H being the height reached by the ball, H0 being the initial height the ball was dropped at, b being the number of bounces and λ being the decay constant. Since the height is proportional to the number of bounces, taking logs to the base e
In H = in H0 - λt as In ex = x
Hence to get the equation to apply to the y = mx + c notation,
In H/In H0 = -λt = In H/H0
However as H0 = 1m then In H = -λt + c
c being the y intercept. From my graph it’s been found that
y=2.5 x – 0.24
18
So λ= 2.5 = 0.14(2dp)
18
Though the coefficient of restitution and the deacy constant will be achieved from analysing the same bit of data they will not be dependant on each other ie. If the ball has a low decay constant at a certain temperature it doesn’t necessarily mean that the coefficient of restitutuin will be low at that temperature. This is true because the decay constant looks at height, which looks at gravity and so is a measure of energy (considering G.P.E.= mgh). The coefficient of restitution however looks at the velocity before and after a collision and so is a measure of momentum [considering momentum (p)=mass (m) x velocity (v)].
However if; G.P.E. = mgh and p=mv
Then; EG=mgh
So; EG x v = mvgh = pgh
And so; EG = pgh
v
Or as; EG + EK = ETOTAL
Then; Ek x m = 1/2 m2v2 = 1/2 p2
And so; Ek = . p2 .
2m
As my second part of the experiment I will be testing out the effect of temperature on the silicone and rubber balls, I’m expecting that the higher the temperature the higher the coefficient of restitution, and the lower the decay constant. Rubber and Silicone are made up of polymer chains, to make up the balls these chains are entwined within each other, and mainly held together by covalent bonds (Intermolecular forces). When the ball hits the surface this exerts a force on these polymer chains stretching them out. The Intermolecular forces between the polymer chains then act against this force to ‘re-entwine’ the polymer chains. So I believe that the hotter the ball the more energy supplied to the polymer chains, making them more stretchy so the Intermolecular forces re-entwine the chains quicker. The quicker the chains re-entwine the more energy conserved by the ball and so the higher its return bounce, so the higher thenumber of bounces achieved and so the lower the decay constant. Therefore making the balls more elastic, and so the collisiions more elastic (higher coefficient of restitution) with increasing temperature.
The Effect Of Temperature
On A Rubber Ball (Preliminary)
This preliminary will be done to test out that indeed there is a noticeable change of the properties of the rubber ball at different temperatures. This will be done by conducting the planned method at the highest and lowest temperatures and investigating its variables. The aim of this investigation is to see how temperature effects different balls and why. The effects of temperature will be established using the decay constant and the coefficent of restitution that will be found from my results using the H = H0 e-λb (decay constant) and speed of seperation/speed of approach = e (the coefficient of restitution). Ofcourse the decay constant and the coefficient of restitution willnot be dependant on each other as the decay constant is a measure of energy and e a measure of momentum Depending on this preliminary I will decide whether or not a rubber ball is suitable for this experiment.
Rubber is made up of linear polymer chains with some cross links. This means that rubber is flexible because the chains can easily slide past each other. Rubbers are ‘elastomers’ – polmers with elastic properties.
Elastomers can be stretched considerably and still return to their original lengths when the stresses are removed. They are linear chain polymers with a degree of cross links between the chains. The cross-links are few enough to allow the chains to slide past each other during stretching, but numerous enough to pull the chains back into position once the stress is removed. Thus it is the presence of the cross-links which accounts for the elasticity. Their high elasticity is due to the fact that the chains are tangled and have relatively large amounts of space between them, so stretching them simply means stretching out the chains.
The raw rubber (latex) with which the squash ball is made of is an elastomer. But it has so few cross-links that doesn’t recoil too easily. To make squash balls, this raw rubber undergoes a process called vulcanisation. This is when the rubber is heated together with sulphur; the sulphur atoms bond covalently with the carbon atoms to form extra cross links between the linear chains, thus producing a more useful material. If large amounts of sulphur are added, the number of cross-links become so great that it becomes rigid. The rubber of this ball is soft and 4% sulphur.
Rubber is an amorphous polymer, in which its chains criss-cross in a random way like tangled strands of spaghetti, these long polymers have freely rotating bonds, that can be coiled into innumerable shapes, all having the same energy.
As shown rubber stretches but returns to its original shapedue to it’s molecular memory. However the rubber ball is empty inside and so this lack of pressure means the rubber ball is not instantaneously returned to it’s original shape.
The air inside the rubber ball maintains its turgidity of the ball. As the temperature of the ball is increases the energy of the molecules in the ball have an increasing amount of energy and so excert a greater force from the inside of the ball increasing the pressure within the ball. Therefore the return height reached by the ball will increase and so the decay constant will decrease.
Method
The rubber ball will be dropped from a height of 1m from the ground. This height will be calibrated with spirit measures and right angled traingles placed at the bottom to keep the metre rule perpendicular to the ground. This rule will be sellotaped to a shorter table so that it is held upright.
To make sure that I do not send the ball down with a velocity i.e/ throw the ball, the ball will be placed on top of the metre ruler and will fall down due to its lack of balance. The ball will hit the wooden surface of the level floor which must be at 20-25°c. A Dr Daq system measures the time between each bounce by the sound produced by the ball. The Dr Daq microphone will pick up these sounds inducing a voltage which will produce a voltage against time graph on the picoscope programme on the laptop.The The Dr Daq system will be stopped after each experiment when a good number of sound readings have been produced. Then using my cusor and lining up a line at the very beging of each peak, the computer will calculate the time difference between each bounce.
Here I’ll be testing the difference of the rubber ball at around -15°C to 60°C. The rubber ball was cooled to -15°c using a freezer, a thermometer measuring the temp to the 0.1°c accuracy so a ± 0.05°c error with range of -20°c to 110°c. will also be kept inside the freezer measure the exact temperature of the ball’s surroundings. The ball will be heated up to 60°c using a water bath. This ball must be kept dry, however, to avoid absorption and so will be kept in a plastic bag along with a metal weight . The metal wieght will keep the ball under water so that successful temperature change is achieved, the weight will also have to be metal, as metal is a ver.y good thermal conductor, transferring as much heat as possible to the rubber ball from the water bath. The same thermometer as before will of course be also within the water bath to measure the temperature.
To ensure the ball is at the temperature desired when conducting the experiment the ball was left at these temperatures for at least half an hour.
The time between the bounces is used to work out the height reached by the ball between each bounce, using the equation height = -0.5 x -9.81 x(t/2)2, however this value is useless when plotted so using the decay constant: H = Hoe-^t the graph of
Ln H/Ho against bounce number is drawn so that the decay constant may be found. The experiment at each temperature will be repeated three times. The graph will be made up of the highest , lowest and average vales produced, to show the maximum margin of error and whether or not the Dr Daq equipment is accurate enough. The results are on the following pages.
Implementation of results
Although the readings of the experiment with the ruber ball at -17°c may not be very many. The do tell us a lot.
- as expected the coeffieent of restitution of the rubber ball at -17°c is 0.264, whereas at 63°c it is 0.694. The closer to +1 the coefficient is the more elastic the collision therefore the bouncier the ball is.
-
Although a graph cannot be constructed for the rubber ball at -17°c, the decay constant (gravity) will obviouly be a lot higher. This can be predicted due to the fact that the In H/H0 will reach the value of –2.66 at the first bounce producing a very steep line and so a much larger gradient and hence a much larger decay constant.
These readings are as expected, as at higher temperatures energy is supplied to the ball, making it’s molecules vibrate more violently making the ball bouncier.
In order for a solid material to be deformed, work has to be done on it. For that work to be done energy must be expended (in the case of a squash/rubber ball, it bounced onto the floor). Some of this energy is dissipated (as heat, etc), but some is stored in the defromed material and is released when the material relaxes. The extent to which a material stores energy under deformation is called ‘resilience’. Some materials (like sprung steel) store a lot of energy and are described as having high resilience; others (like putty) store very little and therefore have low resilience.
The squash ball , being made of a rubber compound, is of fairly low resilience. Unfortunately, the lower the the resilience of an object, the higher the proportion of energy used in deforming it must be dissipated. When the squash ball is heated up, it has two effects on the ball: the air inside the ball (which was originally at normal atmospheric pressure) effectively becomes ‘pressurised’, and the rubber compound from which the ball is made becomes more resilient. For both these reasons the ball bounces higher. Obviously, the ball does not continue to indefinitely become more and more resilient with higher temperatures. Eventually equilibrium is reached where the initial temperature of the ball at the beginning of each experiment is equal to the heat gained from deformation; this temperature I have decided to define as the ‘equilibrium of resilience’. The rubber ball’s manufacturer, WSF has calculated their rebound resilience specification to be at 45°C (which I’ll be proving). Which also explains why squash balls are designed to have too little resilience at room temperature and therefore why they need warming before play.
According to WSF’s specifications however, this equilibrium of resilience depends on the temperature of the room and the height at which the ball is dropped at; or in their words “the temperature of the court and the ability of the players”. The point at which the ball temperature reaches equilibrium is in fact an excess over the ambient temperature of the surroundings. So dye to WSF, if the room is at only 5°C then the ball may only reach 35°C.
From this preliminary I can see that Rubber will give me consistent readings and the resilience of rubber will be explored further.
The Effect of Temperature on the
Silicone Ball (Preliminary)
Silicone polymers are what bouncy balls are made out of. Silicone is a polymer made by addition polmerisation:
Silicone as a monomer Silicone as a polymer
An alternating silicon-oxygen atom pattern form the backbone.
Polymers are produced when molecules within a small molar mass join up to become molecules with very large molecular masses. Silicone dioxide (silica) is made up of silicon atoms and oxygen atoms held together by covalent bonding (the sharing of electrons between two nuclei). Where every Silicon atom is linked to 4 Oxygen atoms, and in turn every oxygen atom is linked to 2 Silicon atoms, this produces a giant covalent structure, another amorphous polymer.
This amorphous polymer is also held together by intermolecular forces betweem the closely packed molecule chains. These intermolecular forces just keep the structure inact and gives the structure its high melting and boiling points.
These polymers exist in coiled and tangled states under the standard conditions of 25°c and 1 atmosphere pressure. This allowes the ball to bounce as these polymers uncoil and untangle when a force is applied ie. when the ball is in contact with the floor. In order for this collision to be elastic however these coils must only deform temporarily and so the ball bounces as the polymer chains recoil and tangle up again working against the forces exerted on them using the intermolecular fores between them. I will conduct this prelimnary experment using the following method.
The silicone ball will be dropped from a height of 1m from the ground. This height will be calibrated with spirit measures and right angled traingles placed at the bottom to keep the metre rule perpendicular to the ground. This rule will be sellotaped to a shorter table so that it is held upright.
To make sure that I do not send the ball down with a velocity i.e. throw the ball, the ball will be placed on top of the metre ruler and will fall down due to its lack of balance. The ball will hit the wooden surface of the level floor which must be at 20-25°c. A Dr Daq system measures the time between each bounce by the sound produced by the ball. The Dr Daq microphone will pick up these sounds inducing a voltage which will produce a voltage against time graph on the picoscope programme on the laptop.The The Dr Daq system will be stopped after each experiment when a good number of sound readings have been produced. Then using my cusor and lining up a line at the very beging of each peak, the computer will calculate the time difference between each bounce.
Here I’ll be testing the difference of the silicone ball at around -15°C to 60°C. The silicone ball was cooled to -15°c using a freezer, a thermometer measuring the temp to the 0.1°c accuracy so a ± 0.05°c error with range of -20°c to 110°c. will also be kept inside the freezer measure the exact temperature of the ball’s surroundings. The ball will be heated up to 60°c using a water bath. This ball must be kept dry, however, to avoid absorption and so will be kept in a plastic bag along with a metal weight . The metal wieght will keep the ball under water so that successful temperature change is achieved, the weight will also have to be metal, as metal is a ver.y good thermal conductor, transferring as much heat as possible to the silicone ball from the water bath. The same thermometer as before will of course be also within the water bath to measure the temperature.
To ensure the ball is at the temperature desired when conducting the experiment the ball was left at these temperatures for at least half an hour.
The time between the bounces is used to work out the height reached by the ball between each bounce, using the equation height = -0.5 x -9.81 x(t/2)2, however this value is useless when plotted so using the decay constant: H = Hoe-^t the graph of
Ln H/Ho against bounce number is drawn so that the decay constant may be found. The experiment at each temperature will be repeated three times. The graph will be made up of the highest , lowest and average vales produced, to show the maximum margin of error and whether or not the Dr Daq equipment is accurate enough. The results are on the following pages.
Using this method the readings of my experiment are as follws: As expected the decay constant of the silicone bale is higher at -17°c with 0.269 than that of the silicone ball at 63% with 0212. The explantion for this diffeence in decay constant is that the silicone ball is made up of coiled and tangled polymers held togther with the addition of intermolcular forces. As the ball is in contact with the ground, the ground exerts a force on the ball untangling and uncoiling the polymer. The intermolcuat forces in between the polymers in the silicone ball then act against this force causing the polymers to recoil and work against the force. As seen from my preliminary results, as the temperature of the ball is increased so is the energy of the ball making the polymers recoil much quicker. This extra energy mades the polymers more flexible. Considering that the polymers recoil faster, then less energy is transferred to the table supplying the ball with a large portion of kinetic energy to by converted into potential energy as the ball seperates from the floor.
Gravitational Potential Energy= mass x gravity x height
so the more gravitational potential energy the ball has the greater the bounce return height will be, as gravitational potential energy is proportional to height. These results are also backed up by the coefficient of restitution being greater at 63°C with 0.889 than at -17°c with 0.86. Though this difference in the coefficient of restitution is not great it proves that with increasing temperature in the ball the greater the energy in the ball to recoil and so the greater the gravitational potential energy that becomes stored, so the greater the return bounce height recieved and so the coefficient increases. This proves that the colision becomes more elastic with temperature as the deformations of the ball become less and less permanent.
Having shown that consistent enough results can be obtained from the silicone ball, I will continue with it in my proper experiment.
The Effect Of Temperature
On A Rubber Ball
Method
The rubber ball will be dropped from a height of 1m from the ground. This height will be calibrated with spirit measures and right angled traingles placed at the bottom to keep the metre rule perpendicular to the ground. This rule will be sellotaped to a shorter table so that it is held upright.
To make sure that I do not send the ball down with a velocity i.e/ throw the ball, I’ll have my wrist leaning on a hard stationary surface 1m above ground level. The ball will hit the wooden surface of the level floor which must be at 20-25°C. A wooden surface was found best as it didn’t absorb too much energy from the ball, allowing a sufficient amount for the ball to bounce back up again and enough to produce sound at each bounce loud enough for the Dr Daq system to pick up but not too loud to obscure readings between bounces.
A Dr Daq system measures the time between each bounce by the sound produced by the ball. The Dr Daq microphone will pick up these sounds inducing a voltage which will produce a voltage against time graph on the picoscope programme on the laptop. The Dr Daq system will be stopped after each experiment when a good number of sound readings have been produced. Then using my cusor and lining up a line at the very beging of each peak, the computer will calculate the time difference between each bounce.
Here I’ll be testing the difference of the rubber ball from around -15°C to 60°C. The rubber ball will be cooled to -15°c using a freezer, a thermometer measuring the temp to the 0.1°c accuracy so a ± 0.05°c error with range of –20°c to 110°c. will also be kept inside the freezer to measure the exact temperature of the ball’s surroundings. The experiments will be carried out at 0°C, room temperature, 40°C, and at 80°C.The ball wil be cooled down to 0°C using an ice bath. The squash ball will be heated up to 40°C, 60°C and 80°C using a water bath placed on top of a flame. A water bath was used as it had been proven perfect from the preliminary. This ball must be kept dry, however, to avoid absorption and so will be kept in a plastic bag along with a metal weight . The metal wieght will keep the ball under water so that successful temperature change is achieved, the weight will also have to be metal, as metal is a ver.y good thermal conductor, transferring as much heat as possible to the rubber ball from the water bath. This plastic bag will be kept in yet another plastic bag to avoid any absorbtion whatsoever as rubber becomes brittle when water is absorbed or when subjected to a flame directly.The same thermometer as before will of course be also within the water bath to measure the temperature. The wider the range of temperatures the better
To ensure the ball is at the temperature desired when conducting the experiment the ball was left at these temperatures for at least half an hour.
The time between the bounces is used to work out the height reached by the ball between each bounce, using the equation height = -0.5 x -9.81 x(t/2)2, however this value is useless when plotted so using the decay constant: H = Hoe-^t the graph of
Ln H/Ho against bounce number is drawn so that the decay constant may be found. The experiment at each temperature will be repeated three times. The graph will be made up of the highest , lowest and average vales produced, to show the maximum margin of error and whether or not the Dr Daq equipment is accurate enough. The results are on the following pages.
Calibration check
The Dr Daq system works using a crystal which is very accurate and extremely sensitive and therefore is in no need of a calibration check. However, a calibration check was necessary on the alcohol thermometer I was using. First I placed the thermometer in boiling water and checked the reading. After a certain amount of time with the thermometer at room temperature the thermometer was then placed in a cup of ice and the thermometer reading again checked. The thermometer was meant to be at 100°C in the boiling water except the thermometer only reacheed a temperature between 96°C and 97°C. In the ice the thermometer reached a temperature between -3°C and 4°C rahter than 0°C. This is rather odd as the results of a thermometer from Selfridges is meant to be well calibrated as one would believe, however due to this calibration check the thermometer is faulty by –3 to –4 degrees. Meaning that every temperature reading I have, 3.5°C is meant to be taken away to achieve the true temperature reading.
The picoscope programme shows it’s readings to the nearest millisecond. Giving a ±0.5 x 10-3second (s) error.However as my readings of time were from two different points, each one from the beginning of each peak or trough, this means that there were two ±0.5ms errors making the total error for each reading taken ±1ms. When the values are converted to seconds ie. half the times of between each bounce, the error becomes ±0.5 x 10-3s. and when this is converted to height (0.5 x 10-3)2 = 0.25 x 10-6s and as the heights shown on the tables are only to 9 d.p.; this error is lost, ie. too small and so too insignificant to the results obtained. This error will obviously be greater when the height is at it’s greatest due to error build up.
Usually when squared error would be doubled:
For example; 100 ± 2% = 100 ± 2
If squared; (100 ± 2)2 = (100 ± 2) x (100 ± 2)
It becomes; 10000 ± 2x100 ±2x100 +4 = 10404
The 4 is insignificant and so might as well be ignored.
Overall it equals; 10000 ± 4%
Showing that the error has been doubled however it is still too small to be plotted.
Considering the greatest value of ie. the first bounce of the silicone ball at 40°C the maximum value of time of the first bounce is 0.4005s which produces a height of 0.78578 which has only a difference in 0.2%, as this error is too small to be plotted , error bars according to this finding alone cannot be added.
The fact that the ball was left to drop off the metre rule due to it’s own lack of balance could have effected my results completely, so I had decided to revise this method and change it. The reason for the effect is that when the ball loses it’s balance it would have to roll off the top of the metre rule giving the balls velocity before they drop off the side and so the initial velocity at which the balls are dropped at would not be zero. I therefore decided that it would be wiser if I dropped the ball my self, as whther or not I add velocity to the ball or not the ball will bounce closer to the same spot every time than if it were to roll off the ruler. As when the ball roles off the ruler the calculations made would have to consider the angles to the horizontal of each bounce and so I would have to use projectile motion calculations to minimise error when working out the height reached. Using projectile motion calculations would be almost impossible to use as I would not know the readings needed to make the calculations too well and so overall allowing the ball to roll off the metre rule would increase the error a lot more than simply dropping the ball my self.
Dropping the ball myself would still have the certain error ofcourse as any movement I do other than dropping the ball would send the ball down with a velocity. Leaning my wrist onto a hard stationary surface 1m above the ground can reduce this risk considerably. However, overall whether or not I send the ball down with a speed or not it will be picked up by the results as error will then build up and there will be a considerable difference between each set of results, although the error of my dropping the ball is too uncertain to use to plot error bars.
Results
My reruns for the experiment at -17°C and 63°C were so similar that I decided to just use the same results as my preliminary. However my results are as follows.
Indeed there is a relationship between the decay constant and coefficient of restitution due to my results for the experiments with the rubber ball. This is shown by the fact that the squash ball had the lowest decay constant at 60°C with a value of 0.636 and so the largest number of bounces were produced at this temperature. The coefficient of Restitution was 0.694 which shows that at 60°C the rubber ball was at its bounciest. The decay constant decreases sufficiently from -17°C to 63°C and the coefficient of restitution increases sufficiently from -17°C to 63°C showing that there is an increasing bounciness and number of bounces with increasing temperature.
When the ball hits the surface this exerts a force on these polymer chains stretching them out. The Intermolecular forces between the polymer chains then act against this force to ‘re-entwine’ the polymer chains. At -17°C only one reading was taken showing very large decay constant. This is when the intermolecular forces had the least amount of energy and so not enough to re-entwine the polymer chains together quickly enough and so a lot of the energy was absorbed by the surface, dramatically reducing the return height reached by the ball. This is also considered when the coefficient of restitution is shown to be at the lowest at this temperature. Also the air molecules within the ball would have such a low amount of energy , therefore a low amount of kinetic energy and so move with a small speed; not creating enough pressure inside the ball for it to bounce properly. The coefficient of restitution being only 0.264 shows that this was more of an inelastic collision than an elastic one as the coefficient of restition is the measure of the elasticity of a bounce. When a bounce is inelastic it means that some kinetic energy is lost rather than simply being converted to sound and / or heat; this coefficient of restitution shows that the rubber ball has been permanently defomed or that the kinetic energy of the ball has significantly decreased after the collision compared to the amount of energy possessed by the ball before the collision.
The rubber is made of amorphous polymers, when a force is applied to this polymer the intermolecularforces act against this force with assisstance from the polymer chain’s ‘molecular memory’. At very low temperatures this molecular memory is lost and the polymer chains deformed. When the intermolecular forces don’t have enough energy to re-entwine the polymers then are no longer free rotating and can only recoil into certain shapes, explaining why only one clear bounce was achieved at -17°C.
Due to WSF specifications the equilibrium of resilience of the squash ball was meant to be at 45°C, however at 63°C the coefficient of restitution was higher than at 45°C showing a still increasing resilience with temperature beyond 45°C.The reason for this lack of evidence may have been due to the fact that I had carried out the experiment at -17°C first which may have obscured my results as I used the same ball for every experiment although the ball was left at each temperature for at least half an hour to make sure the ball was at that temperature when used in the experiment. Or Perhaps if I had explored an even wider range of temperatures for this experiment I may have found that the equilibrium of resilience may have just been higher than what was suggested by WSF.
The fact that I used the same ball could have effected my results. As an attempt to check this fault I waited about four days before conducting the experiment at 40°C and 63°C , however I still got very similar results, that were almost identical to the others. Perhaps the best way I could have checked this fault was to test out each temperature with a different ball each time, however this again may give rise to certain errors.
I had decided to plot on my graphs the ranges between the highest, lowest and average temperatures to show the certain accuracy of each experiment. This has managed to prove useful. From my previous analysis I have shown that the error is too small to plot and so error bars cannot be produced, however, this would mean that the lines of best fit would have to pass through every point, which they don’t showing that there is indeed error. From my graphs of the experiment with rubber, my ranges between thehighest, average and lowest seem to increase with every bounce. This implies error build up, showing that some erroor took place that had no effect on the first bounce but built up with each bounce. This error build up could have been mainly due to how I dropped the ball for even though my wrist was leant against a stationary surface I still could have made a movement other than dropping the ball sending it down with a velocity, this then lead to the error build up.
I have had to increase and sometimes even double the range between the highest, average and lowest result to produce error bars for which the lines of best fit can go through. This is very suitable as the results gained are only a sample; ie. if I were to repeat the experiment 30 times ofcourse the range between the highest, lowest and average readings would increase.
However, I believe that my results are cosistent enough and reliable enough to conclude that between the increasing temperatures of -17°C and 63°C there is a general increase in coefficient of restitution and a decrease in the decay constant. My results could have been even more consistent if there was a way in which I could tell the actual temperature of the ball as the plastic bags could have insulated some of the heat from the ball. Also if there was a way of minimising error by finding a way to ensure the ball was being sent down with a certain velocity of zero and to make sure that the ball bounced on the or close to the same spot every time ie. followed the perfect bounce path set for the model world.
Having only four different temperatures hasn’t proven enough. That is what I see as my most major flaw in the experiment, yet the reason for this limit in the number of different temperatures is that the temperature at 70°C or beyond is very hard to maintain the ball at, as the readings given by the ball when it is at a temperature beyond 63°C had such large ranges between the highest, lowest and average that they were too unreliable. Having tried out the experiment at temperatures above 63°C many times and failed I then could not repeat the experiment at 0°C to get better results due to the lack of time left. However I didn’t feel that it was that necessary to have readings at 0°C as it was obvious that it would just follow the general trend between –17°C and 23°C.
If I were able to approch a method of heating that would successfuly heat the ball to temperatures above 63°C and have the ball maintain the temperature. It would have been interesting to see the ball’s reaction to for example temperatures of 140°C and test for example whether or not the rubber would become brittle and maybe harden and break apart. Other conditions could have also been experimented with, ie. the atmosphere pressure, the saturaton of the ball, or maybe even saturation in certain chemicals such as sulphur to try and further vulcanise the rubber. I could have also tried such experiments with different manufactures to find out the difference between each manufacturer’s type of ball like for example each ball’s equilibrium of resilience.
The Effect of Temperature on
the Silicone Ball
Method
The silicone ball will be dropped from a height of 1m from the ground. This height will be calibrated with spirit measures and right angled traingles placed at the bottom to keep the metre rule perpendicular to the ground. This rule will be sellotaped to a shorter table so that it is held upright.
To make sure that I do not send the ball down with a velocity i.e/ throw the ball, I’ll have my wrist leaning on a hard stationary surface 1m above ground level. The ball will hit the wooden surface of the level floor which must be at 20-25°C. A wooden surface was found best as it didn’t absorb too much energy from the ball, allowing a sufficient amount for the ball to bounce back up again and enough to produce sound at each bounce loud enough for the Dr Daq system to pick up but not too loud to obscure readings between bounces.
A Dr Daq system measures the time between each bounce by the sound produced by the ball. The Dr Daq microphone will pick up these sounds inducing a voltage which will produce a voltage against time graph on the picoscope programme on the laptop. The Dr Daq system will be stopped after each experiment when a good number of sound readings have been produced. Then using my cusor and lining up a line at the very beging of each peak, the computer will calculate the time difference between each bounce.
Here I’ll be testing the difference of the rubber ball from around -15°C to 60°C. The rubber ball will be cooled to -15°c using a freezer, a thermometer measuring the temp to the 0.1°c accuracy so a ± 0.05°c error with range of –20°c to 110°c. will also be kept inside the freezer to measure the exact temperature of the ball’s surroundings. The experiments will be carried out at 0°C, room temperature, 40°C, and at 80°C.The ball wil be cooled down to 0°C using an ice bath. The squash ball will be heated up to 40°C, 60°C and 80°C using a water bath placed on top of a flame. A water bath was used as it had been proven perfect from the preliminary. This ball must be kept dry, however, to avoid absorption and so will be kept in a plastic bag along with a metal weight . The metal wieght will keep the ball under water so that successful temperature change is achieved, the weight will also have to be metal, as metal is a ver.y good thermal conductor, transferring as much heat as possible to the rubber ball from the water bath. This plastic bag will be kept in yet another plastic bag to avoid any absorbtion whatsoever as rubber becomes brittle when water is absorbed or when subjected to a flame directly.The same thermometer as before will of course be also within the water bath to measure the temperature. The wider the range of temperatures the better
To ensure the ball is at the temperature desired when conducting the experiment the ball was left at these temperatures for at least half an hour.
The time between the bounces is used to work out the height reached by the ball between each bounce, using the equation height = -0.5 x -9.81 x(t/2)2, however this value is useless when plotted so using the decay constant: H = Hoe-^t the graph of
Ln H/Ho against bounce number is drawn so that the decay constant may be found. The experiment at each temperature will be repeated three times. The graph will be made up of the highest , lowest and average vales produced, to show the maximum margin of error and whether or not the Dr Daq equipment is accurate enough. The results are on the following pages.
Calibration check
The Dr Daq system works using a crystal which is very accurate and extremely sensitive and therefore is in no need of a calibration check. However, a calibration check was necessary on the alcohol thermometer I was using. First I placed the thermometer in boiling water and checked the reading. After a certain amount of time with the thermometer at room temperature the thermometer was then placed in a cup of ice and the thermometer reading again checked. The thermometer was meant to be at 100°C in the boiling water except the thermometer only reacheed a temperature between 96°C and 97°C. In the ice the thermometer reached a temperature between -3°C and 4°C rahter than 0°C. This is rather odd as the results of a thermometer from Selfridges is meant to be well calibrated as one would believe, however due to this calibration check the thermometer is faulty by –3 to –4 degrees. Meaning that every temperature reading I have, 3.5°C is meant to be taken away to achieve the true temperature reading.
The picoscope programme shows it’s readings to the nearest millisecond. Giving a ±0.5 x 10-3second (s) error.However as my readings of time were from two different points, each one from the beginning of each peak or trough, this means that there were two ±0.5ms errors making the total error for each reading taken ±1ms. When the values are converted to seconds ie. half the times of between each bounce, the error becomes ±0.5 x 10-3s. and when this is converted to height (0.5 x 10-3)2 = 0.25 x 10-6s and as the heights shown on the tables are only to 9 d.p.; this error is lost, ie. too small and so too insignificant to the results obtained. This error will obviously be greater when the height is at it’s greatest due to error build up.
Usually when squared error would be doubled:
For example; 100 ± 2% = 100 ± 2
If squared; (100 ± 2)2 = (100 ± 2) x (100 ± 2)
It becomes; 10000 ± 2x100 ±2x100 +4 = 10404
The 4 is insignificant and so might as well be ignored.
Overall it equals; 10000 ± 4%
Showing that the error has been doubled however it is still too small to be plotted.
Considering the greatest value of ie. the first bounce of the silicone ball at 40°C the maximum value of time of the first bounce is 0.4005s which produces a height of 0.78578 which has only a difference in 0.2%, as this error is too small to be plotted , error bars according to this finding alone cannot be added.
The fact that the ball was left to drop off the metre rule due to it’s own lack of balance could have effected my results completely, so I had decided to revise this method and change it. The reason for the effect is that when the ball loses it’s balance it would have to roll off the top of the metre rule giving the balls velocity before they drop off the side and so the initial velocity at which the balls are dropped at would not be zero. I therefore decided that it would be wiser if I dropped the ball my self, as whther or not I add velocity to the ball or not the ball will bounce closer to the same spot every time than if it were to roll off the ruler. As when the ball roles off the ruler the calculations made would have to consider the angles to the horizontal of each bounce and so I would have to use projectile motion calculations to minimise error when working out the height reached. Using projectile motion calculations would be almost impossible to use as I would not know the readings needed to make the calculations too well and so overall allowing the ball to roll off the metre rule would increase the error a lot more than simply dropping the ball my self.
Dropping the ball myself would still have the certain error ofcourse as any movement I do other than dropping the ball would send the ball down with a velocity. Leaning my wrist onto a hard stationary surface 1m above the ground can reduce this risk considerably. However, overall whether or not I send the ball down with a speed or not it will be picked up by the results as error will then build up and there will be a considerable difference between each set of results, although the error of my dropping the ball is too uncertain to use to plot error bars.
Results
The results of this experiment are surprising, they don’t follow the pattern of the preliminary where the decay constant simply decreases with temperature as the coefficient of restitution simply increases with temperature formulating the simplest explanation of activity increasing with energy input. From my further experiment the pattern becomes a lot more complicated as a lot more explanation is needed for these, here results.
The decay constant seems to be the lowest at 7°C showing that at 7°C the ball bounces for the longest time at 7°C. This is odd as this does not at all follow the simple trend of my preliminary experiment.I therefore plotted a graph showing the decay constant against temperature as well as a coefficient of restitution graph against temperature, just to show my results a little more clearly.
From these new graphs I can see that the silicone ball does not continue indefinitely heat up. Eventually equilibrium is reached where the initial temperature of the ball at the beginning of each experiment is equal to the heat gained from deformation; this temperature I have decided to define as the ‘equilibrium of resilience’. So the resilience is highest when this equilibrium is reached, indeed with the silicone ball the ball has the highest coefficient of restitution at room temperature which would make sense as this is the main temperature the ball will be played at.
The decay constant increases as the temperature increases from 7°C to 40°C this means that too much energy is is being supplied to the ball so as the temperature increases ie.when the ball hits the table at higher temperatures, the monomers within the polymer chains have so much energy that they may shift slightly out of shape rather then just untangle making it harder for the the polymer chains to re-entwine again and so the ‘molecular memory’ is. Meaning that when the silicone ball polymers uncoil as they hit the ground the intermolecular forces which counteract this force don’t work as fast as they should due to their now mishapen polymers. The longer it takes for the polymer chains to re-entwine the the more energy trnasferred to the table and so the lessthe energy converted back to potential energy, therefore the return height of the ball is decreased as its proportional to the potential energy. This also explains the reason for the coefficient of restitution to decrease as the temperature increases from 20°C to 63°C as the collisions become more inelastic as the temperature increases due to the almost permanent deformations experienced by the silicone ball monomers.I would have liked to see the effect of temperatures higher than 63°C on the ball, however readings given by the ball when it is at a temperature beyond 63°C had such large ranges between the highest, lowest and average that they were too unreliable. What is shown here is imprecise and doubtful. When a method of heating that would successfuly heat the ball to temperatures above 63°C and have the ball maintain the temperature is approched, given enough time more experiments from the temperatures 20°C to 80°C have to be carried out to distinguish an exact behaviour relationship at these temperatures as the. I would have preffered to have smaller ranges between each of mty experiments but due to the restricted time I had this was not possible.
The decay constant is by far at its highest at -17°C as this is close to the glass temperature. The glass temperature is the temperature below which the movement of molecules is severely restricted due to the lack of energy being supplied to the intermolecular forces to counteract the force excerted on the ball when it hits the surface and so when the ball hits the ground it shatters. the ball doesn’t even come close to shattering as the molecules still moved enough for the ball to produce a 0.269 coefficient of restition however I’m only implying that a little further by maybe another 50°C and I may have reached the actual glass temperature. This is also mirrored by the coefficient of restitution being the lowest at -17°C showing that energy is indeed not conserved by the ball due to the restricted movement of the polymers at this temperature.
The decay constant decreases as the temperature increases from 40°C to 63°C it was at first considered an anomaly however my results for the experiment at 63°C were almost completely accurate as in the ranges between the highest, lowest and average were almost non-existent. The only explanation for this is that the more energy supplied to the polymer chains, making them more stretchy so the intermolecular forces re-entwine the chains quicker. The quicker the chains re-entwine the more energy conserved by the ball and so the higher its return bounce, so the higher thenumber of bounces achieved and so the lower the decay constant. However this then contradicts my increasing decay constant between 7°C and 40°C which are equally as accurate. Further experiments must be carried out from 40°C to 80°C to find out exactly the changes in the decay constant. Extra experiments must also be carried out between the temperatures -17°C and 23°C to find out the exact changes in the decay constant.
I have repeated the experiments with the silicone ball twice with an interval of at least a day inbetween, and the result were still almost identical I did however put all my data of the experiments the first time round and the second time round to show the truly highest and lowest readings achieved.
I had decided to plot on my graphs the ranges between the highest, lowest and average temperatures to show the certain accuracy of each experiment. This has managed to prove useful. From my previous analysis I have shown that the error is too small to plot and so error bars cannot be produced, however, this would mean that the lines of best fit would have to pass through every point, which they don’t showing that there is indeed error. I have had to increase and sometimes even double the range between the highest, average and lowest result to produce error bars for which the lines of best fit can go through. This is very suitable as the results gained are only a sample; ie. if I were to repeat the experiment 30 times ofcourse the range between the highest, lowest and average readings would increase.
However, my results are consistent enough for me to say that silicone is a complicated material. The ball seems to be at its bounciest and best at room temperature, yet produces the most number of bounces at 7°C. -17°C is the temperature with the highest decay constant by far and the lowest coefficient of restitution, showing that it’s close to the glass temperature where the movement of the molecules is almost completely restricted.
It would have been interesting to see the ball’s reaction to temperatures above 63°C would have been interesting to see whether or not a regular trend is ever achieved. Other conditions could have also been experimented with, ie. the atmosphere pressure, the saturaton of the ball, or maybe even the saturation of sodium silicate in certain chemicals such as different concentrations of ethanol under certain conditions as an attempt to increase the number of cross-links in the sodium silicate, therefore testing out the effect of the increase of temperature on the bounciness with increasing rigidity.
Investigating the Decay Constant, the Coefficient of Restitution
and a possible relationship.
(From before;)
Though the coefficient of restitution and the decay constant will be achieved from analysing the same bit of data they will not be dependant on each other ie. If the ball has a low decay constant at a certain temperature it doesn’t necessarily mean that the coefficient of restitutuin will be high at that temperature (as seen from the results of my experiment with the silicone ball). This is true because the decay constant looks at height, which looks at gravity and so is a measure of energy (considering G.P.E.= mgh). The coefficient of restitution however looks at the velocity before and after a collision and so is a measure of momentum [considering momentum (p)=mass (m) x velocity (v)]. Energy, (especially kinetic and so gravitational as K.E. + P.E. = a constant,) is not always conserved in a collision especially when the collision is inelastic, however momentum is always conserved before and after a collision whether the collision is inelastic or not.
If; G.P.E. = mgh and p=mv
Then; EG=mgh
So; EG x v = mvgh = pgh
And so; EG = pgh
v
Or as; EG + EK = ETOTAL
Then; Ek x m = 1/2 m2v2 = 1/2 p2
And so; Ek = . p2 .
2m
However this part shows that there must be a relationship between the coefficient of restitution and the decay constant. Evidence of this is shown in my results from the experiment with rubber as the graph of the coefficient of restitution against temperature seems to be dependant on the graph of the decay constant against temperature.
The coefficient of restitution of the rubber shows a positive linear relationship when plotted against temperature, however having only four values gives rise to XX% error.
The decay constant has an exponential relationship with temperature, however Ln(DC) has a linear relationship when plotted against temperature; yet again having only four values gives rise to XX% error.
Now if; 1/2mv2 = mgh
v2 = 2gh
So that; v2 ∝ h
Then e can also equal; e = √(H/Ho) (H=h)
Therefore if; e = √(H/Ho) and λ= -In H/H0
b
Then a relationship can be developed.
As; CoR2 = H/Ho (CoR=e)
And; -ln CoR2 = -ln H/Ho
b b
Then; -ln (CoR)2 = λ
b
I have plotted a graph (with error XX%) of;
ln (CoR)2 = -b
λ
to find out exactly what b is. From the equations I have rearranged, b is meant to stand for the number of bounces, this could mean the maximum or average number of bounces achieved at each tamperature. However my graph of ln (CoR)2 against λ (graph –b) has a positive gradient meaning that the overal b value is negative, it therefore cannot stand for the number of bounces. Another alternative is that b could just be a constant with another meaning other than bounce number. So some kind of relationship is being shown by graph (-b) however what it is is not quite understandable. Perhaps if I’d had more than four results to work from I may have come up with a more solid conclusion, however due to the lack of time and suitable apparatus this was not possible.
As shown that both the coefficient of restitution and the decay constant are dependant on temperature in some way it is then too wishful to find a true relationship between the coefficient of restitution and decay constant when the temperature has been discarded completely in graph (-b). In conclusion the readings from the experiment with the rubber show that there is some kind of relationship between the decay constant and the coefficient of restitution as this is shown when ln (CoR)2 is plotted against the decay constant. However, I’m limited in finding out exacly what the relationship is.
