by Jearl Walker
A BOOMERANG is surely one of the oddest devices ever to serve as a weapon or a plaything. It was apparently invented by accident, notably in Australia by the native people but also independently in many other places. If you throw an ordinary stick, it falls to the ground not far away, but a boomerang can travel as much as 200 meters (round trip) and can be aimed so skillfully by an expert thrower that game or an enemy can be hit. The boomerang probably originated as a weapon designed for accurate straight flight, but most people find the returning boomerang, which was mainly just a plaything for the Australians, more interesting. Ironically the straight-flying boomerang is probably more complicated aerodynamically than the returning boomerang. As ancient as both devices are, an amateur investigator can still do a great deal to advance the understanding of their flight.
Although good returning boomerangs are occasionally available in sporting-goods stores, most commercial returning boomerangs are mass-produced and fly poorly. Indeed, many of them do not even return. The plastic boomerang made by Wham-O is one of the best types available in toy stores. Many other types, all of them excellent, are available from Ruhe-Rangs, Box 7324, Benjamin Franklin Station, Washington, D.C. 20044, thanks to Benjamin Ruhe, a boomerang enthusiast formerly with the Smithsonian Institution. During his service there Ruhe helped to organize the Annual Smithsonian Open Boomerang Tournament. The tournament, held in late spring or early summer, is great fun for the 100 or so contestants who enter. This year it is scheduled for June 9 on the mall in Washington.
If you would like to experiment with boomerangs, you need to be able to construct your own. Only then can you make the variations necessary to determine what factors influence a boomerang's flight. The best material from which to cut the basic stock of a boomerang is Baltic birch in a marine or aircraft plywood between 1/4 and 3/8 inch thick, with five or more laminations. This type of plywood is resistant to wear and water; it is also dense, which makes for a boomerang that is heavy for its size.
Herb Smith's "Gem" design for a boomerang
Cut a cardboard pattern for a boomerang of whatever shape you want. An example is the returning boomerang designed by Herb Smith and shown in the illustration at the left. (If you are left-handed, you will have to make a left-handed boomerang, which is the mirror image of the one drawn.) Place the pattern on the plywood and mark off the outline of the boomerang in pencil.
With a coping saw or a handsaw cut out the boomerang blank. The edges and the top must now be shaped in the general form shown in the illustration. (Little is done to the bottom other than putting a slope on what will be the leading edge.) The leading edge must be blunt, whereas the trailing edge must be sharper, with the top surface sloping down to meet the unaltered, flat bottom surface. The arms must have the cross-sectional shape of a classic airfoil, because they must provide lift much as the classic airfoil does.
Clamp the blank in a vise and cut and shape the edges and the top with a rasp that has a curved surface. Smooth out the grooves left by the rasp and finish shaping the wood by rubbing the surfaces with a piece of coarse sandpaper wrapped around a piece of soft wood. Before you finish off the surface with a finer sandpaper you should make a test flight with the boomerang so that you can tune it by more shaping with the rasp or with coarse sandpaper. Tuning the boomerang means that you throw it, cut or sand it more and then throw it again until it flies the way you want it to.
A boomerang cannot be thrown well in a strong wind. If there is a light wind, face toward it, turn 45 degrees to the right and throw the boomerang in that direction. Hold the boomerang vertically by the tip of one of the arms (which one usually does not matter) with the flat side away from you. Reach behind your head with the boomerang and then throw it toward the horizon, snapping it forward when your arm is fully stretched forward. Do not try to throw hard at first. It is the snap that counts, not the strength of the throw. The boomerang stays up in the air because of the spin the snap imparts to it.
The proper orientation of the boomerang (the plane in which it will be spinning) will vary according to the wind conditions and the type of boomerang. To achieve a good flight you might have to throw the boomerang with it plane nearly vertical. Under other circumstances you will have to shift the plane (tilting the top of the boomerang away from you) by as much as 45 degrees. The greater the tilt of the plane in which the boomerang spins is, the more upward lift the boomerang will initially have. If you give it too much lift, it climbs too fast and then plummets to the ground with such force that it may break.
Preparing to throw a boomerang
In a good flight a returning boomerang travels horizontally around an imaginary sphere. On returning it probably will hover or even loop a bit before it drops to the ground at your feet. If you are lucky, it might make one or two additional circles (smaller than the first circle) before it falls. Although you launch the boomerang with its spin plane almost vertical, it probably will return with the plane nearly horizontal. I shall explain below why the plane must turn over in this way if the boomerang is to complete the trip.
If your boomerang consistently lands to your right in a light wind, try throwing it a bit more to the left of the wind. Similarly, if it is landing to your left, try throwing it more to the right of the wind. If it lands behind you, try throwing it with less force. If that does not work, throw it somewhat above the horizon with the spin plane tilted less from the vertical. If the day is windless and you are not getting a full return, tilt the spin plane farther from the vertical in order to gain more lift during the flight.
Be careful not to injure people or damage things with your boomerang. It can be quite a weapon. Throw it only in a large, open space. If people are present, make sure they know what you are doing so that they can be prepared to dodge.
The successful tuning of a boomerang involves both experience and luck. In general if you make the top surface more curved, the boomerang will have more lift, which means that it will return in a tighter circle. Flattening the top surface or curving the bottom surface will give the boomerang less lift because the cross-sectional shape of the arms is then less like a classic airfoil. If the spin decreases too fast, so that the boomerang falls to the ground in mid-flight, the reason may be that excessive air drag on the arms is robbing them of their spin. Some surface roughening might be beneficial to the flight, but any large grooves left by the rasp will surely create additional air drag that will shorten the flight time.
Instead of shaping the arms carefully you might prefer to twist them so that during a flight the leading edge on each arm is tilted to deflect the passing air to the right, giving the boomerang a lift to the left. This type of lift is easy to visual window of a moving car and turn it through various angles, you can feel the lift. To twist a boomerang heat it gradually in an oven at 400 degrees Fahrenheit and then (with gloves on, of course) carefully twist the arms until the wood is cool. If you twist too much, heat the boomerang again and twist the arms back a bit.
A typical flight path
Once you have a properly flying boomerang you might want to finish it-off with a cellulose covering and some decorative designs. Smith's excellent booklet, which is listed in the bibliography for this issue [below], explains how to do this kind of finishing and also gives a number of boomerang designs. If your boomerang breaks, do not throw away the pieces. Glue them together with epoxy, clamp them until they dry and then file and sand the surface back into the desired shape. Although the boomerang will not be as strong as it was before, its flight path might be altered in an interesting way by the small change in its distribution of mass resulting from the break.
A boomerang does not have to be limited to two arms. Indeed, one of the easiest boomerangs to build is a four-blade design consisting of two rulers crossed and fastened at the center. The right kind of ruler has a curved top surface and a fairly flat bottom surface. You can attach two such crossed rulers either by wrapping a strong rubber band around them or by putting a bolt and nut through the center hole they usually have. Throw this type of boomerang the same way that you would a two-armed one. Take care to avoid being cut by sharp edges, and never use rulers that have metal edges.
A simple cross boomerang can be fashioned from a cardboard square about five inches on a side. Cut out a boomerang with three or four blades and twist them slightly so that the boomerang is not quite all in the same plane. Adding weight to the arms of a boomerang increases its range. With the cardboard boomerang it is easy to add weight by attaching paper clips to the end of the arms. This boomerang can be demonstrated in a classroom. If it has too much range for a classroom, you can decrease the range by increasing the twist on the arms so that the boomerang is less in the same plane or by bending the arms along a center line through their length. With the latter technique the arms have an exaggerated airfoil shape: at least one side is sharply convex but the other is not. As usual, throw the boomerang with the convex side toward you. By changing the arms from being almost flat to being more like an airfoil, you increase the lift on the boomerang: its path will be a tighter circle.
When you begin to throw your wood boomerang well, you might be tempted to catch it. The result may be a sharp blow to your fingers. If you are determined to catch a boomerang, hold your hands flat and slap them together to trap the boomerang as it hovers above the ground, still spinning, in the last stage of its flight. Keep your fingers away from the turning blades.
A four-blade design made with plastic rulers and rubber bands
The explanation of the return of a boomerang lies primarily in the cross-sectional shape of the arms and the fact that the boomerang spins. Without these two features a boomerang would behave like any other thrown stick. The cross-sectional shape gives the boomerang aerodynamic lift similar to the lift generated by some airplane wings. The spinning gives the boomerang stability. Through a bit of fortunate rotational mechanics the spinning also causes the axis about which the boomerang spins to rotate in much the same way that the spin axis of a top rotates about the vertical. The lift and the stability keep the boomerang up, and the rotation of the spin axis brings it back to the thrower.
Aerodynamic lift can be explained with a simple model of a classic airplane wing similar to the one I described in this department for February, 1978, to explain the lift of a kite. The classic airfoil has a flat bottom, a blunt front, a sharp rear and a convex top. Air passes around a wing faster along the top of the wing than along the bottom. The reason can be seen by visualizing the passing air as being of two kinds. One kind flows around the wing with no rotation in the stream and with the same speed on the top and the bottom of the wing. The other is a circulation cell: it flows to the rear over the top of the wing and to the front over the bottom. Such a circulation is created by a real wing because the air's viscosity and its adhesion to the surface of the wing force it into this pattern as it flows to the rear off the curved top surface.
In the superposition of the two idealized airstreams the two velocities add above the wing and subtract below it, with the result that the real air speed is greater above the wing than it is below it. The difference is important to the lift because the air pressure in the stream is inversely related to the speed of the stream. Hence the air pressure is less above the wing than below it, and the wing gets a push upward. (A real airplane wing can have a more complicated airflow pattern than this simple model implies. Moreover, when an airplane is traveling at high speed, some of the lift may come from the impact of the passing air on the underside of a wing that is inclined slightly upward in order to deflect the air downward.)
If the classic airfoil is inclined to the airstream in such a way that the airstream is more incident on the curved top side, the lift is of course less. Such an arrangement is termed a negative angle of attack. In a simple model the reduction of lift is due to the downward push the incident stream exerts on the top surface. One might also argue that lift is partially lost because the tendency for the air to circle about the wing is lessened and the speed of the air on the top side of the airfoil differs less from the speed on the bottom side.
Pattern of airflow over the arm of a flying boomerang
Conversely, if the airfoil is inclined so that the airstream is incident somewhat more on the flat bottom side than on the top, a situation that would be called a positive angle of attack, the lift increases because of the upward push from the airstream on the bottom side. The air drag also increases. If the angle is too large, the disadvantages of increased air drag outweigh the advantages of lift. The attack angle of the arms of the boomerang as they turn through the air is important to its flight.
Boomerang arms can have a variety of cross-sectional shapes, but most of them are similar in cross section to the classic airfoil. Usually this shape includes a blunt edge that turns into the air as the boomerang spins and a sharper edge that trails during the turn. One side is usually flat and the other convex. Variations on this basic form are numerous, however, and little systematic work seems to have been done on determining which shapes are best aerodynamically. Some boomerangs are actually flat on both sides but with their arms twisted so that the air is deflected as the arms turn through the wind.
The lift on a boomerang differs in a major way from the lift on the classic airplane wing. In the first stage of a flight the boomerang's "lift" is mostly horizontal, with only enough upward force to balance the weight of the device. Since the boomerang is spinning mostly about a horizontal axis, the curved sides of the arms spin in a plane that is almost vertical and the lift is almost horizontal. For the sake of simplicity in what follows I shall ignore the weight of the boomerang. I shall also assume a boomerang that is thrown outward by a right-handed thrower so that the plane of spinning is initially exactly vertical. The lift will be to the thrower's left, so that the boomerang begins to move to the left as it continues to spin in the vertical plane.
If this were the entire story, the boomerang would never come back. To see why it turns around and returns you must understand what else the lift does to the boomerang. In particular it is necessary to know how the torque due to the lift on the boomerang causes a precession of the spin plane.
Imagine that one of the boomerang's arms has spun to its highest possible position and the other arm is almost in its lowest possible position. (I am discussing the basic banana-shaped boomerang.) The upper arm is turning in the same direction in which the center of the boomerang is moving, whereas the lower arm is moving opposite to the motion of the center. The air passing the upper arm is moving faster (in relation to the arm) than the air passing the lower arm. Therefore more lift is generated on the upper arm than on the lower one. The part of the boomerang higher in the spin will always experience a greater lift and hence a greater push to the side than the part lower in the spin.
Positive and negative attack angles
My first thought was that the difference in horizontal lift (more lift on the upper arm than on the lower one) would cause the spin plane of the boomerang to tilt, thereby angling the lift downward (a disastrous effect). What actually happens, however, is that the difference in lift causes a rotation of the plane about a vertical axis. It is this rotation of the spin plane, commonly called precession, that brings the boomerang back.
To understand what causes the rotation you must examine the torque created by the lift. Take the center of the boomerang as the axis about which it is spinning. (Actually the center of mass around which a two-armed boomerang spins is likely to be well off center, but that does not alter the outcome of the argument.) Take the average lift on the upper arm as being directed horizontally outward from the center of the arm. Similarly, take the average lift on the lower arm as being also directed horizontally outward from the center of the arm. The torque created by one of these lifts, as measured from the center of the boomerang, is the product of the lift and the distance to where the lift is applied, that is, half the length of an arm. Since the upper arm has the greater lift, it also has the greater torque.
If the boomerang were not spinning this difference in torques would merely make the plane of the boomerang tilt over. Since the upper arm has the greater torque, the plane would tilt counterclockwise as seen by the person who has just thrown the boomerang. The fact that the boomerang is spinning, however, makes a big difference, because it then has angular momentum and the tendency to tilt the spin plane results in a rotation of the spin plane about the vertical axis.
Average lift on the upper and lower arms
Angular momentum is the product of the boomerang's rate of spin and a function involving the mass and the mass distribution of the device. For an example in another setting imagine yourself attempting to turn a merry-go-round holding several children. The force you apply tangent to the rim multiplied b the radius of the merry-go-round is the torque you are supplying. When you begin, the torque causes an angular acceleration of the merry-go-round; the spin increases from zero to some final value. How would you arrange the children in order to achieve a given angular acceleration with the least force? Intuitively you would place them near the center. Their mass is the same, of course, but their mass distribution with respect to the center of rotation is different. When the mass is nearer the center, the merry-go-round is easier to turn. The mass and its distribution are taken into account by the function known as the moment of inertia. The greater the mass or the farther from the center it is placed, the greater the moment of inertia and the greater the force you will have to supply in order to achieve a given angular acceleration.
Once the merry-go-round is spinning and you are no longer pushing on the rim, the apparatus has a certain angular momentum because it has spin and a moment of inertia. Angular momentum is usually represented by a vector pointing perpendicularly to the plane in which the object is spinning. Here the vector would be vertical. The direction (up or down) is chosen by convention as being the direction of the thumb on the right hand when the hand is held in a hitchhiker's pose with the fingers curled in the direction of the spin of the object.
The only way you could change the size or the direction of such a vector would be to apply another torque to the object. With a merry-go-round you could push on the rim again. (A convention for choosing how to draw a vector representing the change in angular momentum involves pointing the index finger of the right hand from the center of the rotation toward the place where the force is applied and pointing the middle finger in the direction of the applied force. If you make your thumb on that hand perpendicular to both fingers, it automatically points in the direction of the change in the angular momentum. The new angular-momentum vector is the combination of the old one and the vector representing the change.) With a merry-go-round that you had resumed pushing tangent to the rim the new vector would still be vertical but would be larger or smaller depending on whether your aim was to make the merry-go-round turn faster or slower.
How to determine angular momentum and a change in it
A boomerang that is spinning has two torques acting on the arms, one created by the average lift on the upper arm and one created by the average lift on the lower arm. Since the lift on the upper arm is greater, it determines what happens to the angular momentum, and so I shall ignore the lift on the lower arm. (The argument would not change even if I included the smaller lift.) Imagine that the boomerang is receding from you just after you have thrown it with your right hand. It is spinning in a vertical plane and has an angular-momentum vector pointing to your left. The average lift on the upper arm creates a torque that will change the direction of the vector as the boomerang continues to fly away.
To determine how the vector changes use your right hand, orienting the fingers and the thumb properly. With your index finger pointing from the center of the boomerang to the center of the upper arm and your middle finger pointing to your left in order to be in the direction of the lift on that arm, your outstretched thumb points toward you. Therefore the vector representing the change in angular momentum points toward you. Mentally combining the change vector and the original vector is best done from an overhead point of view. The change vector is perpendicular to the original one and gives a new vector rotated from the old one toward you. The size of the angular momentum is unaltered because the change vector is perpendicular to the old one. Only the direction of the angular momentum is changed, and it is rotated about a vertical axis to point more toward you.
This type of rotation of an angular-momentum vector is precession; it is seen when the axis of a top precesses about the vertical. Another common example of precession is seen in the turning of a motorcycle. The wheels of a motorcycle spin fast enough and have moments of inertia sufficiently large to make their angular momentum large. To turn the motorcycle you cannot just turn the handlebars, as you would when riding a bicycle. Instead you make the motorcycle lean in the direction of the turn. The torques then experienced by the motorcycle cause the angular-momentum vectors of the wheels to precess, turning the motorcycle as a whole.
How precession keeps the attack angle positive
During the precession of the spin plane of a boomerang the boomerang continues to travel along a path with a certain speed but is continuously deflected by the horizontal lift it experiences. The resulting path approximates a large circle. In a successful boomerang flight the spin plane will precess at the same rate at which the device circles in its path. Its angle of attack remains somewhat positive. This match is necessary in order to keep the arms at the proper attack angle.
Suppose the boomerang precesses too slowly. Then as it travels along its circular path its spin plane rotates about a vertical axis at a rate lower than the rate at which the boomerang as a whole travels along its path. When the spin plane lags behind, the attack angle becomes, increasingly negative and the boomerang loses lift.
If the spin plane precesses too quickly, it turns about a vertical axis faster than the boomerang as a whole travels along the large circular path. As a result the attack angle becomes increasingly positive until the spin plane is perpendicular to the oncoming airstream. The air drag would surely ruin the flight by then.
The match between the precessional rate and the rate at which the boomerang travels along the large circular path is not critical and is in fact somewhat automatic, since both rates depend on the lift. Throw your boomerang, sand down and reshape the arms and throw it again until you come near the match and the boomerang returns. I know of no sure way to remedy a persistently unsuccessful boomerang.
The circular path of the boomerang is independent of the speed with which you throw it. Only the moment of inertia and the cross-sectional shape of the boomerang determine the radius of the path. With a given boomerang you will therefore achieve the same large circle (for the same throw of the boomerang in the vertical plane I have been assuming) regardless of how hard you throw the device (provided, of course, you throw it hard enough so that it has sufficient speed to complete its journey). If you want to change the size of the circle, you must ordinarily choose a different boomerang with a different moment of inertia or cross-sectional shape. Next month, however, I shall explain how you can also add ballast to the arms in order to increase their moment of inertia. This technique is used by boomerang throwers intent on breaking distance records.
A Frisbee flies in a quite similar way. It has a curved top surface and is launched with a flick of the wrist to give it spin. The Frisbee gains lift by virtue of the impact of the air or by the difference in air speed across the top and bottom surfaces. A Frisbee properly thrown in an almost vertical plane will return to the thrower the way a boomerang does. Usually, however, a Frisbee is launched to curve slightly to another person, so that the thrower orients the spin plane to provide just enough horizontal lift to achieve the curve.
Both a boomerang and a Frisbee can be skipped across the ground without destroying the flight. Imagine a Frisbee skimming just above the ground with its leading edge tipped slightly downward. That edge then strikes the ground. The force from the ground at the contact point puts a torque on the Frisbee and changes the angular momentum, but because the change vector is almost perpendicular to the original angular-momentum vector the new angular-momentum vector is just a rotation of the original one. The angular momentum does not change appreciably in size, only in direction. Therefore the spin of the Frisbee is not much slowed; the device is merely reoriented and then goes sailing off in a new direction.
Bibliography
BOOMERANGS: MAKING AND THROWING THEM. Herb A. Smith. Gemstar Publications, Arun Sports, Littlehampton, Sussex, 1975.
THE PHYSICS OF FRISBEE FLIGHT. Jay Shelton in Frisbee: A Practitioner's Manual and Definitive Treatise, edited by Stancil E. D. Johnson. Workman Publishing Co., 1975.
MANY HAPPY RETURNS: THE ART AND SPORT OF BOOMERANGING. Benjamin Ruhe. The Viking Press, 1977.
A BOOMERANG is surely one of the oddest devices ever to serve as a weapon or a plaything. It was apparently invented by accident, notably in Australia by the native people but also independently in many other places. If you throw an ordinary stick, it falls to the ground not far away, but a boomerang can travel as much as 200 meters (round trip) and can be aimed so skillfully by an expert thrower that game or an enemy can be hit. The boomerang probably originated as a weapon designed for accurate straight flight, but most people find the returning boomerang, which was mainly just a plaything for the Australians, more interesting. Ironically the straight-flying boomerang is probably more complicated aerodynamically than the returning boomerang. As ancient as both devices are, an amateur investigator can still do a great deal to advance the understanding of their flight.Although good returning boomerangs are occasionally available in sporting-goods stores, most commercial returning boomerangs are mass-produced and fly poorly. Indeed, many of them do not even return. The plastic boomerang made by Wham-O is one of the best types available in toy stores. Many other types, all of them excellent, are available from Ruhe-Rangs, Box 7324, Benjamin Franklin Station, Washington, D.C. 20044, thanks to Benjamin Ruhe, a boomerang enthusiast formerly with the Smithsonian Institution. During his service there Ruhe helped to organize the Annual Smithsonian Open Boomerang Tournament. The tournament, held in late spring or early summer, is great fun for the 100 or so contestants who enter. This year it is scheduled for June 9 on the mall in Washington.
If you would like to experiment with boomerangs, you need to be able to construct your own. Only then can you make the variations necessary to determine what factors influence a boomerang's flight. The best material from which to cut the basic stock of a boomerang is Baltic birch in a marine or aircraft plywood between 1/4 and 3/8 inch thick, with five or more laminations. This type of plywood is resistant to wear and water; it is also dense, which makes for a boomerang that is heavy for its size.
Herb Smith's "Gem" design for a boomerang
Cut a cardboard pattern for a boomerang of whatever shape you want. An example is the returning boomerang designed by Herb Smith and shown in the illustration at the left. (If you are left-handed, you will have to make a left-handed boomerang, which is the mirror image of the one drawn.) Place the pattern on the plywood and mark off the outline of the boomerang in pencil.
With a coping saw or a handsaw cut out the boomerang blank. The edges and the top must now be shaped in the general form shown in the illustration. (Little is done to the bottom other than putting a slope on what will be the leading edge.) The leading edge must be blunt, whereas the trailing edge must be sharper, with the top surface sloping down to meet the unaltered, flat bottom surface. The arms must have the cross-sectional shape of a classic airfoil, because they must provide lift much as the classic airfoil does.
Clamp the blank in a vise and cut and shape the edges and the top with a rasp that has a curved surface. Smooth out the grooves left by the rasp and finish shaping the wood by rubbing the surfaces with a piece of coarse sandpaper wrapped around a piece of soft wood. Before you finish off the surface with a finer sandpaper you should make a test flight with the boomerang so that you can tune it by more shaping with the rasp or with coarse sandpaper. Tuning the boomerang means that you throw it, cut or sand it more and then throw it again until it flies the way you want it to.
A boomerang cannot be thrown well in a strong wind. If there is a light wind, face toward it, turn 45 degrees to the right and throw the boomerang in that direction. Hold the boomerang vertically by the tip of one of the arms (which one usually does not matter) with the flat side away from you. Reach behind your head with the boomerang and then throw it toward the horizon, snapping it forward when your arm is fully stretched forward. Do not try to throw hard at first. It is the snap that counts, not the strength of the throw. The boomerang stays up in the air because of the spin the snap imparts to it.
The proper orientation of the boomerang (the plane in which it will be spinning) will vary according to the wind conditions and the type of boomerang. To achieve a good flight you might have to throw the boomerang with it plane nearly vertical. Under other circumstances you will have to shift the plane (tilting the top of the boomerang away from you) by as much as 45 degrees. The greater the tilt of the plane in which the boomerang spins is, the more upward lift the boomerang will initially have. If you give it too much lift, it climbs too fast and then plummets to the ground with such force that it may break.
Preparing to throw a boomerang
In a good flight a returning boomerang travels horizontally around an imaginary sphere. On returning it probably will hover or even loop a bit before it drops to the ground at your feet. If you are lucky, it might make one or two additional circles (smaller than the first circle) before it falls. Although you launch the boomerang with its spin plane almost vertical, it probably will return with the plane nearly horizontal. I shall explain below why the plane must turn over in this way if the boomerang is to complete the trip.
If your boomerang consistently lands to your right in a light wind, try throwing it a bit more to the left of the wind. Similarly, if it is landing to your left, try throwing it more to the right of the wind. If it lands behind you, try throwing it with less force. If that does not work, throw it somewhat above the horizon with the spin plane tilted less from the vertical. If the day is windless and you are not getting a full return, tilt the spin plane farther from the vertical in order to gain more lift during the flight.Be careful not to injure people or damage things with your boomerang. It can be quite a weapon. Throw it only in a large, open space. If people are present, make sure they know what you are doing so that they can be prepared to dodge.
The successful tuning of a boomerang involves both experience and luck. In general if you make the top surface more curved, the boomerang will have more lift, which means that it will return in a tighter circle. Flattening the top surface or curving the bottom surface will give the boomerang less lift because the cross-sectional shape of the arms is then less like a classic airfoil. If the spin decreases too fast, so that the boomerang falls to the ground in mid-flight, the reason may be that excessive air drag on the arms is robbing them of their spin. Some surface roughening might be beneficial to the flight, but any large grooves left by the rasp will surely create additional air drag that will shorten the flight time.
Instead of shaping the arms carefully you might prefer to twist them so that during a flight the leading edge on each arm is tilted to deflect the passing air to the right, giving the boomerang a lift to the left. This type of lift is easy to visual window of a moving car and turn it through various angles, you can feel the lift. To twist a boomerang heat it gradually in an oven at 400 degrees Fahrenheit and then (with gloves on, of course) carefully twist the arms until the wood is cool. If you twist too much, heat the boomerang again and twist the arms back a bit.
A typical flight path
Once you have a properly flying boomerang you might want to finish it-off with a cellulose covering and some decorative designs. Smith's excellent booklet, which is listed in the bibliography for this issue [below], explains how to do this kind of finishing and also gives a number of boomerang designs. If your boomerang breaks, do not throw away the pieces. Glue them together with epoxy, clamp them until they dry and then file and sand the surface back into the desired shape. Although the boomerang will not be as strong as it was before, its flight path might be altered in an interesting way by the small change in its distribution of mass resulting from the break.
A boomerang does not have to be limited to two arms. Indeed, one of the easiest boomerangs to build is a four-blade design consisting of two rulers crossed and fastened at the center. The right kind of ruler has a curved top surface and a fairly flat bottom surface. You can attach two such crossed rulers either by wrapping a strong rubber band around them or by putting a bolt and nut through the center hole they usually have. Throw this type of boomerang the same way that you would a two-armed one. Take care to avoid being cut by sharp edges, and never use rulers that have metal edges.
A simple cross boomerang can be fashioned from a cardboard square about five inches on a side. Cut out a boomerang with three or four blades and twist them slightly so that the boomerang is not quite all in the same plane. Adding weight to the arms of a boomerang increases its range. With the cardboard boomerang it is easy to add weight by attaching paper clips to the end of the arms. This boomerang can be demonstrated in a classroom. If it has too much range for a classroom, you can decrease the range by increasing the twist on the arms so that the boomerang is less in the same plane or by bending the arms along a center line through their length. With the latter technique the arms have an exaggerated airfoil shape: at least one side is sharply convex but the other is not. As usual, throw the boomerang with the convex side toward you. By changing the arms from being almost flat to being more like an airfoil, you increase the lift on the boomerang: its path will be a tighter circle.When you begin to throw your wood boomerang well, you might be tempted to catch it. The result may be a sharp blow to your fingers. If you are determined to catch a boomerang, hold your hands flat and slap them together to trap the boomerang as it hovers above the ground, still spinning, in the last stage of its flight. Keep your fingers away from the turning blades.
A four-blade design made with plastic rulers and rubber bands
The explanation of the return of a boomerang lies primarily in the cross-sectional shape of the arms and the fact that the boomerang spins. Without these two features a boomerang would behave like any other thrown stick. The cross-sectional shape gives the boomerang aerodynamic lift similar to the lift generated by some airplane wings. The spinning gives the boomerang stability. Through a bit of fortunate rotational mechanics the spinning also causes the axis about which the boomerang spins to rotate in much the same way that the spin axis of a top rotates about the vertical. The lift and the stability keep the boomerang up, and the rotation of the spin axis brings it back to the thrower.
Aerodynamic lift can be explained with a simple model of a classic airplane wing similar to the one I described in this department for February, 1978, to explain the lift of a kite. The classic airfoil has a flat bottom, a blunt front, a sharp rear and a convex top. Air passes around a wing faster along the top of the wing than along the bottom. The reason can be seen by visualizing the passing air as being of two kinds. One kind flows around the wing with no rotation in the stream and with the same speed on the top and the bottom of the wing. The other is a circulation cell: it flows to the rear over the top of the wing and to the front over the bottom. Such a circulation is created by a real wing because the air's viscosity and its adhesion to the surface of the wing force it into this pattern as it flows to the rear off the curved top surface.
In the superposition of the two idealized airstreams the two velocities add above the wing and subtract below it, with the result that the real air speed is greater above the wing than it is below it. The difference is important to the lift because the air pressure in the stream is inversely related to the speed of the stream. Hence the air pressure is less above the wing than below it, and the wing gets a push upward. (A real airplane wing can have a more complicated airflow pattern than this simple model implies. Moreover, when an airplane is traveling at high speed, some of the lift may come from the impact of the passing air on the underside of a wing that is inclined slightly upward in order to deflect the air downward.)
If the classic airfoil is inclined to the airstream in such a way that the airstream is more incident on the curved top side, the lift is of course less. Such an arrangement is termed a negative angle of attack. In a simple model the reduction of lift is due to the downward push the incident stream exerts on the top surface. One might also argue that lift is partially lost because the tendency for the air to circle about the wing is lessened and the speed of the air on the top side of the airfoil differs less from the speed on the bottom side.
Pattern of airflow over the arm of a flying boomerang
Conversely, if the airfoil is inclined so that the airstream is incident somewhat more on the flat bottom side than on the top, a situation that would be called a positive angle of attack, the lift increases because of the upward push from the airstream on the bottom side. The air drag also increases. If the angle is too large, the disadvantages of increased air drag outweigh the advantages of lift. The attack angle of the arms of the boomerang as they turn through the air is important to its flight.
Boomerang arms can have a variety of cross-sectional shapes, but most of them are similar in cross section to the classic airfoil. Usually this shape includes a blunt edge that turns into the air as the boomerang spins and a sharper edge that trails during the turn. One side is usually flat and the other convex. Variations on this basic form are numerous, however, and little systematic work seems to have been done on determining which shapes are best aerodynamically. Some boomerangs are actually flat on both sides but with their arms twisted so that the air is deflected as the arms turn through the wind.
The lift on a boomerang differs in a major way from the lift on the classic airplane wing. In the first stage of a flight the boomerang's "lift" is mostly horizontal, with only enough upward force to balance the weight of the device. Since the boomerang is spinning mostly about a horizontal axis, the curved sides of the arms spin in a plane that is almost vertical and the lift is almost horizontal. For the sake of simplicity in what follows I shall ignore the weight of the boomerang. I shall also assume a boomerang that is thrown outward by a right-handed thrower so that the plane of spinning is initially exactly vertical. The lift will be to the thrower's left, so that the boomerang begins to move to the left as it continues to spin in the vertical plane.
If this were the entire story, the boomerang would never come back. To see why it turns around and returns you must understand what else the lift does to the boomerang. In particular it is necessary to know how the torque due to the lift on the boomerang causes a precession of the spin plane.
Imagine that one of the boomerang's arms has spun to its highest possible position and the other arm is almost in its lowest possible position. (I am discussing the basic banana-shaped boomerang.) The upper arm is turning in the same direction in which the center of the boomerang is moving, whereas the lower arm is moving opposite to the motion of the center. The air passing the upper arm is moving faster (in relation to the arm) than the air passing the lower arm. Therefore more lift is generated on the upper arm than on the lower one. The part of the boomerang higher in the spin will always experience a greater lift and hence a greater push to the side than the part lower in the spin.
Positive and negative attack angles
My first thought was that the difference in horizontal lift (more lift on the upper arm than on the lower one) would cause the spin plane of the boomerang to tilt, thereby angling the lift downward (a disastrous effect). What actually happens, however, is that the difference in lift causes a rotation of the plane about a vertical axis. It is this rotation of the spin plane, commonly called precession, that brings the boomerang back.
To understand what causes the rotation you must examine the torque created by the lift. Take the center of the boomerang as the axis about which it is spinning. (Actually the center of mass around which a two-armed boomerang spins is likely to be well off center, but that does not alter the outcome of the argument.) Take the average lift on the upper arm as being directed horizontally outward from the center of the arm. Similarly, take the average lift on the lower arm as being also directed horizontally outward from the center of the arm. The torque created by one of these lifts, as measured from the center of the boomerang, is the product of the lift and the distance to where the lift is applied, that is, half the length of an arm. Since the upper arm has the greater lift, it also has the greater torque.
If the boomerang were not spinning this difference in torques would merely make the plane of the boomerang tilt over. Since the upper arm has the greater torque, the plane would tilt counterclockwise as seen by the person who has just thrown the boomerang. The fact that the boomerang is spinning, however, makes a big difference, because it then has angular momentum and the tendency to tilt the spin plane results in a rotation of the spin plane about the vertical axis.
Average lift on the upper and lower arms
Angular momentum is the product of the boomerang's rate of spin and a function involving the mass and the mass distribution of the device. For an example in another setting imagine yourself attempting to turn a merry-go-round holding several children. The force you apply tangent to the rim multiplied b the radius of the merry-go-round is the torque you are supplying. When you begin, the torque causes an angular acceleration of the merry-go-round; the spin increases from zero to some final value. How would you arrange the children in order to achieve a given angular acceleration with the least force? Intuitively you would place them near the center. Their mass is the same, of course, but their mass distribution with respect to the center of rotation is different. When the mass is nearer the center, the merry-go-round is easier to turn. The mass and its distribution are taken into account by the function known as the moment of inertia. The greater the mass or the farther from the center it is placed, the greater the moment of inertia and the greater the force you will have to supply in order to achieve a given angular acceleration.
Once the merry-go-round is spinning and you are no longer pushing on the rim, the apparatus has a certain angular momentum because it has spin and a moment of inertia. Angular momentum is usually represented by a vector pointing perpendicularly to the plane in which the object is spinning. Here the vector would be vertical. The direction (up or down) is chosen by convention as being the direction of the thumb on the right hand when the hand is held in a hitchhiker's pose with the fingers curled in the direction of the spin of the object.
The only way you could change the size or the direction of such a vector would be to apply another torque to the object. With a merry-go-round you could push on the rim again. (A convention for choosing how to draw a vector representing the change in angular momentum involves pointing the index finger of the right hand from the center of the rotation toward the place where the force is applied and pointing the middle finger in the direction of the applied force. If you make your thumb on that hand perpendicular to both fingers, it automatically points in the direction of the change in the angular momentum. The new angular-momentum vector is the combination of the old one and the vector representing the change.) With a merry-go-round that you had resumed pushing tangent to the rim the new vector would still be vertical but would be larger or smaller depending on whether your aim was to make the merry-go-round turn faster or slower.
How to determine angular momentum and a change in it A boomerang that is spinning has two torques acting on the arms, one created by the average lift on the upper arm and one created by the average lift on the lower arm. Since the lift on the upper arm is greater, it determines what happens to the angular momentum, and so I shall ignore the lift on the lower arm. (The argument would not change even if I included the smaller lift.) Imagine that the boomerang is receding from you just after you have thrown it with your right hand. It is spinning in a vertical plane and has an angular-momentum vector pointing to your left. The average lift on the upper arm creates a torque that will change the direction of the vector as the boomerang continues to fly away.
To determine how the vector changes use your right hand, orienting the fingers and the thumb properly. With your index finger pointing from the center of the boomerang to the center of the upper arm and your middle finger pointing to your left in order to be in the direction of the lift on that arm, your outstretched thumb points toward you. Therefore the vector representing the change in angular momentum points toward you. Mentally combining the change vector and the original vector is best done from an overhead point of view. The change vector is perpendicular to the original one and gives a new vector rotated from the old one toward you. The size of the angular momentum is unaltered because the change vector is perpendicular to the old one. Only the direction of the angular momentum is changed, and it is rotated about a vertical axis to point more toward you.
This type of rotation of an angular-momentum vector is precession; it is seen when the axis of a top precesses about the vertical. Another common example of precession is seen in the turning of a motorcycle. The wheels of a motorcycle spin fast enough and have moments of inertia sufficiently large to make their angular momentum large. To turn the motorcycle you cannot just turn the handlebars, as you would when riding a bicycle. Instead you make the motorcycle lean in the direction of the turn. The torques then experienced by the motorcycle cause the angular-momentum vectors of the wheels to precess, turning the motorcycle as a whole.
How precession keeps the attack angle positive
During the precession of the spin plane of a boomerang the boomerang continues to travel along a path with a certain speed but is continuously deflected by the horizontal lift it experiences. The resulting path approximates a large circle. In a successful boomerang flight the spin plane will precess at the same rate at which the device circles in its path. Its angle of attack remains somewhat positive. This match is necessary in order to keep the arms at the proper attack angle.
Suppose the boomerang precesses too slowly. Then as it travels along its circular path its spin plane rotates about a vertical axis at a rate lower than the rate at which the boomerang as a whole travels along its path. When the spin plane lags behind, the attack angle becomes, increasingly negative and the boomerang loses lift.
If the spin plane precesses too quickly, it turns about a vertical axis faster than the boomerang as a whole travels along the large circular path. As a result the attack angle becomes increasingly positive until the spin plane is perpendicular to the oncoming airstream. The air drag would surely ruin the flight by then.
The match between the precessional rate and the rate at which the boomerang travels along the large circular path is not critical and is in fact somewhat automatic, since both rates depend on the lift. Throw your boomerang, sand down and reshape the arms and throw it again until you come near the match and the boomerang returns. I know of no sure way to remedy a persistently unsuccessful boomerang.
The circular path of the boomerang is independent of the speed with which you throw it. Only the moment of inertia and the cross-sectional shape of the boomerang determine the radius of the path. With a given boomerang you will therefore achieve the same large circle (for the same throw of the boomerang in the vertical plane I have been assuming) regardless of how hard you throw the device (provided, of course, you throw it hard enough so that it has sufficient speed to complete its journey). If you want to change the size of the circle, you must ordinarily choose a different boomerang with a different moment of inertia or cross-sectional shape. Next month, however, I shall explain how you can also add ballast to the arms in order to increase their moment of inertia. This technique is used by boomerang throwers intent on breaking distance records.
A Frisbee flies in a quite similar way. It has a curved top surface and is launched with a flick of the wrist to give it spin. The Frisbee gains lift by virtue of the impact of the air or by the difference in air speed across the top and bottom surfaces. A Frisbee properly thrown in an almost vertical plane will return to the thrower the way a boomerang does. Usually, however, a Frisbee is launched to curve slightly to another person, so that the thrower orients the spin plane to provide just enough horizontal lift to achieve the curve.
Both a boomerang and a Frisbee can be skipped across the ground without destroying the flight. Imagine a Frisbee skimming just above the ground with its leading edge tipped slightly downward. That edge then strikes the ground. The force from the ground at the contact point puts a torque on the Frisbee and changes the angular momentum, but because the change vector is almost perpendicular to the original angular-momentum vector the new angular-momentum vector is just a rotation of the original one. The angular momentum does not change appreciably in size, only in direction. Therefore the spin of the Frisbee is not much slowed; the device is merely reoriented and then goes sailing off in a new direction.
Bibliography
BOOMERANGS: MAKING AND THROWING THEM. Herb A. Smith. Gemstar Publications, Arun Sports, Littlehampton, Sussex, 1975.
THE PHYSICS OF FRISBEE FLIGHT. Jay Shelton in Frisbee: A Practitioner's Manual and Definitive Treatise, edited by Stancil E. D. Johnson. Workman Publishing Co., 1975.
MANY HAPPY RETURNS: THE ART AND SPORT OF BOOMERANGING. Benjamin Ruhe. The Viking Press, 1977.