The University of Southern Mississippi. Analysis of the Motion of a Falling Maple Seed (Acer species) Ty McCleery

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1 The University of Southern Mississippi Analysis of the Motion of a Falling Maple Seed (Acer species) by Ty McCleery A Prospectus of a Thesis Submitted to the Honors College of The University of Southern Mississippi in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in the Department of Physics and Astronomy December 2008

2 Tyler McCleery 2 Approved by Bill Hughes Department of Physics and Astronomy Klin Maung Maung, Chair Department of Physics and Astronomy David R. Davies, Dean Honors College

3 Tyler McCleery 3 Chapter I: The Problem The seeds of the Maple genus, Acer species, are produced into a shape that allows for an autorotating motion during freefall. This motion causes the sinking speed of the seed s descent to decrease. It is believed that any form of motion that slows the rate of descent of a seed from the parent plant to the ground will increase its chance of being caught by the wind and blown laterally away from the parent, a process known as anemonochory (Fenner, 1985, pp. 32 and 40). For the maple genus which relies on wind dispersal of seeds to increase chances of regeneration, autorotation is a great advantage. The ability of the maple seed to autorotate during freefall is a direct result of the seed s structure. This structure known as a samara can be defined by parameters of the seed s shape and size such as chord length of the wing, the mass, the surface area, etc. There is a limit to the shape and size of a seed that the parent tree can produce. As Fenner (1985) explains, [a maple tree] has a limited amount of energy and nutrients to devote to the production of seeds, a result of the principle of allocation. These limitations define both the seed structure and the number of seeds produced. It is the process of natural selection which has determined the ratio of seed size to seed number (pp. 5 and 16). A seed may be able to travel farther if it were bigger, but if the tree can either produce a small number of large seeds or a large number of small seeds natural selection will lean towards the option that yields more offspring. Natural selection and the genetics of the tree are not the only determinates for seed size. The same tree may produce a variety of seed sizes over a span of numerous years based upon the availability of nutrients and environmental conditions (Fenner 1985, p. 23).

4 Tyler McCleery 4 Studying the autorotation of the maple seed will lead to an identification and relation of the physical parameters of the seed and of the air through which it falls that control this motion. Such relationships can be valuable in ecology as well as physics. For an ecologist studying a grove of maple trees there might be an interest in the projected growth of the grove after a certain period of time. As stated previously, a maple seed caught by a gust of wind can travel away from the parent tree. An ecologist could measure the average parameters of the seeds in the grove and calculate the terminal velocity from the developed relationship of parameters. Measuring the average height of the seeds in the parent trees and dividing the calculated terminal velocity would give the average fall time of the seeds. If the ecologist knows the typical strength of the winds through the grove, he or she could multiply the average fall time by the wind speed to yield an average distance the seeds would fall from their parent trees. The ecologist could then predict the size of the grove after sufficient time for a second generation of maple trees to grow. The equation of motion could also provide useful information for possible future technological developments in physics that might use autorotation. The investigation of the falling autorotation of the maple seed will pursue an identification of the parameters of the seed and medium through which it falls that control the seed s motion and a relationship of these parameters that correlates to data of the actual motion of the seed. Dimensional analysis will be used to determine which parameters and to what degree the parameters affect the motion of the seed by combining the units of the parameters to yield units of motion. A variety of seeds will then be chosen and studied to collect data on typical values of terminal velocity, rotational speed, and coning angle. The data will be correlated with the results of the dimensional analysis

5 Tyler McCleery 5 to determine the relationship of the parameters that best fits the data. This relationship will then be analyzed to ensure that it agrees with the expected laws of physics that govern the autorotation and free fall movement. Chapter II: Literature Review II.1 Early Research The problem of the autorotation of samaras has been reviewed in the past using a variety of methods. Dingler (1889) used a technique of dropping the seeds into fine sand and studying the impressions made. He then made models to test possible shapes and sizes that were necessary to produce similar results. He was able to draw conclusions on the necessary structure for autorotation. His experimental process was, however, inadequate for determining the physical mechanics of autorotation (as cited in Norberg, 1973, p. 563). Ulbrich (1928), observing similar movements of the seeds, mimicked Dingler s classification by motion and illustrated the samara s structure (as cited in Norberg, 1973, p. 563). Schmidt (1939) also briefly described the seed s motion (as cited in Norberg, 1973, p. 563). None of these early researchers were able to completely describe the seed s autorotation as a relationship of the parameters. Hertel (1966) used the momentum theorem to analyze the seed s motion. He studied the seed s as they fell by photographing the fall path. A single screw-like path was captured on film holding the shutter open for the duration of the fall. Studying the data, he was able to calculate related air velocities by the momentum theorem (pp. 90-6). This method of photographing the seed as it falls can be useful in determining the terms of motion such as the terminal velocity, radial speed, and pitch angle. Norberg (1973)

6 Tyler McCleery 6 continued Hertel s research and was able to study the samara motion in much greater depth. II.2 Basics of Fluid Mechanics To understand Norberg s analysis a brief introduction of fluid mechanics is necessary. Ward-Smith (1984), who reviewed Norberg s work, gives a useful brief overview of the important terms and ideas. First, air, as is the case with all matter, is composed of large quantities of molecules which are surrounded by a finite amount of empty space. On the scale of maple seeds, which must collide and interact with the molecules of air, this finite amount of empty space is negligible. It is therefore safe to consider air as continuous medium, a conjecture known as the continuum hypothesis. Second, all fluids, including air, can be described by a property known as viscosity. Roughly, viscosity refers to the thickness of the fluid. Ward-Smith (1984) explains when relative motion takes place between adjacent layers of a fluid internal shear stresses are set up which oppose that motion. This property of resistance to shearing motion due to the shear stresses is termed viscosity (p. 5). There are different kinds of viscosity though. Dynamic viscosity refers to the varying fluid velocity between two parallel sliding plates as shown in Figure 1.

7 Tyler McCleery 7 This changing velocity can be written as a function of distance from one of the plates. u = (y/h)u (1) where u is the varying velocity as a function of y, the distance from the stationary plate, h is the total distance between plates, and U is the velocity of the moving plate. An equation for the shear stress,, can be written as a scalar multiple of the tangent of the velocity equation. = μ(du/dy) (2) The constant of proportionality, μ, used as the scalar in this equation is the dynamic viscosity. Kinematic viscosity,, expands the dynamic viscosity to apply to both viscous and inertial forces together. = μ/ (3) where μ is the dynamic viscosity and is the fluid density. Third, another important property of fluid mechanics is the Reynolds number. The Reynolds number is a unitless ratio of inertial to viscous forces in a moving fluid. Simplified, the Reynolds number can be calculated by the ratio of the product of the airspeed, V, and a characteristic dimension of the object, D, by the kinematic viscosity,. Re = VD/ (4) A low Reynolds number ( < 1) characterizes a steady, continuous fluid movement. A high Reynolds number ( > 1) corresponds to a fluid motion with heavy ripples and turbulence. According to Ward-Smith (1984), the Reynolds number that describes the autorotation of a maple seed is between 10 3 to 10 5 (p. 10). Fourth, for an object moving through a fluid it is sometimes useful to describe the fluid moving past the object rather than the object moving past the fluid. This is an

8 Tyler McCleery 8 allowable method of thinking by simply changing reference. For the maple seed, if the air is allowed to move past the rotating seed it is possible to define the fluid motion in terms of a flow field. At the Reynolds number associated with falling maple seeds, the flow field is divided into two regions. There is a tiny boundary layer adjacent to the surface, and there is a surrounding inviscid layer, meaning it has no associated viscosity. In other words, at a small macroscopic distance away from the surface of the seed, there is no noticeable effect of the seed s motion on the air. Fifth, depending upon how the fluid impacts the surface, the object can experience drag and lift forces. There are two types of drag forces: skin-friction drag and form drag. Skin-friction drag occurs parallel to the surface as the fluid slides over the contact point resulting in friction. Form drag occurs perpendicular to the surface as the fluid collides with the surface. For Reynolds numbers close to or greater than that required for the maple seed, the drag force is proportional to the square of the object s velocity. In mechanics, air resistance that is proportional to the square of a falling object s speed gives a quadratic terminal velocity term. Lift, which is dependent on the shape of the surface that the fluid is moving over, pulls the object towards the side that the fluid moves faster over. This is the result of Bernoulli s Principle. It should also be noted that the autorotating fall of a maple seed is stable. It is able to attain an equilibrium position that can be resumed should there be a minor disturbance in its fall (pp. 2 23). II.3 Analysis and Results of Norberg Norberg (1973) gives a detailed analysis of what causes the autorotation and stability of the falling seed. The importance of this analysis is in the results. The motion of the maple seed is divided into three areas which are depicted in Figure 2. There is a

9 Tyler McCleery 9 sinking speed which is the speed at which the seed descends. There is a pitching angle of the wing blade which is the angle made between the tangent plane of the cone swept out by the wing and the wing blade itself. The pitching angle relates to the rotational speed of the seed. There is also a coning angle which corresponds to the incline of the wing from the horizontal assuming the falling seed path is perpendicular to the ground. The coning angle corresponds to both the sinking speed and the pitching angle or rotational speed. Each of these is studied in great detail, the results of which are summarized below. First, the sinking speed was studied as the relative speed of air through the areal disk. From a top view of the maple seed as it descends, it is easy to see that the wing sweeps through a circular area called the areal disk shown in Figure 3. The ratio of the

10 Tyler McCleery 10 weight of the seed to this disk area is known as disk loading. The sinking speed was found to be proportional to the square root of the disk loading. Second, the pitching angle was found to control the rotational speed of the maple seed. As the pitch angle becomes steeper, the lift on the blade increases while the drag decreases. This causes the blade to rotate faster. The opposite effect occurs when the pitch angle becomes shallower. The pitch angle has a balanced equilibrium state given by the counteracting forces that occur at the center of mass and center of pressure as depicted in Figure 4. These forces pull the blade towards the equilibrium angle if a disturbance is to occur in either direction. Therefore, the rotation of the blade has an equilibrium speed. If the rotation should increase though, the sinking speed will decrease and vice versa. Third, the coning angle is determined by the balance of the weight and the aerodynamic forces. If an imbalance causes the coning angle to steepen, the samara will sink faster, and vice versa. The coning angle is correlated with the rotational speed in the opposite manner. If the coning angle steepens, it is the rotational speed that decreases (pp ).

11 Tyler McCleery 11 II.4 Dispersal Curves In the field of ecology, anemonochoric plants which rely on the wind to disperse their seeds are characterized by dispersal curves. Fenner (1985) explains that these curves display the probable number of seeds likely to be dispersed versus distance from the parent tree as shown in Figure 5. The height of the parent plant, the structure of the seed, the presence and density of surrounding obstacles or competition, and the average wind velocity are all factors that define the dispersal curve (p. 38). Any calculations performed using a developed relationship of parameters, should be able to approximate a dispersal curve for maple trees in a known area. II.5 Dimensional Analysis In studying naturally occurring phenomenon of fluid dynamics it is often useful to use a method of dimensional analysis. Pennycuick discusses how dimensional analysis can be used to determine how the parameters of the phenomenon combine to yield the units of motion (1992). For instance, to determine a relationship for the terminal velocity of an object with parameters of mass, acceleration due to gravity, center of mass, and

12 Tyler McCleery 12 kinematic viscosity, the units of the parameters (kg, m/s 2, m, m 2 /s) must combine in such a way that the units yield meters per second (m/s). In this example, the mass is not necessary to attain the units of meters per second, and so through dimensional analysis one can infer that the object s mass does not affect its motion though other parameters such as the center of mass do. By dividing the combination of units by the term of motion, a dimensionless or unit-less number can be attained. It is a combination of multiple dimensionless numbers which express various terms of motion that will yield the degree and relationship of the parameters necessary to describe the observed motion. This method of combining dimensionless numbers is referred to as the Pi Theorem by Pennycuick (1992, p. 12). The exact combination of these dimensionless numbers is found by correlating physical data of the motion. II.6 Summary In light of the previous research on the topic of autorotating samaras, there are a few considerations to take note of in planning further research. First, it appears that the best way to collect data such as falling speed, rotation speed, and coning angle without the assistance of a wind tunnel is to video record or photograph the seed as it falls similar to Hertel s method. If a photograph is used, a strobe light might be necessary as Norberg used. Second, when calculating the dimensional analysis, kinematic viscosity should be taken into account along with the other physical parameters of the seed. One of the resulting dimensionless numbers found should also resemble Reynolds number. Third, the results of the derived relationship of parameters should at least approximate the data found by Norberg and Fenner s dispersal curves for maple trees. It should also match the physical descriptions Norberg analyzed.

13 Tyler McCleery 13 Chapter III: Methodology III.1 Overview of Methodology The purpose of this research is to identify which parameters of a maple seed and the medium through which it falls determine the seed s motion, and to find a relationship of these parameters that correlates to data of the seed s autorotational motion in free fall. Kileigh Peturis, an undergraduate, will assist in the data collection of this research advised by Dr. Lawrence Mead. A number of seeds will be collected for study. The seeds will be measured for potential parameters, such as mass and dimensions of length. Then the seeds will be dropped to determine their terminal velocity, angular velocity, and coning angle, which define the seed s motion. This data will be correlated with the theory developed through dimensional analysis. The correlation should yield the exact dimensions of the ratios determined through the theory leading to an understanding of what parameters control the seed s motion to be furthered with analysis of the forces present. III.2 Seed Collection and Selection According to the USDA Natural Resources Conservation Service, Maple trees of various kinds can be found throughout the United States (PLANTS n.d.). It is not of concern to the experiment as to the exact species chosen for study. The only necessary criterion for selection is that the seed does autorotate and that there is a large enough assortment available to collect data describing the range of seed shapes and sizes. A variety of species in fact could be more useful. The chosen seeds will be marked to keep

14 Tyler McCleery 14 them separate and identifiable, but the mark must be tested to ensure that the falling motion is undisturbed. III.3 Theory Using the method of dimensional analysis described by Pennycuick (1992) as referred to in Chapter II.5, the seeds will be analyzed to yield a set of dimensionless numbers. According to the Pi Theorem, discussed by Pennycuick (1992), the dimensionless numbers must be combined in multiple ratios to correctly correlate to the data collected. This correlation will yield a relationship of the important parameters that affect the seed s motion. Once the relationship is determined a detailed study of the forces that act on the seed can be pursued. III.4 Data Collection The seeds will be measured for the parameters determined by the dimensional analysis to be significant. It is expected that weight and area would be necessary, so the mass and a dimension of length would be measured for each seed. Then each seed will be dropped in front of a centimeter marked backdrop and filmed with a still camera. An adjustable flash rate strobe light will flash onto the seed at a fractional interval of the seed s period of rotation. This setup is shown in Figure 6. The camera s shutter will be

15 Tyler McCleery 15 left open for the duration of the seed s fall to allow multiple flashes to illuminate the seed s path and orientation on one frame. Comparing each flash to the centimeter markings, the terminal velocity of each seed can be determined. The rotational speed and coning angle will also be approximated by comparing the orientation of each flash. All of the values for each seed will be recorded and used to correlate with the theory. III.5 Correlation of Data The dimensionless numbers found through the dimensional analysis corresponding to coning angles, rotational speeds, and terminal velocities will be evaluated for each seed and plotted versus each other. Figure 7 depicts how the dimensionless numbers are plotted against each other. The data points should have coordinate points of one Pi versus a different Pi, i.e. (, ). A line or curve will be fitted to the data points to determine a proportional relationship between the Pis. This relationship will show which parameters and to what degree the parameters determine the seed s motion.

16 Tyler McCleery 16 References Fenner, M. (1985). Seed Ecology (pp. 151). New York: Chapman and Hall. Hertel, H. (1966). Structure, Form, Movement (M. Katz, Trans.). New York: Reinhold. (Original work published in 1963) Norberg, R. Å. (1973). Autorotation, Self-Stability, and Structure of Single-Winged Fruits and Seeds (Samaras) with Comparative Remarks on Animal Flight. Biol. Rev. 48, Pennycuick, C. J. (1992). Newton Rules Biology. Oxford: Oxford University Press. PLANTS Profile: Acer L. (n.d.). PLANTS Database. United States Department of Agriculture Natural Resources Conservation Service. Retrieved November 10, 2008, from Ward-Smith, A. J. (1984). Biophysical Aerodynamics and the Natural Environment (pp. 172). New York: John Wiley and Sons.

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