Sam Blackburn

Chewing Gum Experiment

 

The aim of this experiment will be to investigate the physical properties of chewing gum: why it sticks to things and how its cohesive and adhesive properties can be affected.

 

Summary (written retrospectively):

 

The question “what do you do if you get chewing gum stuck in your hair?” posed by a certain episode of “The Simpsons” is a difficult one to answer. Finding an answer involved studying cohesive and adhesive forces, finding out how they are affected by various other factors and trying to look for some way to make the cohesive forces great enough that the gum can simply be pulled away. This is easier said than done and the mathematical modeling involved gets quite complicated. Although a little troubled by spurious and inaccurate results, some interesting conclusions are drawn here about chewing gum and its similarities and differences to Blu-Tak, another cohesive/adhesive substance.


Correction.

Contrary to the pretty graphics on Page Two (from www.wrigley.com) I will only be using Spearmint chewing gum for the investigation. Here is a revised version of the graphics that takes this important fact into account:

 

This is also the first time I have been allowed to chew gum in a lesson… let alone charge it to the school!  Clearly my next practical will have to be on champagne. Perhaps on the effect of careful pouring upon preservation of effervescence.

Day One.

Attempts to dissolve chewing gum proved futile in both polar and non-polar solvents (water and methylated spirits). It would not even soften. The water dissolved the sugar from its surface, and if the water was mostly dried off, the gum became quite adhesive.

 

A tiny piece (about 0.5mm long) of chewing gum was placed on a microscope slide with a blob of water. It was examined, drawn and compared with another piece that had been chewed for 5 minutes. The observations are on a handwritten sheet.

 

Actually, this was far from perfect. I could have improved the image by staining the gum with an indicator, maybe iodine, as I was taught years ago in Biology. I’m not sure if this would have worked – chewing gum probably doesn’t contain starch.

 

The conclusion is that (this is the image and drawing it much larger.) and the gum appears to be made of round blobs. It is at this point that the 3D nature of the universe becomes annoying as the whole of the final picture was never in focus at the same time.

 

Chewed gum is very different. All the larger features are gone and the edge appears absolutely smooth until you zoom in a very long way. It is here that I can compare with an experiment I did before: Blu-Tak under a microscope consists of separate fibres, and goo that holds the fibres together, creating a material with interesting properties. Chewing gum has no visible fibres and clear from looking at the pictures…) unchewed gum appears to have a fractal shape: every time you zoom in, more detail emerges, and the details at high factors of magnification look much like those at the lower factors. This is true until we reach about 80,000:1 (No, the school microscopes won’t go that high. I was just looking at one small piece of looks, under a microscope, as if it is made from a homogeneous mixture of substances. This should give it different propertied to the Blu-Tak.

 

Finally, the ingredients list revealed nothing. “Gum base” is obviously what I am looking for, but no clue was given as to its constituents or origin. I’m not even sure if it’s vegetarian.


Day Two.

 First of all, verified that iodine does not stain gum. This means that gum contains no starch and also that I cannot think of a way to improve the pictures I drew yesterday. Chewed gum showed no change either, except that a few dislodged bits of cheek tissue turned black. Nice.

 

I needed to test some physical properties. Tensile strength inside gum will be hard to measure as it will be hard to roll a piece of gum into a cylinder of known and constant diameter and then stretch that cylinder, without it sticking to everything it touches.

 

One property of Blu-Tak that I noticed before was that its adhesive forces are stronger than its cohesive forces at low speeds, while the opposite is true at high speeds. This means that however well a piece of Blu-Tak is stuck to something, it can be cleanly removed if pulled away fast enough. The same is clearly not true with chewing gum. The presence of fibres in Blu-Tak could probably be used to explain this somehow.

 

I first attempted to test the tensile forces of an unchewed piece of gum. It needed a thinnest point at which it would snap, so I punched holes in it using… guess what… a common stationery hole punch. I used 10g masses as clamps to grip the ends of the stick and then fed string through the holes in the masses to exert a force. The idea was that I would know the cross-sectional area of the stick as I could easily measure the width (20mm), subtract twice the diameter of a hole (5mm) and multiply that by its thickness (1.25mm).

Needless to say, it failed. The string cut into the chewing gum and it broke at one end, despite the larger overall stress being halfway along.

The graph shows a more or less linear relationship between force and extension, until it approaches breaking stress when the wrong bit broke anyway.


I tried another experiment purely to test the adhesive forces of unchewed gum. A stick of gum was washed to remove the sugar coating. It was then placed between two sheets of paper and a compression force of 2N was applied by placing masses on top.

 

Using string, a pulley and masses, forces were then applied to the pieces of paper, (the arrows) and the paper slid over the gum. The paper seemed too wet to stick to the gum, so I tried again, using the same stick and new sheets of paper. It was drier the second time because some water had been soaked up by the paper the first time.

 

By the fourth time, the gum was actually sticking to the paper. This was not pleasant so I threw it away. The graphs of these experiments show, again, a linear force/extension graph.

 

However, this graph line changes in odd ways, as the gum gets drier. The overall, but not very strong trend is that the adhesive forces become stronger when the gum is drier, but I would not call this at all conclusive.

 

I did also try using chewed gum between the sheets instead of unchewed gum. This worked but no quantitative results were taken, as this experiment was not very scientifically carried out. One observation I did make was that the adhesive forces were no longer being tested. With chewed gum, the blob of gum split in half leaving some stuck to each sheet, so the cohesive forces were the weaker forces. Clearly, what was going on in chewed gum needed further investigation.


Day Three.

I plan to carry on with the experiment I worked out yesterday: two sheets of paper, some chewed gum in between and some forces. First, I will sort out what variables I can change:

 

·        Horizontal Force

- The force applied to try and shear/stretch the gum.

·        Time stretched

- Sounds like a term to describe the Tardis.

·        Time chewed

- Like a paper jam in a time machine.

·        Temperature

- Well, it affects just about everything.

·        Vertical Force

- The weight of the mass placed on top.

 

There are probably others too (surface of paper…). I think I will start by varying the horizontal force. To speed up measurement taking, I will not write down every value. I will instead place a mark on the string and a sheet of paper under it, drawing the position of the mark on the paper at the required times and measuring afterwards.

 

There were lots of numerical results today. By gradually loading up the above experiment and taking extension readings, I built up a picture of the force/extension graph for chewing gum. The biggest surprise is that it’s not linear. Not even remotely. If you don’t believe it, look at the graph. The curve does not fit an exponential (ex) best fit, or a power (y=xa, drawn onto graph) line either.

 

This turned out to be no error: this graph results every time. The curve should be named after me.

 

Chewing gum behaves very differently when its cohesive forces are being tested – i.e. in odd ways as the graph above suggests – to when its adhesive forces are being tested, when a nice linear graph results.

 

It would seem that some long, boring maths is necessary here. I’ll have a go at it tomorrow.


Day Four.

x                          0.02m

 

R

 
It’s mathematics time.

h
 

 

 

 

 

 

 

 


I approximate the cross section of the piece of chewing gum to a parallelogram. I do this because it makes life easy. Using a ruler I measured the length and width of a piece of gum to be about 20mm, and using a micrometer I measured the thickness to be about 1mm. These measurements were not very accurate as each bit of gum came out a slightly different shape! This was probably the cause of my very inconsistent results. The horizontal force F is trying to shear the gum, which was initially rectangular, to this parallelogram shape. The stress in the gum due to F can be calculated:

 

Where w is the width of the gum. But l has not been measured: this formula will only be useful in terms of x. From the diagram:

           

Which gives us the shear stress in terms of the force and extension. I’m not quite sure what I’m looking for here, but the compressive stress from the vertical force R is probably also involved:

           

Everything until now is definitely true, although some approximations have been made. What I will do now is make a conjecture, which I intend to test to see if it could be true.

           

Plotting x against F gives this graph:

Which, with some fiddling of the constant k, gives a surprisingly good fit. The only big problem with the model is noticed when you see that the theoretical model actually uses the right hand y-axis, and requires a completely different scale! (The experimental result was chewed for 180s, temperature 298K, 2N vertical force.)

 


Now for some practical work: today I plan to measure the effect upon shear strength (i.e. the above experiment) of chewing the gum for longer or shorter periods before shearing it. This is a hard one to predict. While gum is being chewed, two things are going on: the teeth are constantly deforming the gum, and this tends to affect most materials over time. There are also enzymes in saliva that are trying to break down the gum and these may be able to soften it a little.

 

The experiment that I am now used to doing was repeated after chewing for 60, 120, 180, 240, 300 seconds and once for 780 seconds after a… slight timing error. The results are shown below, one graph with a horizontal force of 1N used and the other with a force of 2N. In both cases, the temperature was 298K, vertical force 2N.

 

 

Oh well. As far as I can tell, there is absolutely no correlation there whatsoever. However, if you can join the dots right you might be able to draw a shipwreck.


Day Five.

Today I plan to change a variable that will actually make a difference. I had noticed that if the horizontal force were applied to the experiment for a long time, the extension would gradually increase. However, I am not sure that this increase will be linear. In fact, I think that it is quite unlikely that the speed will be constant.

 

When I studied Blu-Tak, I measured its reaction to tensile stress and predicted an inverse curve: the extension would gradually increase and then peak at a vertical asymptote. The experimental result looked approximately like this but showed a definite linear region before it started to curve upwards.

 

Back to chewing gum, a force was applied to the experiment and a reading was taken every few seconds, the time interval depending upon roughly how fast it was moving (when it looked more or less stationary, this was about a minute; when it was moving it tended to be about five seconds).

 

As the graph shows, the relationship is not at all linear. This is a very strange graph as the speed actually decreases near the beginning. I think it’s mathematics time again, to try and model just what is going on.

 

Time to brush up on a few differential equations… It seems sensible that the creep (rate of change of ext with respect to time) should be proportional to the stress due to shear forces divided by the stress due to compressive forces, and multiplied by the length of the distance being stretched.

Since I can’t make x the subject of the above formula, I used my computer to plot it on a graph. I’ve plotted one of the experimental results too.

 

Oh dear. The model shows no resemblance whatsoever to the experiment! Clearly the conjecture was a little wrong. The problem here is that the speed actually decreases before the failure, which the model was unable to verify. The gum must become stronger with time, creeping less and less, until a certain point when it begins to fail and the line curves upwards (the model agrees on this part at least).

 

This surprises me. In the Blu-Tak experiment I concluded that the fibres in Blu-Tak could be aligned by a tensile force, thus increasing its strength. However, gum has no fibres (according to the microscope observations) and should not do this – but it does!

 

One other result today – by doing the experiment without gum, the coeff. Of friction between two sheets of paper is 0.3.


Day Six.

Time for another variable, I feel, and temperature sounds like an interesting one to do. Blu-Tak was greatly affected by temperature – it softened when hot and became stronger (but not brittle) when cooled. Chewing gum should do roughly the same thing, I think, although quantitative formulae may be very different. I will carry out the usual experiment but at a range of temperatures from 10ºC (the lowest I can easily expect to reach using the school’s supply of not-very-frozen ice) and 50ºC (ditto with not-very-hot boiling water).

 

During the experiment I noticed a difference in the appearance of hot gum after it had been sheared. Yuk, was my thought. I was, unsurprisingly, unable to quantify this in any useful way. Quantitative results were not excellent today – The graphs all show a slight positive tendency but the correlation is terrible. The ones for horizontal forces of 1N and 8N look the most convincing:

(Vertical force 2N, chewed 180s)

 

However there is insufficient evidence to justify or reject that the relationship may be linear – or the same as with Blu-Tak.

Day Seven.

Last day today – I need a finale. The vertical force is the only thing left to change – and I feel fairly confident that increasing the vertical force will make the gum stronger, thus reducing the extension. An inverse relationship between vertical force and extension seems quite likely to me, and this is what both my models predicted, although one of them turned out to be wrong.

 

Sadly, due to time restrictions (school policy is just under 11 hours maximum time for practical) I only got in one value other than the 2N I had been using until now. The value was 8N and the (not very numerous) results are plotted on this empty looking graph. Due to Excel not being able to plot an inverse best fit, I have made the y-axis 1/Extension to look for proportionality. The result is definitely correlated in the right direction, but rather scattered and with too few values to justify proportionality.


Conclusion.

There are many conclusions to be made to this experiment, and many more unanswered questions. I am very pleased that one of my theoretical models fitted so well, but I am puzzled at the results that I failed to accurately model.

 

Anomalous results were a large problem for this investigation. The first cause of these could have been conditions in my mouth, but I think adequate care was taken here: I drank 100ml of tap water 30s before chewing each stick of gum to try and ensure that I was not dehydrated. The most likely cause, I would say, is that the 1mm thickness and 20mm width of a blob of gum squished between two sheets of paper were not always that size. If the gum was slightly flatter, it would clearly have shown greater apparent strength, and I think that this was the cause of the very bad results of the last few days. This idea is backed up by the fact that results that were all taken from one experiment, i.e. the ones taken on the first few days using the same gum and adding masses, were very well correlated and consistent.

 

However, the one important conclusion is that you cannot get chewing gum out of your hair. Even when heated or cooled, chewing gum is more adhesive than cohesive, although the difference is lessened when it is cooled, and so an attempt to pull it away will always leave most of it behind in a sticky mess. The physicists have failed to remove chewing gum, and any further investigation should be done in the chemistry department.

 

Appendix

On the next few pages is a long, boring table. To find what you are looking for, I recommend loading the spreadsheet version instead, to which I have added a brilliant little facility that compares any variables you choose, and it also includes the working behind all the above graphs.


 

ALL RESULTS

 

 

 

 

 

 

Time chewed for /s

Temperature /K

Vertical Force /N

Horizontal force /N

Time stretched /s

Extension /m

Day

 

 

 

 

 

 

 

60

298

2

1

0

0.0010

3

60

298

2

2

0

0.0055

3

60

298

2

3

0

0.0095

3

180

298

2

1

0

0.0030

3

180

298

2

2

0

0.0050

3

180

298

2

1

0

0.0020

3

180

298

2

2

0

0.0040

3

180

298

2

3

0

0.0060

3

180

298

2

4

0

0.0080

3

180

298

2

1

0

0.0015

3

180

298

2

2

0

0.0025

3

180

298

2

3

0

0.0045

3

300

298

2

1

0

0.0030

3

300

298

2

2

0

0.0050

3

300

298

2

3

0

0.0065

3

300

298

2

4

0

0.0080

3

300

298

2

5

0

0.0100

3

60

298

2

3

15

0.0030

3

60

298

2

3

30

0.0060

3

60

298

2

3

45

0.0080

3

60

298

2

3

60

0.0110

3

60

298

2

3

120

0.0190

3

60

298

2

3

180

0.0250

3

60

298

2

3

240

0.0310

3

60

298

2

3

300

0.0370

3

60

298

2

3

360

0.0440

3

60

298

2

3

410

0.0580

3

60

298

2

3

420

0.0650

3

60

298

2

3

425

0.0750

3

180

298

2

2

30

0.0010

3

180

298

2

2

60

0.0020

3

180

298

2

2

360

0.0045

3

180

298

2

2

480

0.0065

3

180

298

2

2

660

0.0085

3

180

298

2

2

1320

0.0100

3

180

298

2

4

10

0.0025

3

180

298

2

4

20

0.0065

3

180

298

2

4

30

0.0100

3

180

298

2

4

40

0.0130

3

180

298

2

4

50

0.0160

3

180

298

2

4

60

0.0180

3

180

298

2

4

70

0.0215

3

180

298

2

4

80

0.0250

3

180

298

2

4

90

0.0320

3

180

298

2

4

100

0.0420

3

180

298

2

4

110

0.0660

3

180

298

2

3

20

0.0060

3

180

298

2

3

40

0.0110

3

180

298

2

3

60

0.0150

3

180

298

2

3

80

0.0180

3

180

298

2

3

180

0.2150

3

180

298

2

3

280

0.2550

3

180

298

2

5

10

0.0015

3

180

298

2

5

20

0.0040

3

180

298

2

5

30

0.0055

3

180

298

2

5

40

0.0085

3

180

298

2

5

50

0.0100

3

180

298

2

5

60

0.0120

3

180

298

2

5

70

0.0145

3

180

298

2

5

80

0.0170

3

180

298

2

5

90

0.0190

3

180

298

2

5

100

0.0210

3

180

298

2

5

110

0.0235

3

180

298

2

5

120

0.0300

3

180

298

2

5

130

0.0370

3

180

298

2

5

135

0.0440

3

180

298

2

5

140

0.0490

3

180

298

2

5

145

0.0550

3

180

298

2

5

150

0.0670

3

120

298

2

1

0

0.0020

4

120

298

2

2

0

0.0050

4

120

298

2

3

0

0.0090

4

120

298

2

1

0

0.0005

4

120

298

2

2

0

0.0015

4

120

298

2

3

0

0.0030

4

120

298

2

4

0

0.0050

4

120

298

2

5

0

0.0095

4

180

298

2

1

0

0.0025

4

180

298

2

2

0

0.0080

4

180

298

2

3

0

0.0110

4

240

298

2

1

0

0.0005

4

240

298

2

2

0

0.0035

4

240

298

2

3

0

0.0045

4

300

298

2

1

0

0.0005

4

300

298

2

2

0

0.0025

4

300

298

2

3

0

0.0040

4

300

298

2

4

0

0.0060

4

780

298

2

1

0

0.0010

4

780

298

2

2

0

0.0025

4

780

298

2

3

0

0.0050

4

120

298

2

3

20

0.0035

4

120

298

2

3

40

0.0070

4

120

298

2

3

60

0.0115

4

120

298

2

3

80

0.0160

4

120

298

2

3

100

0.0170

4

120

298

2

3

120

0.0185

4

120

298

2

3

140

0.0205

4

120

298

2

3

160

0.0230

4

120

298

2

3

180

0.0255

4

120

298

2

3

200

0.0285

4

120

298

2

3

220

0.0300

4

120

298

2

3

240

0.0345

4

120

298

2

3

260

0.0365

4

120

298

2

3

280

0.0385

4

120

298

2

3

300

0.0435

4

120

298

2

3

310

0.0620

4

120

298

2

3

320

0.1090

4

120

298

2

5

5

0.0015

4

120

298

2

5

10

0.0025

4

120

298

2

5

15

0.0045

4

120

298

2

5

20

0.0060

4

120

298

2

5

25

0.0080

4

120

298

2

5

30

0.0105

4

120

298

2

5

35

0.0125

4

120

298

2

5

40

0.0145

4

120

298

2

5

45

0.0160

4

120

298

2

5

50

0.0170

4

120

298

2

5

55

0.0180

4

120

298

2

5

60

0.0190

4

120

298

2

5

90

0.0240

4

120

298

2

5

120

0.0260

4

120

298

2

5

150

0.0340

4

120

298

2

5

153

0.0810

4

240

298

2

3

10

0.0015

4

240

298

2

3

15

0.0020

4

240

298

2

3

25

0.0035

4

240

298

2

3

35

0.0055

4

240

298

2

3

40

0.0065

4

240

298

2

3

50

0.0070

4

240

298

2

3

60

0.0100

4

240

298

2

3

70

0.0110

4

240

298

2

3

90

0.0120

4

240

298

2

3

120

0.0150

4

240

298

2

3

150

0.0165

4

240

298

2

3

240

0.0210

4

240

298

2

3

480

0.0265

4

240

298

2

3

660

0.0330

4

300

298

2

4

10

0.0010

4

300

298

2

4

20

0.0030

4

300

298

2

4

30

0.0055

4

300

298

2

4

40

0.0075

4

300

298

2

4

50

0.0100

4

300

298

2

4

60

0.0120

4

300

298

2

4

70

0.0135

4

300

298

2

4

80

0.0150

4

300

298

2

4

90

0.0160

4

300

298

2

4

100

0.0170

4

300

298

2

4

120

0.0200

4

300

298

2

4

140

0.0215

4

300

298

2

4

160

0.0310

4

300

298

2

4

165

0.0380

4

300

298

2

4

170

0.0550

4

300

298

2

4

175

0.0820

4

780

298

2

3

5

0.0020

4

780

298

2

3

10

0.0035

4

780

298

2

3

15

0.0050

4

780

298

2

3

20

0.0070

4

780

298

2

3

25

0.0095

4

780

298

2

3

30

0.0115

4

780

298

2

3

35

0.0125

4

780

298

2

3

40

0.0135

4

780

298

2

3

60

0.0170

4

780

298

2

3

80

0.0190

4

780

298

2

3

180

0.0225

4

780

298

2

3

240

0.0245

4

780

298

2

3

360

0.0320

4

780

298

2

3

480

0.0370

4

780

298

2

3

720

0.0550

4

780

298

2

3

730

0.0780

4

780

298

2

3

732

0.1430

4

180

298

2

1

0

0.0010

5

180

298

2

2

0

0.0025

5

180

298

2

3

0

0.0045

5

180

298

2

4

0

0.0070

5

180

298

2

5

0

0.0100

5

180

298

2

6

0

0.0125

5

180

298

2

7

0

0.0165

5

180

298

2

8

0

0.0210

5

180

298

2

9

0

0.0280

5

180

298

2

10

0

0.0440

5

180

298

2

1

0

0.0005

5

180

298

2

2

0

0.0010

5

180

298

2

3

0

0.0020

5

180

298

2

4

0

0.0025

5

180

298

2

5

0

0.0040

5

180

298

2

6

0

0.0070

5

180

298

2

7

0

0.0110

5

180

298

2

8

0

0.0150

5

300

298

2

1

0

0.0010

5

300

298

2

2

0

0.0030

5

300

298

2

3

0

0.0045

5

300

298

2

4

0

0.0065

5

300

298

2

5

0

0.0095

5

300

298

2

6

0

0.0140

5

300

298

2

7

0

0.0220

5

300

298

2

8

0

0.1050

5

300

298

2

1

0

0.0005

5

300

298

2

2

0

0.0010

5

300

298

2

3

0

0.0020

5

300

298

2

4

0

0.0025

5

300

298

2

5

0

0.0035

5

300

298

2

6

0

0.0050

5

300

298

2

7

0

0.0065

5

300

298

2

8

0

0.0086

5

300

298

2

9

0

0.0120

5

300

298

2

10

0

0.0170

5

300

298

2

11

0

0.0250

5

300

298

2

1

0

0.0005

5

300

298

2

2

0

0.0015

5

300

298

2

3

0

0.0025

5

300

298

2

4

0

0.0035

5

300

298

2

5

0

0.0055

5

300

298

2

6

0

0.0075

5

300

298

2

7

0

0.0110

5

300

298

2

8

0

0.0160

5

300

298

2

9

0

0.0290

5

180

319

2

1

0

0.0140

6

180

318

2

1

0

0.0030

6

180

318

2

1

0

0.0025

6

180

317

2

1

0

0.0040

6

180

309

2

1

0

0.0015

6

180

309

2

2

0

0.0040

6

180

309

2

3

0

0.0070

6

180

286

2

1

0

0.0005

6

180

286

2

2

0

0.0010

6

180

286

2

3

0

0.0015

6

180

286

2

4

0

0.0025

6

180

286

2

5

0

0.0040

6

180

286

2

6

0

0.0050

6

180

286

2

7

0

0.0060

6

180

286

2

8

0

0.0080

6

180

286

2

9

0

0.0100

6

180

286

2

10

0

0.0115

6

180

286

2

11

0

0.0145

6

180

286

2

12

0

0.0190

6

180

286

2

13

0

0.0250

6

180

286

2

1

0

0.0005

6

180

286

2

2

0

0.0010

6

180

286

2

3

0

0.0015

6

180

286

2

4

0

0.0025

6

180

286

2

5

0

0.0040

6

180

286

2

6

0

0.0050

6

180

286

2

7

0

0.0060

6

180

286

2

8

0

0.0075

6

180

286

2

9

0

0.0085

6

180

286

2

10

0

0.0095

6

180

286

2

11

0

0.0110

6

180

286

2

12

0

0.0125

6

180

284

2

1

0

0.0010

6

180

284

2

2

0

0.0025

6

180

284

2

3

0

0.0030

6

180

284

2

4

0

0.0035

6

180

284

2

5

0

0.0040

6

180

284

2

6

0

0.0045

6

180

284

2

7

0

0.0050

6

180

284

2

8

0

0.0055

6

180

284

2

9

0

0.0060

6

180

284

2

10

0

0.0070

6

180

284

2

11

0

0.0080

6

180

284

2

12

0

0.0100

6

180

284

2

13

0

0.0130

6

180

284

2

14

0

0.0200

6

180

284

2

15

0

0.0350

6

180

319

2

1

3

0.0070

6

180

319

2

1

6

0.0200

6

180

319

2

1

9

0.0410

6

180

318

2

1

5

0.0040

6

180

318

2

1

10

0.0080

6

180

318

2

1

15

0.0265

6

180

318

2

1

20

0.0490

6

180

318

2

1

5

0.0025

6

180

318

2

1

8

0.0070

6

180

318

2

1

11

0.0095

6

180

318

2

1

14

0.0125

6

180

318

2

1

17

0.0155

6

180

318

2

1

20

0.0200

6

180

318

2

1

23

0.0270

6

180

317

2

1

5

0.0020

6

180

317

2

1

10

0.0075

6

180

317

2

1

15

0.0190

6

180

317

2

1

20

0.0550

6

180

317

2

1

22

0.0890

6

180

309

2

3

5

0.0020

6

180

309

2

3

10

0.0040

6

180

309

2

3

15

0.0070

6

180

309

2

3

20

0.0100

6

180

309

2

3

25

0.0120

6

180

309

2

3

30

0.0140

6

180

309

2

3

35

0.0160

6

180

309

2

3

40

0.0180

6

180

309

2

3

45

0.0190

6

180

309

2

3

50

0.0210

6

180

309

2

3

55

0.0225

6

180

309

2

3

60

0.0235

6

180

309

2

3

90

0.0325

6

180

309

2

3

120

0.0410

6

180

309

2

3

150

0.0495

6

180

309

2

3

180

0.0595

6

180

309

2

3

210

0.0705

6

180

309

2

3

240

0.0830

6

180

309

2

3

245

0.0890

6

180

309

2

3

250

0.0940

6

180

309

2

3

255

0.0975

6

180

309

2

3

260

0.1050

6

180

284

8

15

2

0.0170

6

180

298

8

4

0

0.0010

7

180

298

8

8

0

0.0045

7

180

298

8

4

0

0.0040

7

180

298

8

8

0

0.0150

7

180

298

8

4

0

0.0040

7

180

298

8

8

0

0.0090

7

180

298

8

12

0

0.0290

7

180

298

8

4

0

0.0015

7

180

298

8

8

0

0.0030

7

180

298

8

12

0

0.0070

7

180

298

8

16

0

0.0140

7

300

298

8

4

0

0.0015

7

300

298

8

8

0

0.0045

7

300

298

8

12

0

0.0125

7

300

298

8

4

0

0.0010

7

300

298

8

8

0

0.0050

7

300

298

8

12

0

0.0012

7

300

298

8

16

0

0.0250

7

300

298

8

4

0

0.0030

7

300

298

8

8

0

0.0050

7

300

298

8

12

0

0.0140

7

300

298

8

16

0

0.0290

7

 

Appendix

On the last few pages is a long, boring table. To find what you are looking for, I recommend loading the spreadsheet version instead, to which I have added a brilliant little facility that compares any variables you choose, and it also includes the working behind all the above graphs.