The Asymmetric Aerofoil Shape: Generating Lift

Why does a boomerang come back?

A boomerang returns to the thrower due to a fascinating interplay of aerodynamic forces, primarily lift, torque generated by the spinning motion (gyroscopic precession), and its unique asymmetric shape. This seemingly simple toy embodies complex physics principles that have captivated observers for centuries. To truly understand this returning flight, we need to delve into the intricacies of its design, the forces acting upon it, and the resulting motion.

At its heart, the boomerang’s ability to fly, let alone return, stems from its carefully crafted shape. Unlike a flat piece of wood, a traditional returning boomerang possesses an aerofoil cross-section on its arms. This means that the top surface of each arm is typically more curved and longer than the bottom surface, similar to the wing of an aircraft.

When the boomerang is thrown with a spin, air flows over both the top and bottom surfaces of its rotating arms. Due to the curved upper surface, the air traveling over the top has a longer distance to cover compared to the air flowing underneath. According to Bernoulli’s principle, faster-moving air exerts lower pressure. Consequently, the pressure on the top surface of the spinning boomerang’s arms becomes lower than the pressure on the bottom surface. This pressure difference generates an upward force called lift, perpendicular to the direction of the airflow relative to each arm.

2. The Role of Spin: Introducing Gyroscopic Precession

While lift provides the upward force necessary for flight, it is the spin imparted to the boomerang during the throw that is crucial for its return. This spin introduces the concept of angular momentum, a measure of an object’s tendency to continue rotating. The angular momentum vector of the spinning boomerang points along its axis of rotation, following the right-hand rule (if the fingers of your right hand curl in the direction of the spin, your thumb points in the direction of the angular momentum vector).

Now, consider the lift force acting on the spinning arms. As each arm moves through the air, it experiences lift perpendicular to its instantaneous velocity relative to the air. Importantly, the velocity of each arm is a combination of the boomerang’s overall translational velocity (the direction it’s moving through the air) and the velocity due to its rotation.

Due to the spin, one arm of the boomerang will always be moving forward relative to the overall flight path, while the other arm will be moving backward relative to it. The arm moving forward experiences a higher relative airspeed and therefore generates more lift than the arm moving backward, which experiences a lower relative airspeed and generates less lift.

This difference in lift between the advancing and retreating arms creates a torque (a rotational force) on the boomerang. This torque acts perpendicular to both the lift force and the angular momentum vector. According to the principles of gyroscopic precession, when a torque is applied to a spinning object, it doesn’t rotate in the direction of the torque. Instead, the axis of rotation (and hence the direction of the angular momentum vector) precesses, or tilts, in a direction perpendicular to both the torque and the original angular momentum vector.

In the case of a right-handed throw (counter-clockwise spin when viewed from above), the advancing arm generates more lift, creating a torque that tries to tilt the leading edge of the boomerang upwards. However, due to gyroscopic precession, this torque causes the plane of rotation of the boomerang to tilt to the left (for a right-handed thrower). This tilting continues throughout the flight.

3. The Curved Flight Path: Combining Lift and Precession

The combination of lift and gyroscopic precession results in the characteristic curved flight path of a returning boomerang.

• Initial Throw: The thrower imparts both forward velocity and a strong spin to the boomerang, typically at an angle to the horizontal.
• Lift Generation: The aerofoil shape of the spinning arms generates lift, counteracting gravity and allowing the boomerang to stay aloft.
• Differential Lift and Torque: The advancing arm experiences more lift than the retreating arm, creating a torque.
• Gyroscopic Precession: This torque causes the plane of rotation (and the direction of forward motion) to gradually tilt in the direction of the precession (to the left for a right-handed throw).
• Curved Trajectory: As the plane of rotation tilts, the direction of the lift force also changes. This continuously alters the boomerang’s flight path, causing it to curve.
• Return: If the initial throw is executed correctly with the right amount of spin, angle, and force, the precession will cause the boomerang to complete a roughly elliptical or circular path, eventually returning to the vicinity of the thrower.

4. Factors Influencing the Boomerang’s Return

Several factors significantly influence whether and how a boomerang returns:

• Shape and Design: The number of arms, their length, width, thickness, and the degree of aerofoil shaping are crucial design elements. Different shapes and designs are optimized for different flight characteristics and throwing styles. A more pronounced aerofoil will generate more lift. The angle of dihedral (the upward angle of the arms relative to the center) also plays a role in stability and flight path.
• Weight Distribution: The distribution of weight along the arms affects the moment of inertia and the way the boomerang responds to torque. Even weight distribution is generally preferred for stable returns.
• Spin Rate: A sufficient spin rate is essential for generating the differential lift and the gyroscopic effect necessary for precession. A weak throw with insufficient spin will likely result in the boomerang fluttering and falling without returning.
• Throwing Angle: The angle at which the boomerang is thrown relative to the horizontal and the wind direction significantly impacts its trajectory. Typically, returning boomerangs are thrown with a significant upward angle.
• Wind Conditions: Wind can have a substantial effect on the boomerang’s flight path.¹⁸ Skilled throwers often adjust their throwing technique and angle to compensate for wind.¹⁹ A headwind can shorten the return, while a tailwind can lengthen it or even prevent a return if not managed correctly. Crosswinds require careful aiming.
• Throwing Force: The initial force of the throw determines the initial velocity of the boomerang. This velocity, along with the spin, influences the amount of lift generated.
• Aerodynamic Drag: Air resistance (drag) acts against the motion of the boomerang, slowing it down and affecting its trajectory. The design of the boomerang aims to minimize drag while maximizing lift.

5. The Physics in Detail: A More Mathematical Perspective

While a qualitative explanation provides a good understanding, a more detailed look at the physics involves mathematical concepts:

• Lift Equation: The lift force (L) on each arm can be approximated by the equation:L=21​ρv2CL​Awhere:
  • ρ is the air density.v is the relative airspeed over the arm.CL​ is the lift coefficient (dependent on the aerofoil shape and angle of attack).A is the surface area of the arm.
  As mentioned earlier, the relative airspeed (v) is different for the advancing and retreating arms due to the rotation, leading to a difference in lift.
• Torque Equation: The torque (τ) generated by the differential lift can be expressed as:τ=(Ladvancing​−Lretreating​)×rwhere r is the effective distance from the center of rotation to the point where the lift forces act.
• Gyroscopic Precession: The relationship between the applied torque (τ), the rate of change of angular momentum (dtdL​), and the angular momentum (L) itself is given by:τ=dtdL​This equation shows that the torque causes a change in the direction of the angular momentum vector, leading to precession. The angular velocity of precession (ωp​) is related to the torque and the moment of inertia (I) and angular velocity of spin (ωs​) by:ωp​=Iωs​τ​This indicates that a higher spin rate leads to a slower rate of precession.

6. The Evolution of the Boomerang: From Hunting Tool to Sport

It’s important to note that while returning boomerangs are popular as sports equipment and toys, the earliest boomerangs were primarily hunting tools used by indigenous cultures, particularly in Australia. These hunting boomerangs were often larger and heavier, designed to be thrown with force and accuracy to strike prey. They were not necessarily designed to return.

The returning boomerang likely evolved as a variation, perhaps used for bird hunting in flight or as a decoy. The specific aerodynamic principles that allowed for their return were likely discovered through generations of experimentation and refinement.

Today, the returning boomerang is a sophisticated aerodynamic device, with various shapes and designs optimized for different throwing styles and competitive events. These events include accuracy throwing, trick catching, maximum time aloft, and distance throwing (where non-returning boomerangs are used).

7. The Art of Throwing: Mastering the Return

Even with a well-designed returning boomerang, a proper throwing technique is essential for a successful return. This involves:

• Grip: Holding the boomerang correctly, usually with a firm grip on one of the arms.
• Stance: Positioning the body to allow for a smooth and powerful throw.
• Throwing Motion: A fluid motion that imparts both forward velocity and a strong spin. The release angle is crucial and often involves throwing the boomerang at a significant angle to the vertical (often between 45 and 90 degrees).
• Follow-Through: Completing the throwing motion to ensure maximum spin and controlled release.

Learning to throw a boomerang accurately with a reliable return takes practice and an understanding of how the forces and the boomerang’s design interact.

8. Conclusion: A Symphony of Physics and Design

The returning boomerang is a remarkable example of applied physics. Its seemingly magical return is not due to chance but rather the result of a carefully engineered asymmetric aerofoil shape interacting with the principles of lift and gyroscopic precession, all initiated by a well-executed throw. The differential lift created by the spin generates a torque, which, through gyroscopic precession, causes the plane of rotation and the direction of flight to tilt, resulting in a curved path back to the thrower. The design of the boomerang, the spin imparted during the throw, the throwing angle, and even environmental factors like wind all play crucial roles in this fascinating aerodynamic ballet. From its ancient origins as a hunting tool to its modern form as a sport and recreational device, the boomerang continues to intrigue and inspire with its elegant dance through the air. Understanding the physics behind its return offers a profound appreciation for the intricate ways in which shape, motion, and the laws of nature can combine to create such a captivating phenomenon. The boomerang’s flight is a testament to the power of observation, experimentation, and the enduring human fascination with harnessing the forces of the natural world.