FREE FALL TIME OF PAPER CONES

Transcription

1 FREE FALL TIME OF PAPER CONES Project Report of DOE Jiankun SUN, Nan CHEN, Donghui LI, Quan YUAN, Tianyang XU June, 203

2 DIRECTORY Introduction Lecture review Factor decomposition Assumption Response variable Input variable Control variable Nuisance factor Pre-experiment Design of pre-experiment Results of pre-experiment Analysis of pre-experiment Reproducibility of experiment Influence degree of variables Formal experiment Experiment design Variables level setting Full factorial design Experiment process Data analysis Descriptive statistics Scatterplot Main effects Interaction plot General linear model Result analysis and discussion Conclusion... 8

3 Introduction Our project is motivated by a famous experiment in the high school physic textbook. Originally, the experiment only studies the free fall time of the paper cone with the same radius and different angle. We hope to do some further research of this topic by adding more variables and creating a full factorial design, and try to establish a model to describe the relationship between the free fall time and the variables. We think that this more general design of the experiment will help us get a better idea of this physic problem. 2 Literature review We find a research paper relevant to what we want to study. From the paper we get some basic design ideas:. How to make the paper cone? We use ordinary A4 size paper and cut it into a fan with different angle. Then fold the paper into a cone, using tape to connect. Showed in the figure below. Figure The making of the paper cone 2. How to set the variable? According to the paper, the free fall time may be influenced by height, angle, radius and the thickness of paper under our hypothesis. We will design the factor and the factor level under this consideration. 3. How to measure each variable? We will a camera to record the whole experiment procedure. It will bring benefit to our later data collecting and analysis. Also we make sure that every paper cone is under the same quality. We try our best to maintain the quality of the paper cone we make at the same level. 陆文彬. 浅探纸锥下落快慢与纸锥大小的关系 [J]. 物理实验, 200, 30(0): 9-23

4 4. Some guesses of the result and the explanations The paper has done some simple experiment about this topic and come up with the conclusion that the radius has nothing to do with the free fall time. Furthermore, prove it by mathematic analysis. Figure 2 Design with different angle Figure 3 Design with different radius However, the experiment only studies the free fall time of the paper cone with the same radius and different angle. Besides, it derives a complicated equation to describe the relationship between the acceleration and velocity of the paper cone and the impacts from paper cone properties, which is not intuitive and practical to evaluate the free fall time of a given paper cone. Therefore, we hope to explore the influence of more variables on the free fall time and try to establish a relatively simple model describe such influence.

5 3 Factor decomposition 3. Assumption We design the experiment based on the following assumptions: The paper of the same type has the same thickness and quality. The wind speed of the environment is slow and we can ignore its effect. 3.2 Response variable Response variable (units) Normal operating level & range Means, precision Free fall time 0~5s Video analysis, 0. sec Relationship of response variable to objective Estimate mean value to reflect the objective 3.3 Input variables Input variable (units) Normal level & Range Predicted effects (For various responses) Paper thickness 2 levels: thick/thin Thick paper result in shorter free fall time Cone radius 2 levels: small/large Has nothing to do with the time Central angle 0 levels: 90 ~270 Smaller angle result in shorter free fall time 3.4 Control variable We design the experiment at the same height. 3.5 Nuisance factor There are several noises in this experiment. First, the performance of the operator may vary from person to person, but we do some training before the experiment. Therefore, we assume that they will not affect the experiment. Also the error of the time recorded will be neglected.

6 4 Pre-experiment 4. Design of pre-experiment Before the formal experiment, we want to check the repeatability and reproducibility of our experiment and have a general idea of the influence degree of factors, paper radius, cone radius and central angle, on the free fall time. Here, repeatability means that the experiment results are stable among different operations by the same operator, and reproducibility means that the experiment results are stable among operations among different operators, Then we design a pre-experiment to check the free fall time of paper cone at different factor levels by operator Jiankun Sun and Vincent Chen. The variables process parameter and their selected range is shown in table. Table Actual and code values of the variable parameters in pre-expriment Code Paper Thickness Cone Radius Central Angle - Thin 5 cm 50 Thick 0 cm 270 The code of the two operators are - and for Jiankun Sun and Vincent Chen, respectively. Both operators run the experiment for three times, coded, 2 and Results of pre-experiment After analyzing the video and we get the following data. Operator Group No. Cone Radius Table 2 Results of Pre-experiment Paper Thickness Central Angle Free Fall Time

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8 4.3 Analysis of pre-experiment 4.3. Reproducibility of experiment First we check whether this experiment can be repeated, that is to say, whether there is significant difference between data of different operators. We show the descriptive statistics of two groups of data, apply pair t-test and get the following results. Figure 4 Descriptive Statistics The descriptive statistics shows that the two groups of data have very similar mean and standard deviation. The minimum, median and maximum are also very close. Figure 5 Pair t-test results From the pair t-test, we can get that the 95% confidence interval for difference is ( , ), which includes 0. The p-value is 0.03, we cannot reject the null hypothesis at a 0.95 significance level. Hence, we can find that there is no significant difference between different operators and this experiment can be repeated Influence degree of variables From the results of pre-experiment, we can also get a general idea of the influence degree of variables on the free fall time of paper cone. Then we check the main effects plot and interaction plot of the free fall time as follows:

9 Figure 6 Main Effects Plot From the main effects plot, it seems that cone radius has no influence or a very weak positive influence on the free fall time, the paper thickness has a negative influence, while central angle has a significant positive influence. Figure 7 Interaction Plot According to interaction plot, these three variables seem to have no significant interaction since the lines are nearly parallel of different levels.

10 5 Formal experiment 5. Experiment design 5.. Variables level setting Cone radius is coded as - and ; - for 5cm and for 0cm. Paper thickness is also coded in two levels; the actual value, however, is defined indirectly: - is for the A4 paper with the weight of 70 g/m 2 and for the A3 paper with a relatively heavier weight of 200 g/m 2. Central angle is defined as a continuous variable with the range from 90 to 270. In the experiment, we take the angle value with 20 as an interval to approximate it with discrete value. All in all, the variables we set are shown below. In the following analysis, we use radius, thickness, angle and time to represent the three variables respectively and the response variable for simplicity. Table 3 Actual and code values of the variable parameters in formal expriment Cone Radius Paper Thickness Central Angle - - continuous variable 5 cm 0 cm 70 g/m 2 200g/m 2 (90+20k) (k=0,,2 9) 5..2 Full factorial design A full-factorial design is taken considering radius and thickness. Due to the ten value levels of angle, 40 pilots are required.

11 5.2 Experiment process The experiment is undertaken in the hall of Shunde Building at 8:00 pm on Friday, with few people in the hall, thus the experiment moves smoothly. 40 paper cones are made previously for the experiment. Figure 8 Paper cones for experiment For each pilot, 3 replications are made and there are altogether 20 times of free falls of the paper cones. The free fall is taken place at the platform of the stairs in the hall, and each free fall is taken as follows: the operator stretch out the arm horizontally and hold the paper cone with thumb and index finger only, and then open two fingers at the same time to release the cone, after a short interval, the experimenter holds another cone and undertakes it again. A camera is used to record the whole process so as to find out the time needed.

12 Figure 9 Free-fall place Figure 0 Free-fall posture 5.3 Data analysis 5.3. Descriptive statistics First we check the descriptive statistics for our data. Figure Descriptive statistics From the result above, we can see that for variable thickness, it may influence the free fall time; while for the variable radius, it may not be that obvious to conclude that the paper cones with different radius will spend different time for the free-fall. To find the conclusion, further analysis is needed.

13 5.3.2 Scatterplot Then we can use scatterplot to get a general impression of the data. 5 4 Scatterplot of Time vs Angle Radius Thickness Time Angle Figure 2 scatter plot From the scatterplot, we can find there is a linear relationship between time and angel. Also, the thickness will affect the time a lot, but the radius has little effect on time. This scatter plot can also help find outliers. We can see for small angle (90 degree and 0 degree), the variance is very large because these paper cones cannot fall smoothly. The paper cones on the black line (thin paper cones with small radius) also cannot fall smoothly. They may tilt in the air and cannot fall along a straight line perpendicular to the ground. We can remove these outliers and get the following scatterplot.

14 Scatterplot of Time vs Angle Radius Thickness Time Angle Figure 3 Scatter plot of modified data This time the data fits the line better and there are relatively nice linear relation representation Main effects It is necessary to have a general idea of the effect on the response variable of each variable we choose. Thus a main effect graph is plotted, shown below. Main Effects Plot for Time Data Means Radius Thickness Mean Angle - Replication Figure 4 Main effects plot

15 From the graph, one thing that should be pointed out is that we add the replication in this graph to check whether the experiments are stable. Form the main effects plot, we can get that angle and thickness will affect the time significantly. But radius and replication number have little effect. The replication number, in our common sense, does have little effect on the experiment results if the noise is controlled well. As to the variable radius, we want to study further before jumping to the conclusion of wiping it out now Interaction plot Interaction Plot for Time Data Means Radius 4 3 Radius - 2 Thickness 4 3 Thickness - 2 Angle Figure 5 Interaction plot From the interaction plot, we can get there is not significant interaction between radius and thickness or between angle and radius. But there are interaction between thickness and angle. For thin paper, when the angle increases, the time will increase faster than thick paper. We also notice that the two lines, in both the radius versus thickness and that versus angle, almost coincide. This may suggest that the variable radius have little effect on the free-fall time General linear model To study in detail, a general linear model is built. Firstly we put radius, thickness and angle in our model (radius is covariate)

16 Figure 6 GLM regression

17 Normal Probability Plot Residual Plots for Time 0.50 Versus Fits Percent Residual Residual Fitted Value 4 Histogram Versus Order Frequency Residual Residual Observation Order Figure 67 Residual plots of GLM regression From the result above we can find that considering the R, 2 the regression leads us to believe that the time of free-fall of paper cones have strong linear relationship with the factors we choose. Notice that at the 95% confidence level, the correlation of the radius and the response variable, time, is not significant, which proves what we guessed previously. From the residual plot, we find that the residual fits normal distribution quite well. But we can also find that the residual versus fits is abnormal. A bowl-like curve has been observed and we guess this may be caused by the interaction between angle and thickness Add the interaction According to the regression result above, we add the interaction of angle and thickness and do the regression again.

18 Figure 78 GLM regression with interaction Residual Plots for Time Normal Probability Plot Versus Fits Percent Residual Residual Fitted Value 4 Histogram 0.4 Versus Order Frequency Residual Residual Observation Order Figure 89 Residual plot of GLM regression (adding interaction)

19 After adding the interaction of radius and thickness, we get a relatively higher R, 2 and more importantly, we can find that the residual versus fits shows no obvious regulation. The effect of the interaction is significant under the 95% confidence level. We can conclude that there does exist the interaction between angle and thickness. 5.4 Result analysis and discussion From the experiment result, we can find that the thickness and the angle of the paper cone has significant correlation with the time taken for free-fall. With a larger angle (smaller than 270), the free-fall time will increase; this is due to the influence of the air resistance. The air resistance on the paper cone is related to the velocity and the horizontally projected area: a higher velocity and a larger horizontally projected area will lead to a larger air resistance. Here we deduce the process of two paper cones freefall with same radius, same thickness, at the same height and only with different angles (denote s as the one with smaller angle, l the larger). At the beginning, both cones fall at zero speed, and s will face a smaller air resistance thus a larger acceleration speed (notice that s is also slightly lighter in mass), and gets a larger speed, and finally falls to the ground earlier. For paper cones made by different types of paper (denote t as the one made by thin paper, T as the one made by thick paper), if the angle decrease from α to β, the air resistance will decrease by similar amount, or we can say the total force on the paper cones will increase a similar amount. For t, because it has smaller mass, the time will increase more than T. This can explain why angle and thickness have interaction and the free-fall time of paper cones made by thin paper will increase more when angle increases. Also we can notice the coefficient of thickness has wrong sign, thin paper has negative coefficient which means it will drop faster. But we can see the coefficient of thickness fail the t-test, which means it is around zero. This may because the interaction between thickness and angle is much more significant than the main effect of thickness. 6 Conclusion Using general regression of Minitab, we can get our final model. If made by thin paper Time = Angle If made by thick paper Time = Angle

20 Figure 20 Final results Note that the above two equations are only applicable for the specific conditions in this experiment, including the specific height, paper thickness and other paper properties and environmental conditions. However, we ve proved that there does exist some simple way to evaluate the free fall time of a paper cone. As long as we know the central angle and some other parameters, a linear model can help us finish this task. We have to admit that there are some problems remained to be solved for this experiment because of our limited experimental conditions, and thus we cannot explore the detailed impact of possible variables on these parameters (the coefficients got from the model). If conditions permit, it can be explored further in the future.