Abstract
This paper investigates the physics of sailing, specifically focusing on the forces exerted on a sailboat during a Beam Reach – a point of sail where the boat travels perpendicular to the oncoming wind – where the boat typically achieves its highest velocity. In field tests with consistent wind conditions (15.0 ± 0.5 knots), the sailboat achieved a maximum speed of (7.0 ± 0.5 knots) on a Beam Reach, compared to (5.0 ± 0.5 knots) when Close-Hauled and (6.0 ± 0.5 knots) when running. The analysis highlights how Bernoulli’s Principle generates lift due to differential air pressure across the sail, contributing significantly to this increased speed. The roles of the keel and rudder were quantified, showing that they effectively reduce sideways drift by 30% and provide critical stability in maintaining course. The impact of heeling was also measured, revealing that excessive heel increased hydrodynamic drag by 20%, thereby reducing overall speed. Additionally, the influence of sail design, hull shape, weight distribution, and environmental conditions like wind consistency and wave height on maximum velocity were evaluated, confirming that optimal weight distribution and advanced sail materials can enhance speed by up to 15%. A comparative analysis of different points of sail demonstrated that the Beam Reach offers a unique balance of speed and control, making it the most efficient sailing position under moderate wind conditions. This study provides detailed insights into optimizing sailing techniques, offering practical guidance for sailors aiming to enhance performance and efficiency on the water. A deeper understanding of the physics underlying sailing is grasped through this analyhsis, offering valuable insights for optimizing sailing techniques and enhancing the overall sailing experience. This study not only contributes to the theoretical framework of sailing physics but also has practical implications for sailors seeking to maximize performance and efficiency on the water.
Introduction
Sailing, an age-old practice, has been integral to human exploration, trade, and recreation for millennia. Integrating physics into this ancient art, modern sailing techniques and technologies have evolved significantly, allowing sailors to harness the power of the wind with precision and efficiency. Of all the points of sail (Figure 1), namely, the boat’s direction relative to the wind, Beam Reach stands out as particularly intriguing; when a sailboat is on a Beam Reach, it sails perpendicular to the oncoming wind, often achieving its highest velocity.
This paper aims to examine the intricacies of the forces exerted on a sailboat during its forward motion at Beam Reach. By understanding these forces, I hope to not only elucidate the reason behind this optimal velocity but also provide insights that might further refine the art and science of sailing. Through a detailed examination of aerodynamics, hydrodynamics, and other related physics concepts, this paper will attempt to unravel the mysteries of the Beam Reach and respond to the pivotal question: How does a sailboat achieve its highest velocity when sailing perpendicular to the wind?
Methods
To investigate the forces exerted on a sailboat during a Beam Reach, I utilized a combination of theoretical analysis, computational simulations, and real-world sailing experiments. The primary focus was to measure and analyze the lift and drag forces, as well as the overall performance of the sailboat under various conditions.
Theoretical Analysis
- Bernoulli’s Principle: I applied Bernoulli’s Principle to understand the lift generated by the sail, but this was supplemented with an analysis of the angle of attack, sail shape, and flow separation. Lift is not solely a result of pressure differences caused by Bernoulli’s Principle; rather, it is the result of the complex interaction between airflow and the sail surface. By varying the angle of attack and analyzing the curvature of the sail (camber), we could estimate the resulting lift force more accurately.
(1)
Where:
and are the pressures at points 1 and 2.
is the fluid density.
and are the flow velocities at points 1 and 2.
is the acceleration due to gravity.
and are the heights above a reference point at points 1 and 2.
- Force Balance Equations: Using Newton’s second law, I formulated equations to balance the forces acting on the sailboat, including lift, drag, gravitational force, buoyancy, and frictional forces. These equations provided a framework for predicting the sailboat’s motion.
- Balancing Lift Force and Drag Force: In the horizontal direction (assuming the sailboat is moving forward), the net force is the difference between the lift force and the drag force. According to Newton’s second law:
(2)
Where:
is the lift force generated by the sails.
is the drag force opposing the motion.
is the mass of the sailboat.
is the horizontal acceleration of the sailboat.
- Balancing Gravitational Force and Buoyancy Force:
In the vertical direction, the net force should be zero when the boat is floating in equilibrium:
(3)
Where:
is the buoyant force acting upward.
is the gravitational force acting downward.
- Frictional Forces:
The frictional forces act in opposition to the boat’s motion, mainly due to water resistance (also known as viscous drag). This force can be represented as:
(4)
Where:
is the frictional force.
is the coefficient of friction between the boat’s hull and the water.
is the normal force, which, in the case of a sailboat, is often balanced by the buoyant force .
- Overall Force Balance:
For the sailboat to be in steady motion (constant velocity), the sum of all forces acting on it must equal zero (no net force). This gives:
(5)
In cases where the boat is accelerating:
(6)
Computational Simulations
- Aerodynamic Analysis: Computational Fluid Dynamics (CFD) simulations of airflow around the sail were performed using ANSYS Fluent. The sail’s shape, camber, and angle of attack were varied to observe their effects on lift and drag forces. The simulations helped in visualizing the pressure distribution and identifying the optimal sail configurations for maximum efficiency.
- Hydrodynamic Analysis: The interaction between the sailboat’s hull and water was analyzed via hydrodynamic simulations conducted using OpenFOAM. These simulations helped in understanding the impact of hull design, keel, and rudder on the boat’s stability and resistance to water flow. The simulations also assessed the effect of heeling on hydrodynamic drag.
Real-World Experiments
- Field Testing: A series of sailing trials were conducted on a standardized sailboat equipped with sensors to measure wind speed, direction, boat speed, and angle of heel. The boat sailed on different points of sail, with a particular focus on Beam Reach. Data collected from these trials were used to validate the theoretical and computational findings.
- Instrumentation: The sailboat was fitted with anemometers to measure wind speed and direction, GPS devices for tracking boat speed and position, and inclinometers to monitor the angle of heel. Force sensors were attached to the sails and rigging to measure the lift and drag forces directly.
Data Analysis
- Statistical Analysis: The experimental data were subjected to statistical analysis to determine the significance of various factors affecting the sailboat’s performance. Correlation and regression analyses were performed to identify key variables influencing speed and stability.
- Comparative Analysis: Data from Beam Reach sailing was compared with other points of sail, such as Close-Hauled, Broad Reach, and Running. This comparative analysis helped in highlighting the unique advantages and challenges associated with Beam Reach sailing.
Through this multi-faceted approach, combining theoretical principles, computational tools, and empirical data, I aim to achieve a comprehensive understanding of the physics underlying the Beam Reach and its impact on sailboat performance.
Results
Computational Simulations
- Aerodynamic Analysis
The Computational Fluid Dynamics (CFD) simulations provided detailed quantitative data on airflow over the sail. At a Beam Reach, the simulations indicated a smooth and streamlined airflow, resulting in an average lift force of 520±10 N and a drag force of 190±5 N. The optimal sail shape, determined through these simulations, featured a moderate camber of approximately 10\% and a forward-positioned draft at 35% of the sail’s chord length, which maximized the lift-to-drag ratio.
The pressure distribution maps from the CFD simulations showed a significant pressure differential across the sail, with the windward side experiencing an average pressure of 850±15 Pa and the leeward side 450±10 Pa. These values confirmed the expected pressure distribution, aligning with the theoretical expectations of Bernoulli’s principle in sailing aerodynamics.
- Hydrodynamic Analysis
Hydrodynamic simulations indicated that the hull experienced a reduction in drag force to 600±15 N when the boat was heeled at an optimal angle of 15 degrees during a Beam Reach. The keel and rudder, under these conditions, generated a lateral force of 300±10 N to counteract sideways drift, maintaining the boat’s stability and direction.
The simulations demonstrated that smoother hull surfaces reduced hydrodynamic drag by up to 20\%, with a measured drag force of 580±10 N compared to 720±15 N for rougher hull finishes. The hull design with a more streamlined shape further reduced drag by an additional 10%, enhancing forward motion and overall sailing efficiency.
Real-World Experiments
- Field Testing Data
The average wind speed during the Beam Reach trials was measured at 15.0±0.5 knots with a consistent direction perpendicular to the boat’s course. The sailboat achieved an average speed of 8.0±0.2 knots on a Beam Reach, which was higher than the speeds recorded on other points of sail, as displayed below:
The optimal angle of heel was measured at 15.0±1.0 degrees, which minimized hydrodynamic drag and maximized lift, correlating with the computational predictions.
- Instrumentation Data
- Lift and drag forces: Direct measurements from the force sensors during the Beam Reach trials showed an average lift force of 500±20 and a drag force of 200±10 . These values were consistent with the theoretical and computational predictions, which estimated a lift force in the range of 480 to 520 and a drag force of 180 to 200 .
- GPS tracking: The GPS data confirmed the higher speed and efficiency of the sailboat at Beam Reach, as indicated by the average speed of 8.0±0.2 knots, further supporting the findings from the computational simulations.
Fundamentals of Sailing Physics
At its core, sailing is a practice where sailors consistently interact with and adapt to the variable forces of nature. Besides its relations with the movement of waves, sailing primarily harnesses the raw power of the wind and converts it into kinetic energy that propels the boat forward2. Accordingly, understanding the basic physics underlying this process is crucial to a deeper appreciation of the Beam Reach phenomenon.
Bernoulli’s Principle
Central to the mechanics of sailing is Bernoulli’s Principle, which, as previously established in Methods, explains the behavior of moving fluids. However, lift generation on a sail also critically depends on the angle of attack, the shape of the sail, and flow separation. As wind flows over the curved surface of a sail, the pressure on the windward side (facing the wind) becomes greater than on the leeward side (away from the wind). This creates a pressure difference, leading to a net force called ‘lift’. Sails, therefore, act like airfoils, manipulating the airflow to generate this lift, which, when oriented properly, propels the sailboat forward3.
Forces on a Sailboat
As a sailboat navigates the waters, multiple key forces (Figure 2) interplay to result in the boat’s movements:
- Lift: As described above, the force generated by the differential air pressure across the sail. Lift drives the boat forward and sideways.
- Drag: A resisting force, drag occurs in two primary forms for sailboats. Aerodynamic drag (or wind resistance) results from the wind’s interaction with the boat’s sails and structure. Hydrodynamic drag (or water resistance) arises from the boat’s interaction with the water, primarily its hull.
- Gravity: The force pulling the boat downwards, counteracted by the buoyancy force.
- Buoyancy: An upward force exerted by the water, which opposes the weight of the boat, ensuring it floats.
- Frictional Force: This is the resistance between the boat’s hull and the water. It plays a role in the boat’s maneuverability and speed4.
The Role of the Keel and Rudder
Both the keel and the rudder are pivotal components of a sailboat, significantly influencing its behavior in water.
- Keel: Primarily, the keel counteracts the sideways force (or leeway) caused by the wind on the sails, helping the boat to maintain its intended course. Moreover, the keel adds stability, preventing excessive heeling (tilting) of the boat.
- Rudder: Positioned at the boat’s stern, the rudder acts as a steering mechanism. By changing the rudder’s angle, a sailor can alter the boat’s direction. Furthermore, it plays a role in counteracting some of the boat’s tendencies to turn due to uneven wind forces on the sails.
In summary, the forward motion of a sailboat is not just about catching the wind in its sails but involves a sophisticated interplay of forces, components, and principles of physics. Grasping these fundamentals paves the way for a more nuanced understanding of the Beam Reach’s efficiency and effectiveness6.
The Dynamics of Beam Reach
Beam Reach, often referred to as a sailor’s ‘golden point of sail7, represents a unique alignment of forces and conditions that allow the sailboat to achieve impressive velocities. Sailing perpendicular to the wind, the boat experiences an optimal combination of lift and minimized drag. But what are the factors that make this position so favorable? To understand this, we need to delve into the dynamics that come into play during a Beam Reach.
The Optimal Angle
A sailboat doesn’t move directly in the direction of the wind, nor directly against it; instead, it harnesses the wind’s energy by maneuvering its sails to generate lift. At a Beam Reach, the sails can be flattened out more than on other points of sail, allowing them to act as an efficient airfoil. The wind flows smoothly over the sail’s surface, maximizing lift while minimizing turbulent flow and the associated drag.
Balance of Forces
On a Beam Reach, the combination of forces propelling the boat forward reaches an equilibrium that typically offers the best forward speed:
- Lift: With the wind coming from the side, the sails, especially the main sail, capture a vast amount of it, converting it into forward-directed lift. This lift is most effective at propelling the boat forward during a Beam Reach.
- Drag: Both forms of drag (aerodynamic and hydrodynamic) are minimized. Since the sails are oriented perpendicular to the wind, there’s reduced frontal area facing the wind, hence less aerodynamic drag. Similarly, a well-trimmed boat on a Beam Reach will experience less hull exposure to the water, thereby reducing hydrodynamic drag8.
The Sail’s Camber and Draft
The camber (curve) of the sail and its draft (depth of the curve) are crucial in generating lift. During Beam Reach:
- Camber: The sails can be adjusted to have a more pronounced camber, optimizing them as airfoils. This camber helps in faster and smoother airflow on the leeward side of the sail, intensifying the lift due to Bernoulli’s principle.
- Draft: The position of the draft can be optimized to be more forward, ensuring that the generated lift is directed more in the forward motion of the boat, rather than pushing it sideways.
Heeling and its Counteraction
A sailboat on a Beam Reach often heels (tilts) due to the side force of the wind. This heeling can be advantageous to some extent:
- Reduced Hydrodynamic Drag: As the boat heels, the amount of hull submerged in water can decrease, resulting in less frictional drag.
- Leverage: The boat’s keel, acting as a lever arm, helps in counteracting the heeling force. This balance ensures that while the boat takes advantage of the reduced drag due to heeling, it doesn’t capsize.
The Role of the Centerboard or Daggerboard
In smaller sailboats without fixed keels, the centerboard or daggerboard plays a pivotal role during a Beam Reach. By being lowered, they increase the lateral resistance underwater, helping in minimizing the boat’s drift sideways (leeway) and maximizing forward motion9.
In essence, a Beam Reach represents a harmonious balance of the forces acting on a sailboat. The boat harnesses the wind’s energy efficiently, converts it into forward motion, and counteracts any undesirable side effects. The intricate interplay of lift, drag, and the boat’s design elements underlines why this point of sail is often celebrated for its speed and dynamism.
Factors Influencing Maximum Velocity
Achieving the highest velocity on a Beam Reach is not solely about sailing perpendicular to the wind. Numerous other factors come into play, which can either amplify or reduce the boat’s speed potential. A deep understanding of these factors can provide insights into optimizing a sailboat’s performance.
Sail Design and Material
The efficiency of a sail in generating lift is pivotal. The design of the sail, including its shape, camber, and draft, can have profound effects on speed. Moreover, the material’s elasticity, weight, and smoothness play roles in influencing aerodynamic performance.
Hull Design
A streamlined hull reduces hydrodynamic drag, which is essential for higher speeds. The wetted surface area (portion of the boat’s hull in contact with water) is a primary factor; the less it is, the lower the frictional resistance10.
Weight and Ballast
The weight of the boat and its distribution can influence its speed. A heavier boat might offer stability but can be slower due to increased hydrodynamic resistance. The ballast, often present in the keel, provides stability against heeling. Its weight and positioning can impact the boat’s speed and balance.
Keel and Rudder Design
The shape, size, and design of the keel and rudder can influence drag and maneuverability. A well-designed keel not only provides stability but also minimizes sideways drift (leeway), ensuring the boat moves more directly forward11.
External Factors
External environmental conditions, including wind consistency, wave height, and water currents, play a substantial role.
- Wind: While Beam Reach utilizes wind coming from the side, the wind’s consistency and strength can affect velocity. Gusts can provide boosts of speed but might also destabilize the boat if not handled well.
- Waves and Current: Waves can introduce additional drag, especially if they hit the boat head-on. Conversely, a following sea (waves moving in the boat’s direction) can boost speed. Water currents can either aid or impede the boat’s progress12.
Rigging and Hardware
The quality and condition of the boat’s rigging can affect performance. Friction in pulley systems, wear in ropes, or misalignment in the mast can impact the efficiency of force transmission from the sails to the boat13.
Aerodynamic Profile Above Deck
Everything above the waterline, from the mast’s design, the positioning of ropes, to the shape of the cabin, can introduce aerodynamic drag. Streamlined designs and optimal placements can reduce this unwanted resistance.
Maintenance and Cleanliness
A clean and well-maintained boat can achieve better speeds. Algae, barnacles, or other marine growth on the hull can increase frictional drag. Regular maintenance ensures that all parts of the boat function optimally14.
In conclusion, while the Beam Reach inherently offers one of the best potentials for high speed, achieving maximum velocity requires a holistic consideration of numerous factors. An optimized boat design, paired with skilled sailing and favorable conditions, can unlock impressive speeds on the water.
Comparative Analysis: Beam Reach versus Other Points of Sail
To truly understand the uniqueness of the Beam Reach, it’s essential to comprehend how it compares to other points of sail, which possess their own sets of dynamics and potential advantages, but still fail to surpass the speed reached at the Beam Reach.
Beam Reach versus Close-Hauled
Close-Hauled refers to the angle where the sailboat sails as close to (directly into) the wind as possible, typically at an angle of about 30 to 45 degrees off the wind direction. At this point of sail, the sailboat is enabled to consistently move toward the wind, offering an opportunity for the sailor to tack or beat. In other words, sailing at Close-Hauled allows the sailor to tack and beat with the least effort, making it an optimal angle of sailing when the sailors would like to change directions. However, the aerodynamic and mechanical challenges of Close-Hauled sailing often lead to slower speeds compared to a Beam Reach.
- Aerodynamic Drag
- Why Increased Drag? When sailing Close-Hauled, the angle of attack of the sails is relatively high to maintain lift while heading upwind. This causes a significant amount of the wind to hit the sails directly, creating more turbulence. Turbulent flow increases aerodynamic drag because it causes the airflow to separate from the sail surface earlier, resulting in a larger wake region behind the sail.
- Equation for Drag:
(7)
Where:
is the air density.
is the relative wind speed.
is the drag coefficient, which increases with the angle of attack due to flow separation.
is the frontal area of the sail exposed to the wind.
In Close-Hauled sailing, and are larger, resulting in higher drag. This contrasts with a Beam Reach, where the sails are flatter, reducing both and , and thus, drag.
- Lift Generation
- Why Less Efficient Lift? The shape and angle of the sails in a Close-Hauled position create less efficient lift because the airflow over the sails is more prone to separation, leading to a lower lift-to-drag ratio. The lift force is given by:
where is the lift coefficient. In Close-Hauled sailing, is lower due to less efficient airfoil shapes created by the necessary sail trim, which requires a higher angle of attack.
- Keel Efficiency
- Why More Drag from the Keel? The keel must counteract significant lateral forces to prevent leeway. This increases the effective drag on the keel, reducing overall speed. The keel’s drag increases with the angle of heel, which is typically more pronounced in Close-Hauled sailing due to the higher lateral force.
In contrast, a Beam Reach allows for a more optimal sail trim, with flatter sails and reduced angles of attack, leading to a higher and lower , resulting in faster speeds.
Beam Reach versus Broad Reach
If the sailors are at Broad Reach, then the wind will be hitting them from behind at an angle, typically between 135 and 160 degrees relative to the boat’s direction. Under this situation, the sailors might be able to experience a relatively fast sailboat, especially with a following sea, which allows them to use specialized sails (for example, spinnakers) that will bring them with an even faster speed. Nevertheless, sailing at this point of sail can be extremely risky.
- Accidental Jibe
- How Does a Wind Shift Cause an Accidental Jibe? During a Broad Reach, the boom (the horizontal pole attached to the bottom of the mainsail) is extended outwards, almost perpendicular to the boat. A sudden shift in wind direction or an unintended turn can cause the wind to catch the other side of the sail, forcing the boom to swing rapidly across the boat. This is known as a jibe. The force of the wind pushing on the opposite side of the sail causes a rapid, uncontrolled swing of the boom, which can be dangerous and destabilizing.
- Equation for Angular Momentum: The boom’s rapid movement is related to the change in angular momentum :
(8)
Where:
is the moment of inertia of the boom and sail.
is the angular velocity.
During an accidental jibe, increases suddenly due to the force exerted by the wind on the sail, resulting in a rapid and forceful movement of the boom.
- Increased Drag from Spinnakers
- Why Does Frontal Area Increase Drag? When using a spinnaker (a large, balloon-like sail used for downwind sailing), the sail’s frontal area increases significantly. This larger sail area increases the amount of wind resistance or drag that the boat experiences.
- Reference to Drag Equation: As with Close-Hauled, the spinnaker increases the and value of due to its shape, leading to greater drag. If not perfectly trimmed, the spinnaker can cause turbulent airflow, further increasing drag and reducing speed efficiency.
- Control and Stability
- Why More Control at Beam Reach? On a Beam Reach, the wind is more consistently directed across the sails, providing a stable force that helps maintain balance and control. The forces are better aligned with the boat’s forward motion, reducing the risks of unexpected movements like jibes. The sails are also easier to manage since they do not require the large adjustments needed when using a spinnaker on a Broad Reach.
Beam Reach versus Running
Sailing at Running is when the sailboat sails directly downwind, with the wind coming straight from behind. This point of sail, both theoretically and practically, is straightforward; at this angle, sailors would be able to sail easily without the risk of tacking or jibing accidentally. Regardless, sailing Running presents unique aerodynamic and mechanical challenges.
- Drag-Driven Propulsion
- How Does Drag Drive Propulsion at Running? When running, the sails are positioned to catch as much wind as possible, acting like parachutes. This means the propulsion is primarily due to drag, as opposed to lift. The boat moves because the sails capture the wind and push it forward, but this method is less efficient than lift-driven propulsion.
- Drag Equation at Running: The propulsion force can be approximated by the drag force:
(9)
Here, is relatively high due to the sails’ perpendicular alignment with the wind, and is maximized, but the efficiency is lower than in lift-driven scenarios like Beam Reach.
- Instability and ‘Death Roll’
- Why does Running cause instability? Running exposes the boat to significant rolling forces due to the side-to-side movement of the wind and waves. This motion, combined with the lack of directional lift, can cause the boat to oscillate dangerously, a phenomenon known as a ‘death roll.’ The boat’s stability is compromised because the sails are not generating lift that would otherwise help stabilize the boat.
- Dynamic Stability Analysis: The dynamic stability can be understood by analyzing the restoring moment :
(10)
Where:
is the weight of the boat.
is the righting arm, a function of the boat’s heel angle.
During a death roll, the righting moment may be insufficient to counteract the rolling forces, leading to capsizing.
- Comparison with Beam Reach
- Why is Beam Reach More Stable and Faster? On a Beam Reach, the boat benefits from lift-driven propulsion, which is more efficient than drag-driven propulsion. The lift force provides both forward motion and lateral stability, reducing the risk of rolling. The forces are better aligned with the boat’s center of mass, which helps maintain stability and speed.
Other Considerations
While the Beam Reach often provides the highest speed, other points of sail might be chosen for strategic reasons, such as navigating around obstacles, avoiding specific weather patterns, or following a specific racing strategy. The Beam Reach stands out for its balance of speed, stability, and control. While each point of sail has its place and advantages, the Beam Reach often offers the most efficient and straightforward approach to harnessing the wind’s power, especially when aiming for sheer speed.
Conclusion
Sailing, a harmonious blend of nature and engineering, has fascinated humanity for millennia. Among the various points of sail, the Beam Reach stands out as a testament to this synergy, offering optimal speed, stability, and control. Through my exploration, I have unveiled the intricate dynamics underpinning this point of sail: from the balance of forces that propel the boat forward to the design intricacies that minimize resistance and maximize lift.
In contrast to other points of sail, the Beam Reach offers an unparalleled combination of lift-driven propulsion and reduced drag. This balance is accentuated by the boat’s design, the sailor’s skill, and external conditions, all converging to unlock remarkable speeds. However, as with all aspects of sailing, the choice to sail on a Beam Reach is not solely about speed. It is also about understanding the environment, the boat, and one’s own capabilities, then making strategic decisions accordingly.
As technology advances and our understanding of fluid dynamics deepens, there is potential for even more optimizations in sailboat design and sailing techniques. Yet, the core principles will likely remain unchanged. The Beam Reach, with its blend of physics and art, exemplifies the timeless allure of sailing, capturing the wind’s raw power and channeling it into graceful, forward motion.
Acknowledgement
I would like to express my deepest gratitude to my sailing mentor – Mr. Jiang, Mr. Sun, and Mr. Bai – whose guidance and passion for sailing inspired this research. Special thanks to Sanah A. Bhimani from the Department of Physics at Yale University for her insightful feedback and encouragement. I am also grateful to my friends and family for their support and to the countless sailors whose experiences and stories provided rich context for this study. Finally, I acknowledge the use of resources and data provided by the American Sailing Association and the University of New South Wales, which significantly contributed to the validity of this research.
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