Physics and Physiology Define the Hammer Throw

hammerthrowThe Hammer Throw is a Track and Field event which involves throwing a 12-16 pound ball secured on the end of a ~ 3.5 ft wire. Angles, trajectories, and even a unique physiological approach make this sport a precise and complex skill.

Learn the basics of how physics and physiology define the hammer throw.

Articles by Dave Kieda

Curling: The Friction Sport

pic_curlingCurling was first created in 16th century Scotland, where river bottom rocks were slid across ice-covered lochs to a target. In modern times, the “roaring game,” named for the sound of the “stone” sliding across the ice, is more refined. It consists of a 41 pound granite stone sliding across 42 m of ice to a target called a house, utilizing pebbled ice rinks, brooms, curling shoes, and carefully formed stones.

The ice rinks used for Curling are not smooth, but have a pebbled surface made by spraying the rink with water and allowing the tiny droplets to freeze on its surface. This surface is necessary for the stone to have suction and slide across the ice. Two players with “brooms” vigorously sweep the ice immediately in front of the stone to influence the trajectory, or direction it travels. These players wear a “gripping” shoe and a “sliding shoe,” which allow them to use friction to move along the ice. As the stone travels, the interaction between the stone, ice, and sweeping changes the sources of friction, and cause the stone to “curl” in a curved trajectory, giving the sport its name.

Although there are many theories, it is poorly understood as to what causes the stone to curl as it travels along the ice. As the sport gains popularity, further scientific inquiry can be expected to explore the role of friction and different sweeping styles.

Learn the basics or read about the technical aspects of friction in curling.

Articles by Jessica Egan

Curling: The Friction Sport (Basic)

The modern game of curling consists of a 41 pound granite rock sliding across some 42 m of ice to a target called a house. The rock, technically termed a “stone,” is typically preceded on the ice by two players with “brooms,” who vigorously sweep the ice immediately in front of the stone to influence its trajectory (Willoughby et al. 2005, Bradley 2009, Esser 2011). This unique winter sport, affectionately termed the “roaring game” because of the sound the stone makes as it slides on the ice, has its roots in 16th century Scotland, where river bottom rocks were the stones which slid across ice-covered lochs to a target (Clark 2008). Now, play is more refined, utilizing pebbled ice rinks, brooms, curling shoes, and carefully formed granite stones.

The ice rinks are pebbled by spraying water onto the ice to make a slightly bumpy surface as the droplets freeze into little protrusions on the ice surface, without which the stone could not curl (Shegelski 2001). Brooms are now typically made of synthetic materials such as nylon, and curling shoes consist of one shoe with a larger friction coefficient than the ice and another shoe which has a friction coefficient smaller than that of the ice, thus providing players with a “gripping” shoe on one foot and a “sliding shoe” on the other. A friction coefficient is the measure of how much a substance resists sliding on another substance. The granite stones are highly polished, very hydrophobic, and have a small hollow on their underside so that only a small ring on the bottom of the stone actually touches the ice at any one time.

Each of the pieces of equipment needed to play curling is designed to maximize the impact it has on friction coefficients. Style of play also influences friction coefficients. For example, some sweepers use a conventional style of sweeping, which consists of the sweeper standing to the side of the stone while they sweep. The sweepers employ more force closest to their feet, so there is a greater rise in temperature on the side of the stone closest to the sweeper, which causes asymmetric friction acting on the stone. This matters because the sweeping causes the uppermost layer of ice to melt, thus providing a lubrication layer for the stone to glide on. On the other hand, a high-angle style of sweeping consists of the sweepers being behind the stone while it travels, giving a more even distribution of heated ice, and so lessening the curl of the stone (Marmo 2006). Both can be useful, depending on where the stone is desired to go.

There is much scientific debate about the role friction plays in causing the stone to curl. The friction acting on the stone is undoubtedly asymmetric, but how this results in the stone’s curling trajectory is not yet fully understood. Some believe that as the stone twists, it pushes the water layer to the side, creating a lubrication film for the rock to slide on (Bradley 2009). Others say that the rotating stone not only pools water one its one side, but also chipped ice and other debris (Denny 2002). As curling continues to gain popularity, it can be expected that further scientific inquiry will continue to be directed at exploring the role of friction in the curl of the stone.

Learn more about the technical aspects of friction and curling.

By: Jessica Egan, University of Utah
Jessica Egan began her love of chemistry under her mom’s direction with homeschool experiments in middle school and then up into high school through her high school chemistry teacher. She pursued a chemistry and art double major at Hillsdale College and decided to continue her chemical education by attending graduate school at the U. She has recently graduated with a M.S. from the U in Analytical Chemistry and is looking for a career in industry.

References

Bradley, J. L. 2009. The sports science of curling: a practical review. Journal of Sports Science and Medicine 8:495-500.

Clark, D. 2008. The roaring game: a sweeping saga of curling. Key Porter Books, Toronto, Ontario, Canada.
Denny, M. Curling rock dynamics: towards a realistic model. 2002. Canadian Journal of Physics 80:1005-1014.

Esser, L. 2011. Swept away: exploring the physics of curling. Science Scope 35:36-39.

Marmo, A. A., I. S. Farrow, M-P Buckingham, and J. R. Blackford. 2006. Frictional heat generated by sweeping in curling and its effects on ice friction. Proceedings of the Institution of Mechanical Engineers, Part L.: Journal of materials: Design and Applications 220:189-197.

Marmo, B. A., M-P Buckingham, and J. R. Blackford. 2006. Optimising sweeping techniques for Olympic curlers. The Engineering of Sport 6 3:249-254.

Shegelski, M. R. A. 2001. Maximizing the lateral motion of a curling rock. Canadian Journal of Physics 79:1117-1120.

Willoughby, K. A., and K. J. Kostuk. 2005. An analysis of strategic decision in the sport of curling. Decision Analysis 2: 58-63.

Physics and Physiology Define the Hammer Throw (Basic)

The Hammer Throw is a Track and Field event which involves throwing a 12-16 lb ball secured on the end of a ~ 3.5 ft wire. The other end of the wire is secured to a handle which is used to grip the hammer as it is thrown. The hammer is thrown by gripping the handle and swinging the hammer in a circle, then spinning one’s entire body for 3-4 turns and then the handle is released. A men’s championship collegiate hammer thrower will toss a 16 lb hammer 190 ft or more; the current world record distance (2011) is approximately 285 ft.

A primary concept associated with the hammer (as well as the shot-put) is the ballistic trajectory of the object, used to determine the optimal angle to release the device. The optimal  angle is almost independent of the speed of the steel ball when the hammer is released. In a vacuum, the optimal release angle  (angle between the velocity at release and the horizontal plane) for maximum distance would be 45°, but the presence of air resistance slows the horizontal velocity of the ball down, making the optimal release angle closer to 42-43°.

Achieving the proper release angle requires some thought and planning. When the hammer thrower begins the first turn, the plane of the hammer swing is considerably lower than 45°, closer to 10°. At the start of the throw, the velocity of the hammer in the ‘orbit’, combined with the radial distance from the thrower to the steel ball, defines the angular momentum of the hammer.  As the hammer thrower uses his legs to turn and accelerate the ball, he applies an off-axis torque to the angular momentum, and rapidly turns the orbital plane to steeper and steeper angles, achieving the optimal release angle near 42° in the final turn.

Since most hammer throwers will learn to throw near the optimal release angle fairly easily, the most important factor affecting the final travel distance of the hammer is the speed of the steel ball upon release. Because the hammer thrower uses a circular orbit to throw the hammer, the hammer thrower must exert a centripetal force to keep the steel ball moving on the circular orbit. This force is proportional to the square of the velocity of the hammer divided by the radial distance between the steel ball and the hammer thrower’s body (center of mass), and can easily reach 600lbs or more at release. The ability of the hammer thrower to withstand such huge force is the main limitation in the distance that can be thrown; most hammer thrower perform heavy weight lifting exercises in order to increase their ability to withstand this extraordinary force.

Having developed one’s strength to the maximum feasible, the hammer thrower has additional strategies for increasing the final velocity of the hammer while exerting the same centripetal force.  Since the centripetal force depends upon the square of the hammer velocity divided by the radial distance between the steel ball and the hammer thrower’s center of mass, higher velocities can be accommodated (with the same centripetal force ) by increasing that radial distance. Physiologically, this requires allowing one’s arms to extend as far our as possible, so championship class hammer throwers are generally tall, with exceptionally long arms. A particular individual, with a given arm length, can also increase the radial distance by working to keep the steel wire exactly perpendicular to one’s chest throughout the entire throwing motion. In addition, the hammer thrower will substantially increase the  orbit radius by completely relaxing the upper body and arms, allowing  the arms to dangle completely freely and relaxed as they carry the centripetal force.

At the same time the lower body and legs will drive as explosively as possible in order to accelerate the steel ball as quickly as possible to the final speed. The optimal technique for the maximum hammer throw distance is therefore “schizophrenic”: the upper half of the body is completely relaxed and passive, and the lower half of the body is completely energized with explosive power. This seemingly contradictory combination is what makes the hammer throw one of the most unique and spectacular events in track and field!

By: Dave Kieda, Department of Physics and Astronomy, University of Utah

From Tee to Fairway: The Basics of How Physics Affects the Drive, the Club, and the Golf Ball (Basic)

The motion of a golf ball can be thought of as a projectile, whose trajectory is parabolic and acted upon by gravity.  The initial velocity imparted to the ball by the club head can be broken down into both a horizontal and vertical component.  Numerous scientific studies have identified the optimum launch angle as 11-20° to achieve maximum distance (Erlichson, 1983). Though drivers are typically 8-10° in loft, the flexibility of the graphite driver shafts increases the launch angle through a split second whipping action (Zumerchik, 1997).  Golf clubs have grooves added to their faces to add some friction to the club head, so that momentum is transferred to the ball and backspin is created to generate lift. The difference between a ball with backspin and one without can add up to 100 yards after 2 or 3 seconds of additional flight time (Zumerchik, 1997).

There are several forces that act upon the aerodynamics of a golf ball in flight.  The most recognizable force acting upon a golf ball is gravity, which pulls the ball downward and creates the parabolic trajectory common of projectiles.  Another force on a golf ball is lift, the force that opposes gravity.  When backspin is transferred to the ball from the grooves on the clubhead, the velocity of air on top of the ball (which is moving in the direction of the backspin) is higher than the velocity of air on the bottom of the ball.  To counteract this, the Magnus effect generates lift on the ball and pushes it up.

In addition to gravity and lift, another force acting on a golf ball is drag, or air resistance.  As a golf ball is sent flying through the air, the molecules that come into contact with the front of the ball exert a large pressure force on the front of the ball (drag). Drag slows down the forward velocity of the ball. As the air comes into contact with the front of the golf ball, the fluid motion of the air becomes turbulent.  Turbulent flow can be thought of as smoke from a smoke stack- chaotic and wispy. As the turbulent air swirls around the golf ball, the dimples capture some of the swirls and keep them close to the surface of the golf ball. This means that the boundary layer of air stays close and hugs the ball longer, which means that there is a smaller pressure difference between the front of the ball and the back (when compared to a ball without dimples). The dimples thus allow the golf ball to travel farther than a smooth ball because the golf ball experiences less drag.

By: Trevor Stoddard, University of Utah

Learn more about the technical science behind golf.

References:

Benson, T. (2010) Drag of a Sphere.  National Aeronautics and Space Administration, Date Accessed:  8/10/2012 <http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html>

Bird, R. B., W.E. Stewart and E.N. Lightfoot. 2007. Transport Phenomena, 2nd edition, Wiley & Sons, New York.

Cochran, A. (ed.). 1990. Science and Golf.  New York: Chapman and Hall

Cochran, A. (ed.). 1992. Science and Golf II.  New York: Chapman and Hall

Davies, J.  1949. “The Aerodynamics of Golf Balls.”  Journal of Applied Physics 20: 821-828

Erlichson, H. 1983. “Maximum Projectile Range with Drag and Lift, with Particular Application to Golf.”  American Journal of Physics 51: 357-362.

Jorgensen, T. 1994. The Physics of Golf. New York:  American Institute of Physics

McDonald, W. 1991. “The Physics of the Drive in Golf.”  American  Journal of Physics 59: 213-218

Werner, F. and R. Greig. 2000. How Golf Clubs Work and How to Optimize Their Designs.  Jackson Hole, WY:  Origin Inc.

Wesson, J. 2009. Science of Golf.  New York, Oxford University Press

Williams, D. 1959. “Drag Forces on a Golf Ball in Flight and Its Practical Significance.”  Quarterly Journal of Mechanical Applications of Mathematics XII 3: 387-393

Zumerchik. J. (ed.). 1997. Encyclopedia of Sports Science.

Zumerchik J. 2002. Newton on the tee- a good walk through the science of Golf

History of the golf ball <http://www.golfeurope.com/almanac/history/golf_ball.htm> last accessed 8/10/12

From Tee to Fairway: How Physics Affects the Drive, the Club, and the Golf Ball (Technical)

Introduction

According to the PGA there are 27 million golfers in the United States (citation).  By understanding the science behind the game, golfers are more likely to improve their scores because they can better understand the errors that they are making.

From Tee to Fairway: The Physics Behind Golf (Projectile Motion)

Projectile motion is the motion of an object that is imparted with an initial velocity (such as hitting a golf ball with a golf club) that moves in a parabolic trajectory that is caused by the effect of gravity on the object.    Initial velocity can be broken into horizontal and vertical components as in Equation (1).

Where the subscript 0 denotes initial, x and y are the directions and i and j are the vector directions (i is in the x-direction and j is in the y-direction).  The initial horizontal and vertical velocities can be determined if the angle the ball is launched from the horizontal,α, is known, given by Equations (2) and (3).

Acceleration due to gravity pulls the golf ball down during flight and as time goes on actually forces the vertical velocity in the downward direction, the components of the golf ball velocity at any given time, t, during its flight are given by Equations (4) and (5).

Where g is the acceleration due to gravity, which is a constant assumed to be -9.8 m/s2 (or 32 ft/s2), and the time t is in seconds.  The magnitude of the velocity is given by Pythagorean’s theorem.

A diagram of these velocity vectors is given in Figure 1.

Figure 1.  Golf ball velocity

The average golfer drives the golf ball with an initial velocity of over 100 miles per hour (Zumerchik, 2002)! This means that the golf ball, if struck at an angle of 12°, will be initially traveling at 87 mph in the horizontal direction and 13 mph in the vertical direction.

Typically the projectile motion equations used to calculate range and height of a golf ball’s flight do not account for drag, and are therefore only estimates as to the true maximum height and range that the golf ball will travel.

Drag force (or air resistance) is the force that acts opposite to an object that is moving through it. When a golf ball is hit, the air molecules flow past the golf ball as the golf ball flies through the air, thus creating a retarding force on the forward motion of the ball known as drag.

In 1949, Davies conducted experiments to determine the magnitude of drag and lift forces that occur on a golf ball by dropping rotating golf balls into a wind tunnel.  Davies found that “drag increased nearly linearly from about 0.06 lb for no spin to about 0.1 lb at 8000 rpm” and that “lift varied with the rotation speed” (Davies, 1949).

In 1959, Williams used the previous findings of Davies and conducted an analysis on golf ball carry as a function of velocity.  Williams showed that the drag force varied linearly with the velocity.  The drag force on a golf ball can be calculated by Equation (7).

Where CD is the drag coefficient, A is the cross-sectional area of the ball, ρ is the air density, and v is the ball velocity (in ft/s).

Williams showed that the drag coefficient can roughly be estimated as 46/v, showing that at higher speeds the drag coefficient drops significantly.  Williams found that the drag force on the golf ball varied linearly with the speed, he found D to be as given in Equation (8).

Where D is in pounds, and 0.000783 represents the constant c, to be discussed.   One note is that Williams’ (1959) calculations used a British ball with a diameter of 1.62 inches compared to the American ball of 1.68 inches.  The easiest calculation is to consider the case of a nonspinning golf ball (no lift) with linear air resistance in calculations of range and height.  The reader is directed to the paper by Erlichson for a more in depth analysis of range calculations that incorporate lift forces into the derivations.

Erlichson (1983) gives the equations for range and height as in Equations (9) and (10).

Where c is 0.000783 lb/(ft/s) and m is in [lb] and g is 32 ft/s2 and t is time, initial velocities are given in ft/s.

Using these equations a golfer could estimate how long they will hit the ball and where it will end up.  Variables such as wind and weather can affect these numbers in reality so they are best used for only estimations on distance.

Several studies have been conducted on the optimum launch angle.  Scottish physicist Alastair Cochran calculated that the optimum launch angle of 20° achieves the longest carries (Cochran, 1990). One needs to take this finding with a grain of salt, however, because the ball will land at a high angle and will have less bounce and roll, unless the grass is wet.

Erlichson (1983) found that the optimum launch angle was around 12-13°, only slightly more lofted than the drivers available on the market, and with the additional loft generated from shaft flex. Zumerchik concludes that anywhere from 12°-20° will give the ball maximum range with only a few yards difference between the different angles. (Zumerchik, 2002)

Golf Clubs: Loft and Grooves

Several forces act on the golf club, such as torque (exerted by the golfer on the club), centrifugal acceleration, and gravity.  Typically golfers can average a whole 4-5 hP of power generated from their golf swings. (Wesson, 2009)

At the top of the backswing, the club head coils because the shaft is flexible and the center of mass is in the clubhead. As the swing moves forward, the shaft of the club coils, unloads, and recoils as the club head attempts to catch up to the wrists during the swing.  At impact between the clubhead and the ball there is a final forward oscillation that creates a snapping effect that increases the velocity of the club head through the ball, and by extension also increases the initial velocity imparted to the golf ball.  It has been found that a “forward shaft flex of about 3.3 degrees can add 8.7 percent to the velocity of the club head” (Jorgensen, 1994).  During the course of the swing, the club oscillates 1.5 times (Zumerchik, 1997).

Werner et al. (2000) conducted a full suite of simulations and experiments on optimizing club designs. Their findings showed that there is an optimum combination of loft angle and center of gravity location, and that an extra-large face provides advantages (Werner et al., 2000) But does the addition of more surface area on the club head create an increased drag on the club head, diminishing initial velocity?

It’s been found that air drag on the shaft and clubhead results in an energy loss of 10% of the energy of the club at impact with the ball.  This results in a reduction of roughly 15 yards in the range of the ball (Wesson, 2009).

When the club head strikes the ball, the grooves on the club head increase the amount of friction at the ball-club head interface, allowing for the ball to have backspin, which increases the lift force.  Without friction, the ball would merely glide up and off the club face.

When the club head comes into contact with the golf ball, some of the energy is lost and is not entirely transferred into velocity of the golf ball. The ratio of the velocity transfer between the club face and golf ball surface is given by the coefficient of restitution.

If a ball hits a surface at a right angle with a speed v and leaves the surface with a speed v’ then the coefficient of restitution is defined by

When bounced off a hard surface, a typical golf ball has a coefficient of restitution of around 0.7.  This means that if the golf ball were dropped from 100 feet above the ground, it would rebound and bounce back 70 feet. The coefficient of restitution is about 1.46 for a driver, 1.3 for a 5 iron, and 1.12 for a 9 iron.  (Zumerchik, 1997).

Why all the dimples: The Fluid Mechanics behind a golf ball in flight

The aerodynamic forces that act on a golf ball in flight are shown in Figure 1.

Figure 1. Forces that Act on a Golf Ball in Flight

Typically golf balls have between 330 and 500 dimples on their Surlyn covers.  Some dimples are round while most golf ball manufacturers have started making their dimples in a variety of hexagonal shapes.  As stated by Euler’s principle (Bird et al, 2007) separation of the boundary layer (the layer of air molecules next to the surface of the golf ball) is likely to occur in regions where the pressure increases in the direction of the flow. The following youtube video gives a good look at a simulation of the boundary layer.

The reason for the large number of dimples on a golf ball is to ensure that the boundary layer does not separate until the back part of the ball.  A smooth sphere will generate a large wake behind the ball, as there is much lower pressure behind the ball than in front.  The air will move to rush into the area of low pressure, exerting a pressure drag force on the ball.  However, as the turbulent air swirls around the golf ball, the dimples capture some of the swirls and keep them close to the surface of the golf ball. Dimples force a turbulent trip at the surface, ensuring a later separation of the boundary layer which decreases the amount of drag force on the ball. Dimples on the golf ball do not reduce the drag at the front of the ball, because the cross-sectional area of the ball is always the same. However, they do decrease the size of the low pressure wake behind the ball, lowering the overall drag force, and improving flight.  An example of this is shown in Figure 2, a simulation run in Fluent between a sphere and a dimpled golf ball.

Figure 2. Fluent simulation showing the pressure field around a smooth sphere and dimpled golf ball. Balls are traveling to the left.

As shown in both cases, the pressure contacting the front face of both balls is large. However, the magnitude of the pressure difference (front side versus back side of the ball) in the sphere case is much greater than in the golf ball case. The large amount of high pressure (red and yellow) pushing against the flight of the ball slows it down much faster than in the golf ball case, where there is less pressure behind the ball.  In addition to this pressure calculation it is of interest to look at the shear stress distribution on the face of both the sphere and modeled golf ball, as shown in Figure 3.

Figure 3. Wall shear stress of both a dimpled golf ball and a sphere modeled in Fluent

As shown in Figure 3, the shear stress on the surface of the sphere (red/orange) covers a greater surface area, showing where the boundary layer separates from the sphere.  The dimples on the golf ball keep the turbulent flow boundary layer close to the golf ball wall and greatly reduce the shear stresses over the golf ball’s surface area.  This means that the golf ball is slowed less.  The boundary layer separation in laminar and turbulent flow is shown in Figure 4, a COMSOL simulation of a sphere in flow.

Figure 4. Boundary layer separation

As shown in Figure 4, in the laminar case with a low Reynolds number (Re=0.1) (top) has no boundary layer separation as the streamlines hug the outside of the ball. This situation is unrealistic for a golf ball because of the high velocities that the golf ball travels at (~200 ft/s) once it is hit  (Zumerchik, 2002).

The second case (middle) shows a smooth ball at a high velocity. There is a large pressure difference between the front and back side of the ball, and the boundary layer separates much earlier than the dimpled sphere case (bottom).  The dimples allow for the boundary layer to hug the ball and boundary layer separation does not occur until the very tail end of the ball. The smaller the blue zone of low pressure behind the ball, the farther the golf ball will fly.

Similar to an air foil, the dimples on a golf ball also allow for a lift force to be exerted on the ball.  Backspin, as generated from the loft of the clubface that strikes the ball, deforms the airflow around the ball and creates a lift force due to the Magnus effect.  The Magnus effect is a phenomenon where a spinning object flying in a fluid creates a whirlpool around itself and experiences a force (in this case, lift) perpendicular to the line of motion.  Because the top of the ball is spinning with the direction of the air, the air on top of the ball moves more rapidly than the air at the bottom of the ball.  The air at the bottom of the ball moves against the wind, and this shifts the pressure behind the ball downward, in the direction of the backspin.  Because of the differing velocities on top and bottom of the ball, there is a resultant force upward known as lift.

The magnus effect can have a large impact in golf. If two golf balls are hit with the same velocity, a ball hit with backspin will stay in the air 2 or 3 seconds longer and may travel 18 to 30 meters farther (Zumerchik, 1997).

Despite the benefits of backspin, the use of too much spin can be a problem because some of the momentum is imparted into the high spin of the golf ball.  As a ball travels faster, it needs less backspin to generate lift. (Zumerchik, 2002).

Conclusions

Golf equipment has evolved over the centuries to increase the distance a ball can travel. By understanding the forces acting on the golf ball, a golfer can maximize his/her performance

By: Trevor Stoddard, University of Utah

 

References:

Benson, T. (2010) Drag of a Sphere.  National Aeronautics and Space Administration, Date Accessed:  8/10/2012 < http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html>

Bird, R. B., W.E. Stewart and E.N. Lightfoot. 2007. Transport Phenomena, 2nd edition, Wiley & Sons, New York.

Cochran, A. (ed.). 1990. Science and Golf.  New York: Chapman and Hall

Cochran, A. (ed.). 1992. Science and Golf II.  New York: Chapman and Hall

Davies, J.  1949. “The Aerodynamics of Golf Balls.”  Journal of Applied Physics 20: 821-828

Erlichson, H. 1983. “Maximum Projectile Range with Drag and Lift, with Particular Application to Golf.”  American Journal of Physics 51: 357-362.

Jorgensen, T. 1994. The Physics of Golf. New York:  American Institute of Physics

McDonald, W. 1991. “The Physics of the Drive in Golf.”  American  Journal of Physics 59: 213-218

Werner, F. and R. Greig. 2000. How Golf Clubs Work and How to Optimize Their Designs.  Jackson Hole, WY:  Origin Inc.

Wesson, J. 2009. Science of Golf.  New York, Oxford University Press

Williams, D. 1959. “Drag Forces on a Golf Ball in Flight and Its Practical Significance.”  Quarterly Journal of Mechanical Applications of Mathematics XII 3: 387-393

Zumerchik. J. (ed.). 1997. Encyclopedia of Sports Science.

Zumerchik J. 2002. Newton on the tee- a good walk through the science of Golf

History of the golf ball <http://www.golfeurope.com/almanac/history/golf_ball.htm> last accessed 8/10/12

 

For more information:

  1. DeNevers, Noel, Fluid Mechanics for Chemical Engineers, McGraw-Hill, New York, 2005.
  2. Libii, Josue Njock (2007)  “Dimples and Drag:  Experimental demonstration of the aerodynamics of golf balls.”  American Journal of Physics, 75, 764.
  3. Millne, R. and J. Davies (1992)  “The Role of the Shaft in the Golf Swing.”  Journal of Biomechanics 129:  975-983
  4. Zander, J. and A. Chou (February 1999)  “Max out your ball:  Increasing your launch angle and decreasing your spin rate will help you hit farther.”  Golf Digest 50: 76-80

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