Practical investigation on teminal velocity of a sphere in oi

Physics CAT One Extended Practical Investigation Report Student Number: Purpose The Purpose of this investigation is to explore how the terminal velocity of a sphere falling through glycerol varies with the temperature of the glycerol and the size of the sphere. Introduction In the early stages of the project it was intended to investigate how the speed of a sphere falling through glycerol varies with the size of the sphere. However, after analysis it was decided that the investigation would be more callenging if a second variable was incorporated. There are many constants that could have been manipulated such as, amount of glycerol used, distacnce over which times were taken, distance sphere was allowed to fall before timing was taken and the temperature of the glycerol. After much consultation it was decided that the temperature of the glycerol should be varied. Once this had benn incorporated into the investigation some scientific concepts related to the viscosity of a liquid had to be attained. (Refer to article). In conducting the experiments an attempt was made to attain results that could, produce graphs that showed the terminal velocity of a sphere related to the temperature of the glycerol and the terminal velocity of a sphere related to its size. Apparatus used • 600 ml of glycerol (density 1.26/ml. Assay 98.0 - 101.0%) • Small ball bearings of radius: 3.175mm 3.960mm 5.000mm 6.000mm 7.000mm • 900 ml measuring cylinder • Stop watch • Thermometer • Some type of heating and cooling device to varie the temperature of the glycerol • Tweezers Variables and Constants The variables that have been used in this investigation are the size of the ball bearings and the temperature of the glycerol. The constants that have been used in this investigation are the amount of glycerol used, the size of the measuring cylinder, the intervals at which time were taken, the distance the sphere was allowed to drop before times were taken and the number of tests taken. Method To begin experimentation the distance over which the sphere accelerates to reach terminal velocity had to be determined. This was done by systematically varying the distance over which the sphere was allowed to fall then finding the point at which the spheres acceleration is zero. It was found that for the sphere to reach terminal velocity it had to be allowed to fall 6 - 7 centimeters before an accurate, constant reading could be taken. It was found that the distance needed for a sphere to reach terminal velocity is only slightly changed when the temperature of the glycerol is varied (+/- 0.2cm). To attain that the sphere had reached terminal velocity by varying the distance that the sphere fell before timing began, the distance was varied from 2cm to 10cm. Starting at 2cm the measuring cylinder was marked at 2cm intervals and times were taken for each interval. the times taken were analysed to determine if the rate of descent of the sphere was constant for each reading. To ensure that the sphere had reached terminal velocity a full 10 cm of descent was allowed. Using 'Stokes law for the terminal velocity of a sphere falling under gravity' and the relationship of mg = U + F at terminal velocity the above result is proven. These calculations can be seen in the results section. For all experiments room temperature was recorded at 20oc. The first part of the experiment was to vary the size of the ball bearing but not the temperature. A sphere of 3.175mm in diameter was dropped from just above water level and allowed to fall 6 cm before timing began. Once the sphere had fallen the initial 6 cm timings were taken at intervals along the measuring cylinder every 200ml (10cm). This experiment was repeated 4 times and an average was taken. The experiment was then repeated using ball bearings of sizes 3.960mm, 5.00mm, 6.00mm, 7.00mm, 9.00mm. Each individual experiment was repeated 4 times and an average was taken. All results are shown in the results section. The second part of the experiment was to vary the temperature of the glycerol but not the ball bearing size. A sphere of 3.175mm was chosen to be used in all experiments, due to its extremely slow descent rate. The same procedure as above was used except five temperatures of 7oc, 12 oc, 15 oc, 17 oc and 20 oc for the glycerol were used. Results The averaged results obtained from the experiment are presented in the following tables and graphs. (For full documentation of all the results obtained refer to appendix 1.) Size of Sphere Timing Interval No Averaged Results Averaged Velocity Temperature Timing Interval No Averaged Results Averaged Velocity 3.175mm 1 2 3 1.4371.6001.637 6.178 cm/s 7oc 123 5.5758.1158.095 1.24 cm/s 3.960mm 1 2 3 0.7500.9820.970 10.240 cm/s 12oc 123 2.6504.4754.400 2.24 cm/s 5.000mm 1 2 3 0.5000.6900.680 14.598 cm/s 15oc 123 2.3753.6573.755 2.68 cm/s 6.000mm 1 2 3 0.7850.6550.627 15.600 cm/s 17oc 123 2.1252.9352.977 3.36 cm/s 7.000mm 1 2 3 0.3400.3600.360 27.777 cm/s 20oc 123 1.4801.6021.627 6.17 cm/s Chart One Note that there is a reflex error for all the recordings of +/- 0.1 seconds. Also, the first timing interval cannot be used for any calculations as the sphere has not yet reached terminal velocity. This is a graph representing how the velocity of a 3.175mm sphere varies with the temperature of the glycerol. This is a graph representing how the velocity of a sphere varies with the diameter of that sphere. Analysis of Results Chart One demonstrates that as expected the terminal velocity of the sphere increases as the temperature of the glycerol and the size of ball bearings increase. Graphs one and two visually illustrate this point and it can be seen by the positive gradient shown. It is interesting to note that the change of velocity with the temperature is signifigantly greater as the temperature becomes higher (15oc to 20.5 oc). The reason for this is directly related to the change in viscosity as the temperature is varied. As the temperature increase the viscosity becomes less and so the sphere is able to move freely through this less viscous liquid thus having a greater terminal velocity. A chart of temperatures and their relative viscosities for glycerol is shown in appendix two. A hypothetical relationship can be developed between velocity and temperature. The shape of the graph, although not smooth, is a curve and therefore it is reasonable to suggest that the relationship would invole T to the power of something: ie) v = kTn (where k is a constant). Thus, Log10v = Log10k + n Log10T, where n takes the gradient value. If a graph of Log10v vs. Log10T is plotted it may be possible to form a relationship.(Graph 3) A line of best fit for the above graph gives a gradient of 2.69. Therefore a hypothesis for the relationship between velocity and temperature is V = kT2.69. Of course for the results to be most accurate the sphere would ideally have reached terminal velocity when the times in graph three were taken. An attempt has been made to calcualte the terminal velocity at 20oc using stokes law and the relationship mg = U + F at terminal velocity so that it can be compared to the velocity found at this temperature. THIS GRAPH SHOWS HOW VELOCITY VARIES WITH TIME Refering to the graph the velocitites of the ball bearings for each temperature are shown. These results can be proven using Stoke's Law (for a detailed description of Stoke's Law and other related physics concepts refer to the article), but due word limit restrictions these calculations have been removed. From chart one a relationship between the size of a ball bearing and its velocity can also be formed. Studying graph two it can be seen that there is a gradual curve which indictes that it is reasonable to suggest that the relationship would once again involve T to power of something. Therefore a relationship could be formed using a Log-Log graph, shown below. Using a line of best fit the gradient can be found as 0.638. Therefore the relationship between the Log of Velocity vs. Log of Diameter is V = kD0.638. All discrepancies in calculations for graph five and the same as for graph three. Difficulties Difficulties encounted during this investigation are: • Trying to establish weather the sphere had reached terminal velocity before timing began. • Trying to maintain the temperature attained once the glycerol has been heated or cooled. • Human errors when timing. • Human errors in general. • Transfering the glycerol from the measuring cylinder to bottles without loosing any. • Trying to hold the ball bearings just above the glycerol without dropping them in. • Trying to perform as many tests as possible (in an effort to get a more accurate average) within the time allocated in class. Although every difficulty was hard work to around, trying to establish weather the sphere had reached terminal velocity before timing began was the main difficulty encountered. Errors % error in distance = 0.15cm x 100 = 1.5% 10cm 1 % error in time = 0.36s x 100 = 4.4% This is in regard to human error in 8.1 1 responding with the stopwatch. % error in velocity = 8% % Error in temperature = 7 x 100 = 32% This allows for a possible increase 20 1 or decrease in temperature whilst the experiment was taking place or for the chance that the thermometer wasn't calibrated correctly Error in radius = 1% This accounts for human error in 1 reading the measurements or that the radius' of the spheres used was not uniform. % Error in velocity calculations using Stoke's Law and mg = U + F = 1% Success of The Investigation The aim of this investigation was show that the terminal velocity of a sphere falling through glycerol varies with the temperature and the size of the sphere. From the results shown I believe that the investigation was a success. Conclusions As a result of this investigation it can clearly be concluded that as the temperature of glycerol increases, viscosity decreases and therefore any sphere falling through the glycerol will experience an increase in terminal velocity. Also the rate of increase in velocity is greater as the temperature rises. This is because the less viscous the state of the glycerol, the more freely the sphere is able to fall. It can also be concluded that as the diameter of the sphere increases the weight of the sphere increases and therefore its terminal velocity increases.