Maths circus: boomerangs

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1 Teaching Mathematics and its Applications Advance Access published January 4, 2006 Maths circus: boomerangs Steve Humble, Derel Briarley, Christina Mappouridou, Gavin Duncan, David Turner and Jodi Handley Submitted December 2004; accepted November 2005 TEACHING MATHEMATICS AND ITS APPLICATIONS, of10 doi: /teamat/hri033 Abstract This paper presents an example of boomerang motion in mathematical terms available to students studying A-level mathematics. The theory developed in the paper postulates possible mathematical models that are veri ed by experimental results. The paper centres on the three-wing boomerang invented by Professor Yutaka Nishiyama. 1. Introduction I have long been interested in boomerangs and their mechanical behaviour (1), yet believed that experimental work with them in the classroom was not possible. However, after meeting Professor Yutaka Nishiyama at ICME-10 s Maths Circus in 2004, and throwing his paper boomerang (2), I renewed my efforts to bring students into contact with these dynamical curiosities. 2. Historical background Imagine throwing a piece of wood in the air, seeing it fly away, then with time its path curving to the left. It follows a wide, more or less circular loop, eventually coming back to you. As it loses speed it hovers some 3 m above your head, then slowly descends like a helicopter to land back into the hand that launched it. Preposterous! Yet this is exactly what a boomerang does, provided that it has the right shape and is thrown correctly. The boomerang originated with the aboriginal inhabitants of Australia, but boomerangs have also been found in Egypt and India. Most boomerangs were not made to return, and were used as weapons of war, due to their having an effective range of some 140 m. These are very dangerous weapons, having been known to pass completely through an adversary. The banana-like shape of the classic boomerang has very little to do with its ability to return to the thrower. Boomerangs of other shapes such as X, S, T, Y can also be made to return. 3. Professor Yutaka Nishiyama s paper boomerang To make Professor Nishiyama s boomerang trace the shape in Fig. 1 onto a sheet of paper and enlarge the image using a photocopier to give a three-wing boomerang with a wingspan ß The Author Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please journals.permissions@oxfordjournals.org

2 2of10 TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006 Fig 1. Three-wing paper boomerang. Fig 2. Fold down the edges of the wings and bend slightly upwards. of 13 cm. Trace this onto a piece of card in the range mm thick and weighing around 5 g. Then cut out this enlarged version of Fig. 1. Place a ruler along the dotted lines as shown in Fig. 1, and fold down the edges to give the wings the classic airfoil profile. Then place the boomerang on a flat surface and bend the three wings slightly upwards as shown in Fig. 2. Your boomerang is now ready to fly. Hold it vertically in your hand and give it a forward flick, thereby imparting as much spin as possible to the boomerang on release. See Fig Why does a boomerang fly? We can use Bernoulli s equation (3) to explain the pressure difference on either side of a boomerang. If and p are the density and the pressure of the air near the wing of the boomerang and v is the speed of the air, then p þ 1/2 2 ¼ k where k is a constant. It can be seen from Fig. 4 that the air has to travel further over the top of the wing and therefore faster than the air below. Bernoulli s equation tells us that if you increase the speed you decrease the pressure, and conversely by lowering the air speed you raise pressure.

3 TEACHING MATHEMATICS AND ITS APPLICATIONS, of10 Fig 3. Throwing the boomerang. Greater air speed Lower pressure lower air speed higher pressure Fig 4. Distribution of air around an airfoil. L dl L + dl Fig 5. Change in angular momentum. This shows how the pressure difference on the wings will cause the boomerang to tilt to one side. As the boomerang is thrown with a flick of your wrist to induce spin, the pressure difference starts to cause the angular momentum (L) to change (dl) in direction but not in magnitude. From Fig. 5 it can be seen that this change in angular momentum causes the boomerang to precess and return to the thrower. This idea is very similar to the way a coin precesses, when it is rolled at a slight angle to the vertical across a table (4).

4 4of10 TEACHING MATHEMATICS AND ITS APPLICATIONS, Background The main work carried out on boomerangs has been performed by Felix Hess (5) and Jearl Walker (6). Their articles are a great starting point when looking for how and why a boomerang flies. I was also fortunate enough to find a copy of Felix Hess s PhD thesis on boomerangs in Newcastle University Library. This gives much more detail on the Mathematics and Physics of the dynamics of the boomerang s motion. Yet the theory was too hard to explain to my A-level students, so I needed something more simple. 4.2 Mathematical theory Dr Hugh Hunt from Cambridge University has a website (7) with simple demonstrations on boomerang flight. Adapting his two-wing boomerang theory has enabled me to work out the theory for Professor Yutaka Nishiyama s three-wing paper boomerang. Figure 6 shows a three-wing boomerang of radius a, moving forward at speed v and having an angular speed of!. As the boomerang spins, the upper arm moves forward by an angle #. From Fig. 7 it can be shown that the upper arm of the boomerang has a speed v cos # þ x! in the direction of rotation. The lower arm to the left moves down an angle #. From Fig. 8 it can be shown that the lower arm to the left has a speed of!x v sin (# þ /6) in the direction of rotation. Fig 6. Three-wing boomerang.

5 TEACHING MATHEMATICS AND ITS APPLICATIONS, of10 The lower arm to the right moves up by an angle #. From Fig. 9 it can be shown that the lower arm to the right has a speed of!x v sin (/6 #) in the direction of rotation. The aerodynamic lift force perpendicular to the plane of rotation of an airfoil of area A, moving at speed v, in air with density, is given by F ¼ v 2 C L A, where C L is defined as the lift coefficient. The total lift over the three-wing boomerang, averaged over one rotation, is given by ð =2! ð a ð a F ¼ 4bC L ðvcos# þ!xþ 2 dx þ!x vsin # þ 2dx ð a þ!x vsin 2dx 6 6 # dt v wx ϑ Fig 7. Upper arm of the boomerang. v ϑ + π 6 wx Fig 8. Left lower arm of the boomerang. v wx p J 6 Fig 9. Lower arm to the right of the boomerang.

6 6of10 TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006 You can see that the lift force is greater in the wing that is moving forwards, and less in the other two wings. This gives a larger lift force at the top of the boomerang, which explains why there is a change in the angular momentum. This change causes the force of procession which creates circular motion, and hence the return of the boomerang. To simplify the calculation the following assumptions were made: (a) The rate of turning was linear, so # ¼!t (b) The area of each wing can be approximated by rectangular rods A ¼ b a (c) The path taken by the boomerang was circular with radius R This gives the total lift as F ¼ (ab/!)(2! 2 a 2 þ 3v 2 ) C L. A constant centripetal force on a mass m, produces circular motion with speed v on a radius R, hence F ¼ mv 2 /R. Combining these two equations gives R ¼ m!/ab (3 þ (2! 2 a 2 /v 2 )C L ). Assuming that for a short flight the angular speed! is nearly constant, and v!a, we get: R / m/a. This can be interpreted as: Increasing the mass of the boomerang gives a greater radius of flight. Increasing the wingspan of the boomerang reduces the radius of flight. We now have a Mathematical theory which can be tested by experiments. My students and I performed three experiments to check the findings of this theory Experiment 1 In this experiment the weight was gradually increased towards the end of the wings of the boomerang, using paperclips and blue tack. The boomerangs were weighed on electronic scales to

7 TEACHING MATHEMATICS AND ITS APPLICATIONS, of10 the nearest tenth of a gram. The experiment was repeated five times for each weight and the average of these can be seen in the table that follows. Weight (g) Diameter of flight (cm) Figure 10 shows the relationship obtained, which verified that as the weight was increased, the radius of flight increased. Using the Coefficient of Determination test (8) gave r 2 ¼ 96%. So 96% of the data s variations can be explained by the regression equation. This test can be found on most graphic calculators, and is a useful regression analysis technique to determine the validity of fitting data to a curve Experiment 2 Paperclips and blue tack were also used to increase the weight at the centre of the boomerang wings Weight (g) Diameter of flight (cm) Figure 11 shows the relationship obtained, which verified that as the weight was increased, the radius of flight increased. This equation also accounted for 96% of the variation Experiment 3 The final experiment looked at boomerangs with ever increasing radii and the same weight. The results give the following inverse relationship, which accounts for 99% of the variation. 5. Conclusion It has been shown that experimental results validated the theory. Increasing the mass of the boomerang gave a greater radius of flight, and increasing the wingspan of the boomerang reduced the radius of flight.

8 8of10 TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006 Fig 10. Relation between diameter and weight i.e. d ¼ 1.2 (w 2.7 ). Fig 11. Relation between diameter and weight i.e. d ¼ 60.8 (w 0.7 ).

9 TEACHING MATHEMATICS AND ITS APPLICATIONS, of10 Fig 12. Relation between diameter of flight and radius of the wing i.e. d ¼ (r 0.35 ). The students greatly enjoyed experimenting with boomerangs and this enriched their experience of maths. Note on the article s title At ICME-10 in Copenhagen in 2004 there was a Maths Circus with a wide variety of different activities going on every day. You would have always found me down at the Circus enjoying the wonderful ideas that people had brought from all over the world. I have described the Circus as a fire that I feel we all need to relight at the heart of our subject. So often we get so caught up with teaching to the syllabus that we forget the real reason we all teach Mathematics. I run a Maths Circus at Newcastle College and we research into various areas of Mathematics, such as boomerangs. My aim is that for each area we investigate my students and I will publish a paper of our findings. Acknowledgements I wish to thank Professor Yutaka Nishiyama and Professor Robin Johnson for their help and suggestions. References 1. Humble, S. (1994) Anyone for Tennis. Teaching Mathematics and its Applications, 13, Nishiyama, Y. (2002) The World of Boomerangs. Bulletin of Science, Technology and Society, 22, N de Mestre, The Mathematics of projectiles in Sport. Cambridge University Press, UK. 4. Humble, S. (2001) Rolling and spinning coin. Teaching Mathematics and its Applications, 20, Hess, F. (1968) The aerodynamics of boomerangs. Scientific American, 219,

10 10 of10 TEACHING MATHEMATICS AND ITS APPLICATIONS, Walker, J. (1979) The amateur scientist: Boomerangs! How to make them and also how they fly. Scientific American, 240, Hunt, H Owen, F. & Jones, R. Statistics. Polytech publishers Ltd Stockport, UK. Steve Humble is Head of Mathematics at Newcastle College. He has worked in a number of state and private Schools. Steve is the author of the book The Experimenter s A to Z of Mathematics which develops an experimenter s investigative approach to Mathematical ideas. Always having had great fun playing with maths, he enjoys teaching this to others. Address for correspondence: Steve Humble, Head of Mathematices, Newcastle College, Rye Hill Campus, Scotswood Road, Newcastle Upon Tyne, NE4 7SA UK. DRMaths@hotmail.co.uk