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

Creating the Perfect Golf Ball with Chemistry

white-golf-ballThe composition of golf balls has evolved through the years. Two-layered balls, which are inexpensive and popular,  have come a long way.  Polymers combined with natural compounds from rubber have been used to create golf balls that have good distance, high abrasion resistance, and optimal firmness.  Scientists are beginning to research ways to prolong the life of balls after they are exposed to moisture.

Brief the basic chemistry principles or read the more technical chemistry explanation.

Articles by Jessica Egan

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: How Physics Affects the Drive, the Club, and the Golf Ball

Golf Ball Velocity

Golf Ball Velocity

The average golfer drives the golf ball with an initial velocity of over 100 miles per hour.  If the player uses a club with a flexible shaft, the act of swinging adds an additional measure of torque as the head of the club also propels forward to connect with the ball.  The head of the club has grooves that increase the friction between the club and the ball, allowing the club to more effectively focus the area of contact.

The optimal angle to hit the ball ranges from about 12 to 20 degrees.  Putting a backspin on the ball increases lift and can add significant distance to the drive.  The dimples on the golf ball itself help reduce drag from the air stream by reducing turbulent air pressure around and behind the ball, shifting the wake further behind the ball, thus allowing for smoother, less resistant flight.   Any combination of these variables contributes to how well the ball overcomes the forces of gravity and air resistance.

Learn the basics of how physics affects golf or read the more technical details here.

Articles by Trevor Stoddard

Creating the Perfect Golf Ball with Chemistry (Basic)

Gowf, as the Scottish first called it, had humble beginnings in eastern Scotland as a game played with wooden balls and clubs (Mallon, 2011).  Since its humble beginnings, golf has grown to be a sophisticated sport growing ever more so as researchers pump all of their expertise into the game in a quest to make the ideal ball.

The first modern golf ball consisted of a small, hard core wound with a long string of rubber and then coated with tree gum called gutta-percha (Mallon, 2011).   Since then, many improvements have been made on golf balls to give them the perfect “feel” (i.e. a certain resilient, soft feel that golfers look for when their clubs make contact), but also great durability and wear and tear resistance. These two qualities tend to be mutually exclusive in a single material; however, scientists have developed balls with several layers, each layer addressing a specific need that the ball has.

Researchers have targeted polymers, a long molecular chain made of many smaller subunit molecules linked together, as the best materials for golf balls. This is because polymers are very flexible – by changing even one atom in the subunit or twisting the subunit slightly, a polymer can go from being used for the hard golf ball cover to the more elastic inner layer.

Figure 1: A – chemical structure of natural rubber, B – chemical structure of gutta-percha

An example of how a slight difference between polymers makes a big difference in the ball is seen between gutta-percha and natural rubber, as seen in Figure 1 (Goodman, 1974; Yikmis,Steinbüchel, 2012). The lower structure is gutta-percha, which was replaced in golf history by natural rubber, the upper structure. As can be seen, rubber has more kinks in its structure, making it a better molecular spring than the more rigid gutta-percha. This spring-like quality made the ball compress more when struck, thus transferring more energy to the ball’s flight when the molecules “pushed off” the club.

Golf balls have other important pieces to them, such as the cover. The cover of a golf ball must be able to withstand up to 10,000 N (Penner, 2003) of force without cracking and also be able to take repeated hits without wearing down. The entire ball must be able to snap back into its original shape without any damage to itself or its properties from the momentary deformation that occurs when it is hit with the club. The ideal ball would have a perfect transfer of energy between the club and the ball, so that none of the golfer’s force is wasted.

How can a perfect ball be manufactured? A perfect ball would be easy to manufacture, being made from polymers that easily release from their mold, thus preventing damage to the ball if it had to be pried from its mold. The best golf balls to date are multi-layer balls, which have four layers devoted to a specific purpose: middle layers that are compressible and elastic, the cover that is hard and resilient, and an inner core that is rigid to maintain form (Giffin, 2002). Even with such high-tech polymer balls, other challenges remain: can scientists develop a polymer to do all that has been required of it previously and prevent moisture from seeping into the ball and compromising its integrity?

Learn the technical chemistry in creating the perfect golf ball.

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.

Creating the Perfect Golf Ball with Chemistry (Technical)

History of the Golf Ball

Gowf, as the Scots called the game, originated in Eastern Scotland in the 1500s, although precursors of golf existed in Belgium and the Netherlands as early as 1257. Despite the fact that King James II,III, and IV of Scotland each banned the game to encourage skill in archery instead, the sport surged in popularity among royalty and nobles once the ban was lifted—King James IV even purchased his own set of golf equipment (Mallon and Jerris 2011). At this time, players would play with balls and clubs fashioned out of wood. In the 1600s, the game was improved by the introduction of a “featherie,” a small leather ball stuffed with feathers. The seams on the ball provided needed turbulence to make the ball fly farther than the simple wooden balls. Making a featherie was a time-consuming and expensive process, since an experienced golf ball maker could only produce four in a day (Seltzer 2008). These featheries thus restricted the game to an elite group that could afford the expensive balls. The expense of the game, however, caused the next breakthrough in golf balls – a university student at St. Andrews began experimenting to make a cheaper ball, since he could not afford to buy his own featherie. He used some tree sap called gutta percha, molding it into a ball and then baking it (Seltzer 2008). Thus the next revolution in golf was launched; “gutties,” which could be mass-produced, became the dominant ball (Goodman 1974). The modern ball came into existence when American Coburn Haskell began using the polymer rubber to create a new ball by coating a rubber-wound ball core with a gutta-percha covering and dimpled the outside surface. (A polymer is a molecule that has repeated smaller subunits joined into a large molecular chain.) Shortly thereafter, players began to break dozens of records with the new ball (Mallon and Jerris 2011). Since then, many modern golf balls have been developed through extensive research in a quest to find the best polymer with superb properties to create the best ball.

The Modern Ball: Properties

Figure 1: A – chemical structure of natural rubber, B – chemical structure of gutta-percha

The modern ball, of which there are many varieties, vastly outperforms previous golf balls because it offers more desirable qualities than the older golf balls: good “feel,” resilience, high abrasion resistance, high coefficient of restitution, and good release from the mold during the manufacturing process. A good golf ball must “feel” soft to the golfer, but must be resilient enough to rebound to its original shape and hardness after the momentary deformation that necessarily results after being struck repeatedly with a force that can equal 10,000 N (Penner 2003). The ball covering must not get small cracks upon the recurring impacts, because then performance is decreased not only because the cracks affect the way the air moves around the ball, but the tiny cracks may also provide an opening for moisture to get inside the ball. The “feel” is thought to be attributed primarily to the ball’s flexural and tensile moduli (a material’s tendency to bend and the stiffness of an elastic material; Morken and Talkowski 2011).Golfers want balls with high abrasion resistance, balls that take a long time to wear down from repeated hits from the club and skittering on grass and sand. A coefficient of restitution (COR) is the ratio of the velocity of the ball relative to the club before impact over the velocity after impact (Penner 2003). A COR of 1.00 would be a perfectly elastic collision, meaning that a ball’s maximum possible initial speed is limited only by what the golfer can put into his swing. Finally, there is the side of the manufacturers to consider; do the polymers in a ball release easily from a mold? If not, not only is it inefficient and causes a decrease in productivity for manufacturers, but it can result in damage to the ball itself (Shimosaka et al. 1998). These golf ball qualities differ not only from ball to ball, but within the components of a golf ball itself.

Figure 2: Summary of the three different types of golf balls.10

Golf balls are made up of two parts: a cover and a core. The core typically takes up a vast majority of the volume of the ball, often exceeding 80% (Morken and Talkowski 2011). Variations in the number of layers in golf balls form the three main types of balls: traditional wound, two-piece, and multilayer golf balls (Figure 2; Gorss 2005).

 

The Modern Ball: Polymers

The first wound ball had three layers made by winding a small, hard core with elastic, rubber thread stretched many times its length and then coating it with a layer of gutta-percha. It was also the first modern ball, although it did not take long for manufacturers to replace the more rigid gutta-percha with rubber. Figure 1 shows rubber (A) and gutta-percha (B). Both are polyisoprene, but rubber is cis and gutta-percha is trans (Goodman 1974, Yikmis and Steinbüchel 2002). This slight difference causes rubber to be more elastic because its more kinked structure makes for a larger molecular spring, unlike the more brittle gutta-percha.

In a search to find something simpler and better (and cheaper to produce), manufacturers went to the two-piece ball, after discovering that a one piece ball simply could not bring all of the desired properties to a golf ball – a composite had to be made to obtain the optimum qualities. One polymer simply could not have both great durability and a high COR (Cochran 2010). Those qualities are more or less mutually exclusive in one polymer. But by focusing on a cover with high durability, but a core with a high COR, a simpler, better ball was born that now holds about 75% of the market: the two-piece, solid-core ball (Cochran 2010).

The most recent in golf ball innovation shows that multi-layer balls produce drastically better results than any ball previously developed. These balls, which cost $40-50 per dozen, ushered in a revolution of distance in golf (Gorss 2005, McKay et al. 2008). Elements such as titanium may be found in the outer layer of these balls, to ensure near-perfect transmission of energy from the club to the ball, providing great distance (Giffen et al. 2002). The cores of these balls vary greatly, being compressed air, metal, or even honey.

The keys to the modern golf ball are the different polymers used in the ball. By slightly varying the length of the chain or the number or type of atoms in each subunit, everything from Styrofoam to Kevlar can be created. Different polymers bring different strengths to the golf balls as well: while middle layers may be compressible and elastic, the cover may be hard and resilient, and a core may be rigid to maintain form.

One of the most common polymers in use is balata, a resin from the latex of a tree native to South America. Balata can create a cover for golf balls that is softer than most, providing a high spin rate and precision when the club grips the ball (Giffen et al. 2002). This provides the desired “feel” that golfers want in their balls. Indeed, even though balls covered in balata are cut easily if they are mis-hit and they have a short lifetime, the “feel” produced by balata is what researchers seek to imitate in new polymers (Nesbitt 1984, Sullivan et al. 1990).

One solution to creating the “feel” of balata, but also doing away with its lack of durability, is to combine it with another polymer such as the commonly used polybutadiene (Proudfit 1994). An added advantage to combining balata with other polymers is that the cost of production goes down. Polybutadiene, a synthetic rubber, is a thermally linked elastomer (heating causes the elastic-like polymer chains to link together). Since polybutadiene has such a high wear resistance, it can be found in car tires, as well as golf balls. In this way, a golf ball may be endowed with a soft “feel,” but also have great durability. Polybutadiene may also be used in golf ball cores. A combination of two different polybutadienes has been shown to increase the distance of a golf ball by 1 ft. when compared to just a single type of polybutadiene, thus earning the polymer combination a patent (Gendreau and Cadorniga 1990).

Finally, ionomers (polymers with some subunits that carry an electric charge) are also common polymers found in golf balls. Ionomers are a harder resin, providing great durability, wear and tear resistance, and a substantial amount of resilience in golf ball covers (Egashira and Takehana 2010, Morken and Talkowski 2011). These covers are almost cut-proof, although their durability comes at the cost of having a hard “feel.” Additionally, ionomer resins offer an excellent barrier to moisture seeping into the ball.

Future Research

With so much research and high-tech know-how going into golf balls to make them go farther than ever before, what else remains for scientists to develop in golf? Besides the ever-present search for the perfect blend of polymers for optimal durability and “feel,” the effect of water adsorption into golf balls has come to the forefront. Over time and use, the COR decreases as weight increases, whether the ball has actually been in water or is only in a humid environment, thus decreasing the distance a ball can fly (Sullivan et al. 1998). Even if the ball has been in the water for only a week, the ball’s distance capability is decreased by 6 yards (Performance Indicator 2012). In order to combat this problem, some researchers are striving to develop balls that possess a moisture barrier at the core or an ability to change color if their integrity has been compromised (Sullivan et al. 1998, Performance indicator 2012). Either way, the difficulty lies in developing a coating that will not alter the properties of the other polymers in the ball. As long as there are golfers in search of the perfect ball to better their game, researchers will be there with them.

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

Cochran, A. J. 2010. Science and golf: proceedings of the first world scientific congress of golf. Tayor & Francis, United Kingdom.

Egashira, Y. and E. Takehana. “Golf ball material, golf ball and method for preparing golf ball material.” Patent 7,803,874 B2. 28 September 2010.

Gendreau, P.M. and L. Cadorniga. “Polybutadiene golf ball product.” Patent 4,955,613. 11 September 1990.

Giffen, G., S. Boone, R. Cole, and S. McKay.  2002. Modern sport and chemistry: what a chemically aware sports fanatic should know. Journal of Chemical Education. 79(7): 813-819.

Goodman, A., et al.1974. The thermomechanical properties of gutta-percha. Oral Surgery. 37(6): 954-961.

Gorss, J. 2005. Golf balls: polymer chemistry has played a key role in the evolution of the golf ball. Chemical and Engineering News. 83(29): 34.

Mallon, W., and R. Jerris. 2011. Historical dictionary of golf. Scarecrow Press, Lanham, Maryland, USA.

McKay, S., T. Robbins, and R. Cole. 2008. Modern sport and chemistry: what a golf fanatic should know. Journal of Chemical Education. 85(10): 1319-1322.

Morken, P. and C. Talkowski. “Golf balls containing ionomers and polyamides.” Patent Application 13/033,875. 30 June 2011.

Nesbitt, R.D. “Golf ball and method of making same.” Patent 4,431,193. 14 February 1984.

Penner, A.R. 2003. The physics of golf. Reports on Progress in Physics. 66: 131-171.

Performance indicator. (2012). Retrieved from http://www.performanceindicator.com/golf/

Proudfit, J.R. “Golf ball with improved cover.” Patent 5,314,187. 24 May 1994.

Seltzer, L. 2008. Golf: the science and the art. Tate Publishing, Mustang, Oklahoma, USA.

Shimosaka, H., K. Ihara, Y. Masutani, M. Inoue, and A. Kasasima. “Golf ball and method of making same.” Patent 5730665. 24 March 1998.

Sullivan, M.J., T. Kennedy, and M. Binette. “Golf ball and method of making same.” Patent 5,820,488. 13 October 1998.

Sullivan, M.J., T. Melvin, and R.D. Nesbitt. “Golf ball cover of neutralized poly(ethylene-acrylic acid)copolymer.” Patent 4,911,451. 27 March 1990.

Yikmis, M., and A. Steinbüchel. 2012. Historical and recent achievements in the field of microbial degradation of natural and synthetic rubber. Applied and Environmental Microbiology.78(13): 4543-4551.

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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