 8.48±0.07

30.38 70.1±0.1 4.92±0.02 51.94 92.9±0.3 8.63±0.05

31.36 71.3±0.5 5.08±0.06 52.92 94±0.5 8.84±0.08

32.34 72.6±0.4 5.27±0.06 53.90 95.6±0.8 9.14±0.1


A full table of raw results showing calculations is attached in the appendix.

Like the first experiment, the constant gradient of the line of best fit for the graph of frequency2 against

tension in figure 6 suggests direct proportionality between these two variables. Again, the line of best fit

intersects the axes close to zero and a line of best fit that crosses the origin is within the limits of

experimental error, visually shown by error bars.



shown in the appendix (figure 10). This is greater than when l = 0.6.

4.4.EXPERIMENT 3


I set the length of the vibrating wire to 0.5m using the sonometer’s uprights, then preceded in the same way

as the previous experiment.




0

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60

Fre

qu

ency

2(k

Hz2

)

Tension (N)



Tension (N) ± 1.5%

Frequency (Hz)

Frequency2 (kHz2)

Tension (N) ± 1.5%

Frequency (Hz)

Frequency2 (kHz2)

11.76 58.0±0.7* 3.36±0.08* 33.32 93.3±0.3 8.70±0.06

12.74 58.8±0.4* 3.46±0.05* 34.30 94.8±0.4 8.99±0.07

13.72 60.7±0.5* 3.68±0.06* 35.28 95.7±0.1 9.16±0.02

14.70 62.4±0.8* 3.89±0.1* 36.26 96.2±0.3 9.25±0.05

15.68 64.7±0.5 4.19±0.06 37.24 97.5±0.3 9.51±0.05

16.66 66.6±0.4 4.43±0.06 38.22 99.2±0.3 9.84±0.07

17.64 67.8±0.4 4.60±0.05 39.20 100±0.2 10.0±0.04

18.62 70.4±0.4 4.96±0.06 40.18 101±0.3 10.3±0.06

19.60 72.3±0.4 5.22±0.06 41.16 103±0.5 10.5±0.1

20.58 73.4±0.5 5.39±0.07 42.14 104±0.2 10.8±0.04

21.56 74.9±0.4 5.61±0.06 43.12 105±0.4 10.9±0.07

22.54 77.3±0.6 5.98±0.09 44.10 106±0.8 11.3±0.2

23.52 78.9±0.2 6.23±0.03 45.08 108±0.3 11.6±0.05

24.50 80.8±0.3 6.53±0.04 46.06 108±0.2 11.8±0.04

25.48 83.3±0.3 6.94±0.04 47.04 109±0.4 12.0±0.08

26.46 84.3±0.3 7.11±0.05 48.02 110±0.5 12.2±0.1

27.44 86.1±0.1 7.41±0.03 49.00 112±0.4 12.5±0.1

28.42 87.0±0.1 7.57±0.03 49.98 113±0.3 12.8±0.06

29.40 88.6±0.7 7.86±0.1 50.96 114±0.3 13.0±0.06

30.38 90.0±0.6 8.11±0.1 51.94 115±0.3 13.2±0.07

31.36 91.1±0.4 8.30±0.07 52.92 117±0.3 13.6±0.07

32.34 91.9±0.2 8.44±0.04 53.90 118±0.3 14.0±0.06


A full table of raw results showing calculations is attached in the appendix.

Like the first two experiments, the constant gradient of the line of best fit for the graph of frequency2

against tension in figure 7 suggests direct proportionality between these two variables. Again, the line of

best fit intersects the axes close to zero and a line of best fit that crosses the origin is within the limits of

experimental error, visually shown by error bars.



shown in the appendix (figure 11). This is greater than when l = 0.6 and when l = 0.5. From this, it can be

deduced that as length decreases, the sensitivity of frequency2, and therefore frequency, to a change in

tension increases.

4.5. EXPERIMENT 4

This was measuring f, where T was fixed at 49N and T was changed.

This experiment was conducted by loading 49N of tensile force on using the force-multiplying lever. I then

plucked the string lightly and continued in the same way as the previous experiments, changing the length

of the wire using the sonometer’s uprights as necessary.



FIGURE 8: SCATTER GRAPH PLOTTING FREQUENCY -1 AGAINST LENGTH FOR EXPERIMENT 4

0

2

4

6

8

10

12

14

16

18

0 100 200 300 400 500 600 700 800

Fre

qu

ency

-1(m

s)

Length (mm)



Length (mm) ± 1mm

Frequency (Hz)

Frequency-1 (ms)

Length (mm) ± 1mm

Frequency (Hz)

Frequency-1 (ms)

160 282±0.4 3.55±0.005 440 102±0.1 9.80±0.01

180 250±0.3 4.01±0.006 460 97.7±0.1 10.2±0.02

200 224±0.3 4.46±0.005 480 93.8±0.2 10.7±0.02

220 204±0.1 4.90±0.004 500 90.6±0.1 11.0±0.01

240 188±0.1 5.32±0.004 520 87.3±0.2 11.5±0.02

260 173±0.2 5.79±0.005 540 84.2±0.4 11.9±0.06

280 161±0.2 6.22±0.006 560 80.9±0.1 12.4±0.02

300 150±0.2 6.66±0.01 580 78.2±0.1 12.8±0.02

320 141±0.4 7.11±0.02 600 75.7±0.1 13.2±0.03

340 132±0.2 7.57±0.01 620 73.4±0.1 13.6±0.03

360 124±0.4 8.03±0.02 640 70.6±0.5 14.2±0.1

380 118±0.3 8.50±0.02 660 68.7±0.1 14.6±0.02

400 112±0.1 8.93±0.01 680 65.9±0.4 15.2±0.08

420 107±0.3 9.37±0.02 700 62.7±0.4 15.9±0.1

Although all values deduced from recorded results are quoted to three significant figures in the table above,

no rounding took place during the calculation process itself. A full table of raw results showing calculations

is attached in the appendix.

I have chosen to plot frequency-1 against length on the scatter graph of figure 8 because this was the best

manipulation of the two variables of length and frequency that produced an apparent linear relationship.

Figure 8 shows a straight line of best fit between frequency-1 and length that intersects the origin. This is

consistent with my logical assumption that a theoretical wire of infinite length would produce a frequency

of zero and therefore conversely, a theoretical wire of zero length would produce a frequency that tends

towards infinity (this concurs with the extrapolated point (0, 0) on my graph, where the frequency would be

infinity (1 ÷ 0 → ∞) zero length).

It can therefore be deduced from these results that length is directly proportional to the time period

(equivalent to frequency-1).

The constant gradient of this line of best fit is 22.2±0.9µsmm-1 to three significant figures. The uncertainty

in this value was, derived using the maximum and minimum gradients of the line of best fit, shown in the

appendix (figure 12).

5.OVERALL CONCLUSION AND

ANALYSIS Within the limits of experimental error, data from my practical instigations have produced the following

conclusions:

tension (T) and frequency2 (f2) are directly proportional

length (l) and time period or frequency-1 (f-1) are directly proportional

there is a negative correlation between length (l) and the sensitivity of the change in frequency to a

change in tension (∆f/∆T).

The first point gives the following mathematical relationship:



𝑇 ∝ 𝑓2

∴ 𝑇 = 𝑘𝑓2

∴𝑇

𝑓2 = 𝑘 ↔𝑓2

𝑇= 𝑘

The second point gives this mathematical relationship:

𝑙 ∝ 𝑓−1

∴ 𝑙 =𝑘

𝑓↔ 𝑓 =

𝑘

𝑙

∴ 𝑙𝑓 = 𝑘

Combining these to deduce a relationship between all the variables of length, frequency and tension:

𝑇 = 𝑘𝑓2 𝑓 =𝑘

𝑙

𝑘√𝑇 = 𝑓

𝑓 = 𝑘√𝑇𝑘

𝑙

𝑓 = 𝑘√𝑇

𝑙

I think that the constant, ‘k’, includes my control variables and therefore is related to factors including the

stiffness constant of the wire, the density of the string, the mass of the string or the diameter, volume or

cross sectional area of the string.

5.1.COMPARISON WITH MERSENNE’S LAWS

One of Mersenne’s laws (Jeans, 1969: 64-65) concerning the frequency of a stretched string presents the

following formula for the fundamental frequency of a string (Nave, 2001; Wikipedia, 2014):

𝑓 =

√𝑇𝜇

2𝑙

(Where µ is the linear density (mass per unit length) of the string.)

This concurs with the relationships between T, f and l that I derived from my experimental results. Because

µ was a control variable across all four experiments and was therefore a constant throughout, it can be

accounted for in the unknown constant, ‘k’.

Letting

𝑘 =1

2√𝜇

my derived formula for fundamental frequency is mathematically equivalent to that of Mersenne’s theory.

My prediction about the physical factors in ‘k’ was partially correct. Linear density, and therefore mass,

volume and diameter are involved, however, the stiffness of the string is not included in Mersenne’s law.

Assuming this theory is correct, my prediction that stiffness is a factor is incorrect.



5.2.ERRORS AND ANOMALIES

I recorded a total of 11 anomalous results, all of which had been easily identified during the experiments,

and therefore, they have each been replaced by a re-test to ensure that there are five valid readings for each

result. There were two main reasons for this. The first was a plucking of the wire that was too violent or

strong, causing an initially musically sharp (erroneously higher frequency) sound. The second was the fact

that the mass hangers were not directly in the slots of the force multiplier, therefore producing an incorrect

turning moment about the force multiplier’s pivot and therefore an incorrect tension on the wire.

There were other possible systematic errors that could have reduced the accuracy of my results. The force

multiplier’s arm needed to be parallel with the wire in order to make the system work correctly. This was

judged by eye (I knelt down so that my eye level was consistent with the arm) and changed using a screw at

the other end of the sonometer. I checked this after every change in the independent variable. Furthermore,

I had only calibrated my tuning instrument (mobile phone) to frequencies in the range 330Hz to 660Hz –

outside the range of my collected data. If the device was not calibrated in the range I recorded, there could

have been a recording error due to this. Also, the “control” equipment I used for calibration was a set of

low-quality guitar tuning forks, which may have been imprecise on inaccurate.

5.3.LIMITATIONS OF EQUIPMENT

The length of my sonometer dictated the maximum length of string I could use. Its force multiplying

mechanism’s construction strength also dictated the maximum tension I could use. The sonometer’s lack of

acoustic holes meant that the amplitude of the sound produced was low. This made it more difficult to

collect results but had no effect on them. My phone’s microphone, for functionally practical reasons, had a

low frequency response to sounds below 60Hz and therefore made it difficult to record very low

frequencies. I got round this by producing the second harmonic frequency by lightly touching the string at

its midpoint.

6.REFERENCE LIST JEANS, Sir J. H., 1969. Science and Music (New Edition), Mineola, NY: Dover Publications

NAVE, R., 2001. Standing Waves on a String [Online]. Available at: http://hyperphysics.phy-

astr.gsu.edu/hbase/waves/string.html [Accessed 17 November 2014]

WIKIPEDIA, 2014. Vibrating string [Online]. Available at: http://en.wikipedia.org/wiki/Vibrating_string

[Accessed 17 November 2014]

7.APPENDICES



FIGURE 9: MAXIMUM AND MINIMUM GRADIENTS OF GRAPH FOR EXPERIMENT 1.

Maximum gradient is 0.121kHz2N-1, minimum gradient is 0.118kHz2N-1.



0

1

2

3

4

5

6

7

0 10 20 30 40 50 60

Fre

qu

ency

2(k

Hz2

)

Tension (N)

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60

Fre

qu

ency

2(k

Hz2

)

Tension (N)






Maximum gradient is 23.3µsmm-1, minimum gradient is 21.6µsmm-1.

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60

Fre

qu

ency

2(k

Hz2

)

Tension (N)

0

2

4

6

8

10

12

14

16

18

0 100 200 300 400 500 600 700 800

Fre

qu

ency

-1(m

s)

Length (mm)



Grey-filled spaced are calculated results derived by halving the second harmonic frequency. All averages are

means of valid recorded results. Raw results for experiment 1:

Tension (N)

Frequency recorded (Hz) Average frequency (Hz)

Frequency uncertainty (Hz) Anomalous results (Hz) #1 #2 #3 #4 #5

11.76 37.8 37.5 37.9 37.9 37.7 37.76 0.2

12.74 39.3 39.1 39.3 39.4 39 39.22 0.2

13.72 40.6 40.6 40.6 40.8 40.6 40.64 0.1

14.7 42.1 42 41.9 42 42 42 0.1

15.68 43.1 43.3 43.2 43.2 43.1 43.18 0.1

16.66 44.5 44.6 44.9 44.7 44.6 44.66 0.2

17.64 46.7 46 45.8 46.3 46.1 46.18 0.45 46.7

18.62 46.7 46.9 46.9 47 47.1 46.92 0.2

19.6 48.3 48.4 48.5 48.5 48.5 48.44 0.1

20.58 49.2 49 49.1 49.2 49.2 49.14 0.1

21.56 50 50.1 50.4 50.3 50.5 50.26 0.25

22.54 51.5 51.6 51.7 51.5 51.4 51.54 0.15

23.52 52.8 52.6 52.6 52.7 52.5 52.64 0.15

24.5 53.6 53.5 53.6 53.5 53.7 53.58 0.1

25.48 54.4 54.5 54.5 54.3 54.5 54.44 0.1

26.46 55.2 55.2 55.3 55.1 55 55.16 0.15

27.44 55.6 56.2 56.2 56.3 56.3 56.12 0.35

28.42 57.6 57.3 57.3 57.4 57.3 57.38 0.15

29.4 58.1 58.5 58.4 58.3 58.7 58.4 0.3

30.38 59.4 59.1 59.2 59.5 59.2 59.28 0.2

31.36 60.6 60.4 60.7 60.3 60.1 60.42 0.3

32.34 60.8 61.4 61.5 61.1 61.5 61.26 0.35

33.32 62.9 62.7 62.4 62.5 62.5 62.6 0.25

34.3 64 63.6 63.8 63.1 63.7 63.64 0.45

35.28 64.1 64.2 63.9 64.4 64.3 64.18 0.25 63.7

36.26 64.9 65 65.1 64.8 64.6 64.88 0.25

37.24 65.7 65.4 65.9 65.7 65.6 65.66 0.25

38.22 66.6 66 66.1 66.5 66.3 66.3 0.3

39.2 67.4 66.9 66.9 67 66.8 67 0.3

40.18 68.3 68.4 68.8 68 67.8 68.26 0.5

41.16 68.8 68.9 68.8 68.9 68.8 68.84 0.05

42.14 69.4 69.7 69.6 69.9 69.8 69.68 0.25

43.12 70.4 70.5 70.5 70.4 70.4 70.44 0.05

44.1 71 71.2 71.3 70.9 71.4 71.16 0.25

45.08 71.8 71.7 72.2 72.3 71.9 71.98 0.3

46.06 72.8 73 72.7 72.4 72.6 72.7 0.3

47.04 73.8 73.7 73.6 73.5 73.2 73.56 0.3

48.02 74.4 74.7 73.9 74.2 74.8 74.4 0.45

49 74.9 75.1 74.6 74.9 75 74.9 0.25 78

49.98 75.4 75.8 75.9 75.7 75.7 75.7 0.25

50.96 76.5 76.4 75.8 76.2 76.2 76.22 0.35

51.94 77.5 77.2 77.4 77.1 77.5 77.34 0.2

52.92 77.4 77.4 77.8 77.6 77.7 77.58 0.2

53.9 78.9 78.6 78.9 79 78.6 78.8 0.2

Raw results for experiment 2:

Tension (N)


Frequency uncertainty (Hz) Anomalous results (Hz) #1 #2 #3 #4 #5

11.76 45.5 45.6 45.6 45.7 45.5 45.58 0.1

12.74 47.7 47.5 47.5 47.6 47.5 47.56 0.1

13.72 49 49.1 49.1 49 49 49.04 0.05

14.7 50.5 50.8 50.6 50.5 50.7 50.62 0.15

15.68 52.8 52 52.1 52.3 52.4 52.32 0.4

16.66 53.5 53.5 53.5 53.4 53.3 53.44 0.1

17.64 55 54.6 54.9 55.1 55 54.92 0.25



18.62 56.6 56.5 56.6 56.7 56.7 56.62 0.1

19.6 58.5 58.3 58.4 58.7 58.2 58.42 0.25

20.58 59.2 59.1 59 59 59 59.06 0.1

21.56 59.9 59.8 60 60.1 60 59.96 0.15

22.54 60.8 60.8 61.3 61.6 61.4 61.18 0.4

23.52 62.6 62.5 62.7 62.3 63 62.62 0.35

24.5 64.3 64.2 64 64.8 63.9 64.24 0.45

25.48 65.1 65.2 65.3 65.2 65.2 65.2 0.1

26.46 65.9 66 66.1 66 66.3 66.06 0.2

27.44 66.9 67 67.1 67.3 67.3 67.12 0.2

28.42 67.6 68 67.9 67.7 68 67.84 0.2 69

29.4 68.8 69.1 69 68.8 68.8 68.9 0.15

30.38 70.1 70 70.1 70.3 70.1 70.12 0.15 71.7, 71.8

31.36 71 70.8 71.4 71.7 71.4 71.26 0.45

32.34 72.9 72.5 72.1 72.6 72.8 72.58 0.4

33.32 73.9 73.5 73.6 73.4 73.9 73.66 0.25

34.3 74.5 74.9 74.7 74.6 74.9 74.72 0.2

35.28 76.1 76.2 75.9 75.7 76.1 76 0.25

36.26 77 77.2 77.1 76.7 76.8 76.96 0.25

37.24 78.6 78.7 78.8 78.9 78.6 78.72 0.15

38.22 79.3 79.7 79.6 79.6 79.5 79.54 0.2

39.2 81.2 80.7 80.6 80.9 80.8 80.84 0.3

40.18 81.9 81.5 81.8 81.7 81.7 81.72 0.2

41.16 82.2 82.4 82.6 82.8 82.6 82.52 0.3

42.14 83 83.2 83.4 83.3 83.3 83.24 0.2

43.12 83.9 84.1 84 84.2 84.3 84.1 0.2

44.1 85 85.2 85.2 85.1 85.4 85.18 0.2

45.08 86 86 86.1 86.5 86.2 86.16 0.25

46.06 87.1 86.9 87.1 86.6 86.9 86.92 0.25

47.04 87.5 87.4 87.5 88.1 87.7 87.64 0.35

48.02 89.2 88.4 88.7 88.8 89 88.82 0.4

49 90.4 90.5 90.6 90.4 90.5 90.48 0.1

49.98 91.4 91 91.3 91.2 91.1 91.2 0.2

50.96 91.7 92.2 92.1 92.5 91.8 92.06 0.4

51.94 92.6 92.9 93 92.8 93.1 92.88 0.25

52.92 93.6 94.2 93.6 94.2 94.5 94.02 0.45

53.9 96.5 96.2 95 95.2 95.1 95.6 0.75


Tension (N)


Frequency uncertainty (Hz)

Anomalous results (Hz) #1 #2 #3 #4 #5

11.76 57.8 58.1 58.6 57.3 58 57.96 0.65

12.74 58.2 59 59.1 58.9 58.8 58.8 0.45

13.72 60.2 61.2 60.7 60.8 60.4 60.66 0.5

14.7 61.3 62.4 62.7 62.7 62.9 62.4 0.8

15.68 64.1 64.8 64.7 65 65.1 64.74 0.5

16.66 66.5 67 66.9 66.1 66.4 66.58 0.4

17.64 67.7 68.3 67.6 67.6 68 67.84 0.35

18.62 70 70.2 70.8 70.4 70.6 70.4 0.4

19.6 72.3 72.2 72.8 72.1 72 72.28 0.4

20.58 73.2 73.9 73.8 73 73.1 73.4 0.45

21.56 74.5 74.7 75.1 74.9 75.3 74.9 0.4

22.54 77.3 78 77.2 76.8 77.2 77.3 0.6

23.52 78.8 79 78.7 79 79.1 78.92 0.2

24.5 81 80.7 80.9 80.5 80.9 80.8 0.25

25.48 83.6 83.6 83.1 83.2 83.1 83.32 0.25

26.46 84.4 84.4 83.9 84.5 84.5 84.34 0.3

27.44 86.1 86.2 86.2 86.1 85.9 86.1 0.15

28.42 86.9 87 86.9 87.2 87.1 87.02 0.15

29.4 89.4 88.4 88.6 88.1 88.7 88.64 0.65

30.38 90.4 90.2 89.2 90.5 89.9 90.04 0.65



31.36 90.8 91 91.5 90.7 91.4 91.08 0.4

32.34 92 91.9 92 91.6 91.8 91.86 0.2

33.32 93 93.1 93.6 93.5 93.1 93.26 0.3

34.3 94.7 95.2 94.5 95 94.8 94.84 0.35

35.28 95.7 95.8 95.6 95.7 95.8 95.72 0.1

36.26 96.2 96.5 96 96 96.1 96.16 0.25

37.24 97.6 97.2 97.5 97.7 97.7 97.54 0.25

38.22 99.3 99.2 99.6 98.9 99.1 99.22 0.35

39.2 99.9 100.1 100.3 100 100.2 100.1 0.2

40.18 101.2 101.2 101.1 101.7 101.4 101.32 0.3

41.16 102.8 102.9 102.6 102.7 101.9 102.58 0.5

42.14 104 103.9 103.6 103.7 103.8 103.8 0.2

43.12 104.3 104.6 104.2 104.9 104.8 104.56 0.35

44.1 106.7 106.8 106.8 106.2 105.2 106.34 0.8

45.08 107.6 107.4 107.8 107.3 107.6 107.54 0.25

46.06 108.3 108.4 108.6 108.2 108.5 108.4 0.2

47.04 109.7 109.8 109.5 109.1 109.2 109.46 0.35 111.7

48.02 109.7 110.5 110.4 110.8 110.6 110.4 0.55

49 112.3 112 111.8 111.4 112.3 111.96 0.45

49.98 113.1 113.4 113.2 112.9 113 113.12 0.25

50.96 114 114.2 113.8 113.7 113.9 113.92 0.25

51.94 114.9 115.5 115 115 114.9 115.06 0.3

52.92 116.8 117 116.7 116.4 116.5 116.68 0.3

53.9 118.2 118 118 118.5 118 118.14 0.25


Length (m)


Frequency uncertainty (Hz)

Anomalous results (Hz) #1 #2 #3 #4 #5

0.16 281.2 282 281.5 281.4 281.7 281.56 0.4 292

0.18 249.5 250 249.5 249.7 249.3 249.6 0.35 257.8, 258.9

0.2 224.1 224.2 224.5 224 224.1 224.18 0.25

0.22 203.8 204 204.1 204 203.9 203.96 0.15 212.2

0.24 188 188 188 188.1 187.8 187.98 0.15

0.26 172.8 172.9 173 172.7 172.8 172.84 0.15

0.28 161 160.7 160.8 160.9 160.7 160.82 0.15

0.3 150.4 150 149.8 150.3 150.4 150.18 0.2

0.32 140.7 140.9 140.2 140.7 141 140.7 0.4

0.34 132 132.3 132.1 131.9 132.1 132.08 0.2

0.36 124.8 124.6 124.3 124.1 124.6 124.48 0.35

0.38 117.9 117.5 117.4 117.3 117.8 117.58 0.3

0.4 112 111.8 112 111.8 112.1 111.94 0.15

0.42 106.8 106.7 106.5 107 106.7 106.74 0.25

0.44 102 102.1 102.2 101.9 102.1 102.06 0.15

0.46 97.7 97.8 97.5 97.6 97.8 97.68 0.15

0.48 93.8 94 93.7 93.6 93.9 93.8 0.2

0.5 90.7 90.6 90.7 90.6 90.5 90.62 0.1

0.52 87.2 87.3 87.1 87.4 87.3 87.26 0.15

0.54 84.3 84.1 83.9 84.8 84 84.22 0.45

0.56 80.8 80.9 80.9 80.9 81 80.9 0.1

0.58 78.1 78.3 78.2 78.3 78 78.18 0.15

0.6 75.7 75.6 75.5 75.8 75.7 75.66 0.15

0.62 73.6 73.4 73.5 73.3 73.4 73.44 0.15

0.64 70.4 70.7 71.2 70.2 70.5 70.6 0.5

0.66 68.7 68.8 68.7 68.7 68.6 68.7 0.1

0.68 65.5 66 66.2 66 65.8 65.9 0.35

0.7 62.4 62.8 62.6 63.3 62.5 62.72 0.45

Documents

OCR GCE Physics B G496 Coursework (experiment)