ball and clubface is so great it quickly begins to spin (roll) off the top of
the club. This generates the tremendous amount of spin necessary to keep the
ball a loft for drives at or above 230 yards. It is these three factors together
that the quantity known as effective loft is derived from. The effective loft of
any club is given as EL = L + a(i) – B(i) ? Y. L is the loft of the club a(i)
and B(i) are angles that are dependant upon each swing and each person
performing the swing and Y is the back swing angle of the arm. From effective
loft of the club, one can estimate the components of drag and lift on the golf
ball. The following table expresses the variations that are possible during the
swing. B(0) – B(i) – EL + Spin + Lift + B(0) + B(i) + EL – Spin – Lift – Y – B(i)
+ EL – Spin – Lift – Y + B(i) – EL + Spin + Lift + TS + B(i) – EL + Spin + Lift
+ Al + B(i) – EL + Spin + Lift + As an example, the third line reveals that when
the back swing angle of the arms is decreased, the effective loft is decreased,
the spin is decreased, and the lift is decreased. As one can see through the
material presented above, the golf swing is a multi-stage process. It is not
simply the swing, or the transfer of energy, or the flight of the ball that is
subject to the laws of physics. The first aspect of the golf stroke, which is
based upon physical principals, is the downswing of the golf club. The golfer
must do two things in order to have a successful shot. He must first generate
enough energy to hit the ball a significant distance. And then he must transfer
this energy into the golf club. The energy is derived from the muscles in the
golfers body. As was previously stated it takes at least 32lbs of muscle to
generate the necessary two horsepower for hitting the golf ball. Most of this
energy comes from the legs and back of the individual. Then, the golfer uses his
body and arms, along with the shaft of the golf club like a whip. Just as a whip
transfers energy from its large mass at the handle down to the tip causing a
dramatic acceleration, the golfer transfers the energy through his body into the
shaft of the golf club, which flexes. When the golfer snaps his wrist at the
point of impact, all of the energy is transferred into the club head allowing it
to achieve a velocity of 100mph or even greater. At the point of impact, more
physical properties take over. As the club comes in contact with the ball, two
important factors are most prevalent. First, the ball is semi- elastic and
therefore the ball flattens somewhat when it comes in contact with the face of
the club. This allows the ball to spring away at a tremendous velocity, which is
also based on the principal of conservation of momentum. The other important
factor that happens at impact is the generation of spin. At first, the ball
begins to slide up the face of the club toward the top, however, because of the
large coefficient of friction; the ball stops sliding and begins rolling. This
action gives the ball a rotation around its horizontal axis, which creates lift
and drag. Lift and drag are the final aspects of how physics relates to golf. As
the ball spins, it creates lift by disturbing the flow of air around the ball.
The dimples help greatly with this. However, drag is also produced, which
threatens to pull the ball back toward the earth. It is the job of the golfer
and the golf ball manufacturer to generate enough lift either through the swing
or the dimpled design of the golf ball so that the upward lifting force
counteracts the downward forces of gravity and drag. As anyone who has played a
round of golf has observed, the spin created with modern clubs and ball design
more then compensates for drag and gravity and allows the ball to stay aloft for
a long time. Because of the unique challenges that physics present during a game
of golf, it will be a long time before anyone is able to master the game. In an
endeavor to improve scores many miracle products have claimed to lower ones
score, however it is evident that only those ideas and products, which have a
basis in science, have stayed on the market. The golf ball is a prime example of
this. It has made dramatic changes from being made of dried goose feathers to
the two piece dimpled design of today. All of the improvements on the ball were
based around trying to give the golfer and edge in lowering his score and
working around some of the laws of physics, which prevent him from reaching
perfection. Appendix 4 The following terms will be defined based on their
relevance to the physics of golf: momentum, moment of inertia, torque,
centripetal force, and centrifugal force Momentum: Newton?s first law defines
a property of a body called inertia, which describes what happens to a body when
no force acts on it; the inertia of a body is said to be measured by its mass.
When acted upon by a constant unbalanced force, the body will experience
acceleration proportional to the mass of the body. The mass of a body is
proportional to its weight. Momentum is then defined as the mass of a body
multiplied by its velocity. Like velocity, momentum, has a direction as well as
magnitude, making it a vector quantity. From the definition of momentum, for
constant mass the rate of change of momentum is the product of the mass and its
acceleration. Newton?s second law suggests that an unbalanced force on a body
is associated with its acceleration. For the purpose of this paper, Newton?s
second law states that the mass of a body multiplied by its acceleration is
proportional to the force acting on it, and the acceleration is in the direction
of the force. The way in which momentum applies to golf is through the transfer
of momentum from the golf club to the golf ball. Before the collision, the club
head is moving at a speed of 100mph along the horizontal. After the collision,
for a club without loft, the ball is moving off at a high velocity, and the club
head continues in the follow-through at a somewhat reduced velocity. For a club
without loft, these velocities will also be horizontal. The momentum is such
that the total momentum before the collision is equal to that of the momentum of
the club head after the collision plus the momentum of the ball. Moment of
Inertia: The linear acceleration of a body when acted upon by a constant force
depends on its mass, which as already stated is quantity proportional to its
weight. The larger the mass is, the smaller the acceleration will be for a given
force. Similarly, when a constant torque acts on a body, its angular
acceleration will depend on the mass of the body and on how the mass is
distributed in the body. The combination of mass and its distribution in the
body is called its ?moment of inertia?. When the axis of rotation of the
body is chosen such that more of the mass is far from the axis, the moment of
inertia will be larger. Thus the moment of inertia will depend of the choice of
axis. This concept is easily demonstrated with the help of a golf club. When the
club is help at the grip end between two fingers and let hang so that the shaft
is along a vertical line, it is very easy to rotate the club along a vertical
axis. But when the club is held near the center of the shaft, where it balances
between the same two fingers, the same torque produces a much smaller angular
acceleration. The moments of inertia in the two cases differ by a factor of 10.[
] This same affect can be observed when a club is first waggled about the grip
in the usual way and then waggled while holding the head. Torque: Torque is the
term used to describe twist in a quantitative manner. Two factors, the amount of
force applied and the distance over which it is applied determine torque. The
size of the torque is found by multiplying the size of the force by the length
of the lever arm, the lever arm being the shortest distance from the line along
which the force acts to the axis about which the body may rotate. The force must
be in a plane perpendicular to the axis of the rotation. Centrifugal Force: This
force can be observed when a golf ball is placed on the dashboard of an
automobile just inside the windshield and is observed while the vehicle travels
around turns. One will notice that the ball will always roll to the outside of
the curve and rolls more quickly the tighter the turn. Actually, the ball does
not accelerate; it appears to accelerate since there is no centripetal force to
make it turn in the same path as the car. Its motion is the result of a lack of
centripetal force rather than the result of an outwardly directed force being
applied to the ball. Centripetal Force: According to Newton?s Second Law, the
centripetal force on a body moving in a circle is proportional to the mass of a
body multiplied by its centripetal acceleration. The centripetal acceleration
increases with the radius of the circle on which it moves and with the square of
the angular velocity of the motion. Appendix 1 The following curves were drawn
based on the information gathered and analyzed with the use of a computer. The
curves are calculations for the energies present during a swing. Curve A shows
the total kinetic energy as it develops throughout the swing. Curve B shows how
the kinetic energy of the arms varies throughout the downswing. Curve C shows
how the kinetic energy of the club alone varies throughout the swing. Curve D
shows the work done by the golfer as he applies the torque by his arms to the
system. (graph taken from source #5) Appendix 2 (original drawings but concept
from source #5) These drawings illustrate the forces on a golf ball during its
flight. The first set of pictures shows how the air moves around the ball during
its flight. The first pictures show that when there is some spin, the air
pressure around the ball is changed because of the turbulence created by the
rotation. The picture below that shows how the air would move if there were no
spin. The other two pictures demonstrate how using spin can change the flight of
the ball. For example, the top picture is showing that a ball spinning on a
vertical axis in a clockwise direction will travel to the right because of the
airflow around the ball. The bottom picture on that side is illustrating another
example of how air can flow around a ball with no spin. The other two
illustrations show a three-dimensional (on the top) and a two-dimensional (on
the bottom) view of some of the vectors involved with the flight of a golf ball.
The illustrations show the effective loft of the club, the lifting vector as
well as the path of the golf club and the angle at which the face points.
Appendix 3 This is a graph of the five-torques acting on the arms as they vary
throughout the downswing. Curve A shows the constant torque TS of the golfer on
the system. Curve B shows the torque that depends mainly on the acceleration of
the wrist-cock-angle. The torque represented by curve C depends mainly on the
square of the velocity of the wrist-cock-angle. Curves D and E show the torques
resulting from action of gravity and the golfers weight shift respectively. The
torque T shows how the sum of the five-torques on the arms varies during the
downswing and becomes very large just prior to the club colliding with the ball.
(graph from source #1)
1. Abrahams, Jonathan (1994). Club Smarts. New York: Lyons & Burford. 2.
Andrisani, John. (1997). The Tiger Woods Way. New York: Random House. 3. Beard,
James (1982). Turf Management for Golf Courses. New York: McMillan. 4. Jones,
Trent (1993). Golf By Design. New York: Little, Brown, and company. 5. Kroen,
William. (1992). The Why Book Of Golf. California: Price Stern Sloan.