STAT2000 Group Project – Claire, Erin, Lauren and Sam
“The effects of wing span and number of clips on how a paper helicopter falls through space”
When an object falls through air it is subject to two different forces. One of those forces is gravity which was studied extensively by Galileo Galilei in the 16th Century. Gravity is the downward pull of an object from a force that is proportional to their weight. Galileo correctly hypothesised that objects of different masses accelerated at the same rate. However, this can only be accounted for in vacuums where there is only gravity acting on the force. In other mediums, such as air or water, there are other forces acting on the object and Galileo theorized that this may lead objects to fall at different speeds. This concept is reflected in theoretical physics as Newtons First Law of Motion: “The velocity of a body remains constant unless the body is acted upon by an external force”, where velocity is the speed of an object at any given direction. In the example of air, the other force acting upon substance is air resistance.
Air resistance or drag is when an object’s surface area interacts with air molecules, causing an upward force on the object. A larger surface area would mean the object would interact with more molecules and a smaller surface area would interact with little amounts of air. When the accelerated downward pull of gravity is concerned, the air resistance increases causing the acceleration of the object to decrease. But what happens when these objects are different weights? At a particular time, the drag force should equal the objects weight. This means that the more weight an object is, the slower it takes for a higher drag to occur, than for something smaller, causing it to fall faster. After the weight and drag have equalised, the object will then fall at a constant speed called terminal velocity where larger objects should fall slower than smaller objects as their terminal velocity should be greater.
From all these principles of motion, we were curious to see how these interacted in the area of aerodynamics, specifically in relation to helicopters. Due to limitations we have chosen to replicate the basic movements of the helicopter using paper. Hence we are testing whether the number of clips on the base of the plane and wing span or rotor length of the helicopter effects the descent time of the paper helicopter. We aim to find which length and which height keeps the helicopter in the air the longest, as this is what helicopters are designed to do. According to what was previously mentioned, the longer and heavier the paper helicopters should take longer to fall as the drag should be greater for these aeroplanes. However due to the fact we plan to test these paper helicopters at a small height, we are unable to predict beforehand whether these paper helicopters will be able to reach terminal velocity which would result in the above. If they don’t reach terminal velocity then it should mean that the lighter helicopters would be the best as drag would initiate faster and the longer rotors should enable it to stay in the air longer. Due to this complication, we will assume the null hypothesis to determine whether the number of paper clips and length of rotors on paper helicopters even affects the descent rate of paper helicopters from a low height.
Method
Apparatus and Design
Each helicopter was made from standard A4 paper, and based on the template shown in Appendix A (www.primaryscience.ie/media/pdfs/col/paper_helicopters.pdf). From the template the wing span of each of the rotors were varied (6cm, 8cm, 10cm). Each helicopter also had placed on it a variation of bull clips (one, two or three clips). Therefore this was a 3 (length; 6cm, 8cm, 10cm) x 3 (number of clips; 1, 2 or 3) within subjects design. We measured the effect of these different conditions on the descent rate (in seconds); our dependent variable.
Procedure
Nine templates of the paper helicopters were printed using a laser printer. Then, separately, three of these were adjusted to have the rotors measuring 6 cm in length. This was then repeated for the 8cm and 10cm conditions. After this was complete, all helicopters were cut out using a standard pair of craft scissors and folded in the appropriate places according to Figure 1. Then within each of the three groups, each helicopter was assigned one clip, two clips or three clips. The clips were placed at the base of the plane like the paper clip shown in the far right of Figure 1.
To conduct the experimental component, a controlled environment was established in a study room of the University of Newcastle Library at the Ourimbah campus. The doors and windows were closed in order to control any possible wind variable. A height of 150cm was measured from the floor using a tape measure. Throughout the experiment, the height was monitored and maintained by one member of the research team to ensure constancy and eliminate any potential height variation. Another member of the team was assigned the role of dropping the helicopters using two 30cm rulers. The 30cm rulers were used to balance each of the rotors and eliminate any manual interference. In order to instigate a spinning motion, the rulers were pulled horizontally away from the base of the helicopter. All time measurements (in seconds) were taken between the base of the paper helicopter and its impact with the floor (total distance of 150cm). Time measurements were recorded by another team member using a stopwatch that was able to record hours, minutes, seconds and milliseconds. After each drop, one member of the group entered the descent rates of each helicopter into a statistical software package (JMP V9). There were nine variations of helicopters (3 (length; 6cm, 8cm, 10cm) x 3 (number of clips; 1, 2 or 3)) that were dropped randomly 11 times, with a total of 99 drops.
Results
Model Test
Ho: αi = βj = αβij = 0
There is no effect of length (αi), weight (βj), or the interaction of length and weight (αβij) on the descent rate of paper helicopters.
HA: at least one effect is not zero
There is an effect of at least length (αi), weight (βj) or the interaction of length and weight (αβij) on the descent rate of paper helicopters.
The high test statistic (F8, 90=85.5961) and low p-value (P<.0001) from the model test shown in Appendix B provides sufficient evidence to reject the null. Thus we will reject the null hypothesis at the 5% significance level, and conclude that there is a statistically significant effect of at least length, weight, or their interaction on the descent rate of paper helicopters (p<.0001).
Interaction Test
Ho: αβij = 0 for all i, j
There is no effect of the interaction (αβij) of the different levels (i) of length of the rotors (α) and the different numbers (i) of clips (β) on the descent rate of paper helicopters.
HA: αβij ≠ 0 for at least one i, j pair
There is an interaction (αβij) between the different levels (i) of length of the rotors (α) and the different numbers (i) of clips (β) which have an effect the descent rate of paper helicopters.
From Appendix B, the test statistic (F = 7.1157) and p-value (<.0001) provide enough evidence for us to reject the null hypothesis at the 5% significance level. Hence we conclude that there is a statistically significant effect of the interaction of the different lengths of the rotors and the different number of bull clips on the descent rate of paper helicopters (p<.0001). In particular the means plot shown in Appendix C showed that as the weight increased for each of the helicopters, the difference in mean descent rate for the different lengths subsequently decreased. Thus the different weights, represented in the number of clips, effected (by decreasing) the influence of the different lengths of the rotors on the descent rate. The Tukey’s HSD post hoc test identified many significant differences and similarities between groups where of particular interest was that similar sizes, e.g. medium length and medium number of clips, and small number of clips and small length, were found to be statistically similar in their descent rates. This would mean that when the weight is the same proportion as the length, the descent rate is quite similar. This test was found to have a power of 0.9936 where our conclusion of rejection was valid.
Number of clips test
Ho: αi = 0 for all i
There is no effect of the different lengths of the rotors (αi) on the descent rate of the paper helicopters.
HA: αi ≠ 0 for at least one i
There is an effect of the different lengths of the rotors (αi) on the descent rate of the paper helicopters.
From the high test statistic (F = 179.7) and low p – value (p<.0001) provided in Appendix B, we decided to reject the null hypothesis and conclude there is a statistically significant difference in descent rate for the different length of the rotors of the paper helicopters. In addition the means plot and the Tukey’s HSD post hoc test shown in Appendix D shows that the descent rate for the paper helicopters was significantly different for each of the different lengths (6cm, 8cm, 10cm) where they increased the greater the rotor length. Also from Appendix D, this test was found to have a power of 1which means that our conclusions of rejection were valid.
Rotor Length Test
Ho: βj = 0 for all j
There is no effect of the varying number of clips (βj) on the descent rate of the paper helicopters.
HA: βj ≠ 0 for at least one j
The varying number of clips (βj) does not have an effect on the descent rate of the paper helicopters.
From the high test statistic (F = 148.5) and low p – value (p<.0001) provided in Appendix B, we decided to reject the null hypothesis. From this we conclude that there is a statistically significant effect of the number of bull clips on the paper helicopters descent rate. In particular, shown in Appendix E in the means plot and the Tukey’s HSD, there is a significant difference between all of the different number of clips where the less the weight created by the clips the greater the descent rate. This test had a power of 1 which means that our conclusion of rejection is valid.
Assumptions
As a result of conducting this type of analysis on the data, one of the assumption’s that is required to be met is the assumption of independence. Independence states that the outcome of one trial must not influence the outcome of any other. Two random variables X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the occurrence of any event involving Y alone. The model undertaken to analyze the data in this report, assumes independence when the random variables describe outcomes that appear unrelated to each other. In this data set we assume that independence has been met.
Random sampling of the paper helicopters validates our statistical method allowing us to make inferences about populations from samples. Random sampling also minimised allocation bias and ensured that all of the types of paper helicopters were tested with equal chance. This assumption was met.
The assumption of normality, supposes that there is a normal distribution of the paper helicopters descent times between the sample groups. From the Shapiro-Wilk test of normality conducted in Appendix H indicates that there was no statistically significant deportation from normality of the residuals (p = 0.2931). This assumption was met.
The assumption of equality of error variances allows an accurate analysis and comparison of these variances in the Two-Way ANOVA. From the residual plot shown in Appendix G, there appeared to be an equal variance for all groups each with an equally large spread. Also the lack of pattern in the plot and a large R2 value (0.88) shows the suitability of this test to the data is high with no problems. Therefore this assumption appeared to be met and the Two-Way within subjects ANOVA appears to be a suitable test for the data.
Discussion
The results we found did not support what was hypothesised. There was found to be an effect of the length of rotors and number of bull clips on the fall (in sec) of a paper helicopters. The longer the wings meant the longer the helicopter had taken to reach the ground. The heavier the helicopter, created by a greater number of bull clips applied, meant the faster the helicopter had reached the floor. There was also an interaction found where the greater the weight meant the more similar the descent rate was for the different lengths. Thus weight had an inhibiting effect on the effect of the different lengths on the descent rate of the paper helicopters.
From these results, this means that heavier objects falling from a moderately short distance will fall at a faster rate than objects of a lesser weight. In addition we found that objects with a greater surface area will fall slower due to the greater drag they create.
The interaction found between the weight of the paper helicopters and the wing span of the paper helicopters, was that when the helicopter is at its greatest weight, the wing span is not going to have as much of an effect on the descent rate, than when there is less of a weight.
However, the limitation of our study was that we only conducted the experiment from a small height of 1.5m. It would have been more effective if we used a variation of heights and also greater height because this would mean that we would get a bigger range for it to reach terminal velocity. This is because the results we found indicate that the helicopters where in the process of equalising their weights with the drag where greater weights take longer to equalise. Whereas at greater heights we should reach terminal velocity as after the weight and drag have equalised, greater weights should have greater drags which means they should fall slower. Thus this should be an interesting study to partake in in the future.