Internal Assessment in Math: Golf Swing Mechanics and Physics

This report delves into the physics and mathematics of a golf swing, providing a comprehensive analysis of the various factors involved. It begins with an introduction to the mechanics of the golf swing, emphasizing the role of mathematics in understanding shot selection and execution. The report then examines the material properties of golf balls and clubs, discussing how these materials influence controllability, flight height, and kinetic energy transfer. A significant portion of the report is dedicated to analyzing golf ball trajectory, including the effects of loft, air resistance (drag), lift, and spin, illustrating these concepts with graphs and diagrams. The classic pendulum model is introduced as a means to simplify the golf swing for analysis, with discussions on the impact of arm length, club length, wrist angle, and club head mass on swing efficiency. Mathematical formulas are presented to calculate torque, kinetic energy, and momentum within this model. The report further examines the velocity of the club head, detailing the factors contributing to it, and provides equations for calculating acceleration. Finally, it discusses the effects of height and arm length on the swing, concluding that taller golfers may benefit from longer clubs. The report aims to provide a detailed understanding of the physics involved in a golf swing.

Internal Assessment in Math

TABLE OF CONTENT
INTRODUCTION...........................................................................................................................1
Mechanics of golf swing..................................................................................................................1
Material of ball and club .............................................................................................................1
Trajectory of golf ball .................................................................................................................1
Classic pendulum modelling .......................................................................................................4
Velocity of club head during golf swing......................................................................................6
Effect of height and length of arms .............................................................................................6
CONCLUSION ...............................................................................................................................7
REFERENCES ...............................................................................................................................8

INTRODUCTION
Mechanism of golf stroke and swing assist golfers to make various decisions such as
selection of shots or clubs and to execute them. Mathematics play an important and integral role
in understanding this mechanism (Smith and et.al., 2017). The report will discuss and present
various mathematical formula and concepts to understand the concepts of golf swing, club speed
and ball trajectory. It will also describe the role of arm length and pendulum model for analysing
maths involved in golf swing.
Mechanics of golf swing
Material of ball and club
The chemistry and material of golf balls affect their controllability and flight height. For
instance in terms of controllability soft balls give better performance but they travel slower than
hard balls. In present day golf balls rubber mantles surrounds butadiene rubber centre and are
topped off with tough skin. Outer layer is made up of ionomeric resins which have ionic cross-
linking for high performance (Balzerson, Banerjee and McPhee, 2016). This design supports
soft, compressible hard core. Some companies also produce golf balls with polymers like
neodymium polybutadiene in core so that impact energy can be effectively converted into kinetic
energy and flying distance can be increased. The dimpling in balls gives better control of flight,
spin and trajectory. The golf club which is employed for hitting golf ball have a club head and
shaft with grip. For tee shots wood is preferred material for shaft while for other variety of shots
iron is used. However presently most of the shafts are made up of titanium, steel and carbon
fibre. The grip provided is made up of rubber or leather. The shorter shafts along with the higher
number of lofts are helpful in shorter and higher trajectory (Morrison, McGrath and Wallace,
2018).
Trajectory of golf ball
The angle between vertical plane and face of club is called loft and plays an important
role in determining the ascending trajectory of ball. Loft also refers to amount which slopes back
by club (Zhang and et.al., 2016). Thus sand wedge clubs which have higher loft does not hit ball
very far but they pop ball higher in the air. The optimum angle for projectile is 45 degree which
makes ball fly the farthest. However 45 degree is also known as pitch wedge angle which hits
ball less than half as far as driver (Cole and Grimshaw, 2016). Though resistance due to air
friction (drag), lift and gravity all affect the projectile but theoretical outcome of 45 degree
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accounts only for the gravity (Han, 2016). Drag is quadratic and is directly proportional to square
of ball velocity. Thus harder hitting result in rapid increment in drag.
(Source: Salzberg, 2013)
From the graph it is observed that best angle is less than 45 degree and distance is also low (112
yards) (Steven and Celermajer, 2016). Thus it is observed that backspin keeps ball in air for long
duration and thus produces lift. The lift force is directly proportional to ball speed and spin rate.
(Source: Salzberg, 2013)
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Illustration 1: Trajectory for different projection angles and
quadratic drags
Illustration 2: Ball flight at different projection angles when
lift force is taken into account

Thus with consideration of lift, ball flies around 200 yards and its optimum angle is 16 degree.
Drive can take place only when ball is struck past the swing bottom. It provides higher launch
angle than the loft (Balzerson, Banerjee and McPhee, 2016).
Trajectory shape, distance and ball flights are influenced by air resistance or drag which
slow down ball and also lowers distance. As the ball moves in air three forces work on ball
which are drag, lift and weight.
The lift is at 90 degree to path of ball and spin is inclined to left. The lift is necessary for
obtaining maximum or best distance from the ball and it is created by backspin (Ivanov and
Javorova, 2017). However for optimum distance the amount of spin varies corresponding to
speed of ball as well as launch angle.
(Source: Tutelman, 2019 )
For instance from the above example it is observed that with zero spin no distance is obtained.
When a slight spin is provided there is enough lift which allow ball to go farther. However with
continuous increase in spin when lift exceeds to certain limit then despite increment in spin does
not result in increased distance (Morrison, McGrath and Wallace, 2018). This effect is known as
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Illustration 3: Golf ball trajectory with spin and

ballooning. At this point (trajectory at 6000 rpm in above example) ball continues at even higher
angle. However when ball climbs, lift holds ball back causing steeper trajectory (Matsumoto and
et.al., 2016). As a result of this ball stays longer in air and travel time is mostly involved in
climbing instead of downrange.
(Source: Tutelman, 2019 )
The adjustment to drag and lift by making changes to dimples on surface of ball can also affect
trajectory. The variation in depth, pattern and area of dimples affect lift to drag ration by 3:1
(Hume and Keogh, 2018). When clubhead speed is greater then amount of lift to keep ball in air
is reduced.
Classic pendulum modelling
The classical double pendulum model provides effective solution to reduce and represent
the golf swing in simplest elements so that their effects can be analysed and swing efficiency can
be improved. In this model one pendulum is tacked on to end of other pendulum. The swing in
upper pendulum occurs from fixed pivot joint while the swing in lower pendulum takes place
from the end terminal of upper pendulum. This arrangement can be represent as equivalent
model in golf and swing techniques. The shoulder can be considered as fixed pivot, hands and
arms as upper pendulum and club head and shaft as lower pendulum (Balzerson, Banerjee and
McPhee, 2016). The efficiency of golf swing is influenced by variety of factors such as length of
player's arms, club length, wrist clock angle , backward bent of wrist during swing, wrist torque
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Illustration 4: Trajectory with variation in lift to
drag ratio

and mass of club head (Morrison, McGrath and Wallace, 2018). The best swing is possible only
when swinging speed of shoulder and arms is good and wrists are uncocked at right moment.
The lack of wrist cock can make swing inefficient by restricting energy transfer form shoulder
and arms to club.
(Source: Mgrdichian, 2006)
In the above model ß is called downswing angle which calculate how far arms of golfer move
and θ is called wrist cock angle which gives angle between arms and club shaft. The rotational
equivalent of F = ma is given by τ = I α where τ is torque, I is moment of inertia and α is
angular acceleration. Hence for enhancing club head speed and constant angular velocity, bigger
torque is required.
Kinetic energy = I * (angular velocity)^2
Momenta = I * (angular velocity)
the symbols i and f denotes initial and final velocity.
As per this model instantaneous motion is consider in following two situations:
 Moment before release of string, when everything is rotating and nothing is flying
 Moment of impact in response of complete release which is linear curve from rotation
centre down the arm of golfer and club to ball (Matsumoto and et.al., 2016).
These snapshots are possible only when kinetic energy and angular momentum are same.
T = 0.5 (m1 L²1 ß²i + m2R²ß²i ) = 0.5 [m1 L²1 ß²f + m2 (L1 + L2)² α²f ]
L = m1 L²1 ßi + m2R²ßi = m1 L²1 ßf + m2 (L1 + L2)²αf
On solving these equations by using equations of collision simplified equations are:
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Illustration 5: Double pendulum model

= (m1 L²1 + m2R²) ß² = m2 (L1 +L2)² α²f
= (m1 L²1 + m2R²) ß = m2 (L1 +L2)² αf
m1 L²1 + m2R² = m2 (L1 +L2)²
R² = L²1 + L²2 - 2 L1 L2 cos θ
Thus necessary condition for optimum club-head mass is given as:
m2 = {m1 / [2*(1 + cos θ)]} * {L1 / L2}
Real life example:
Let us consider a golfer of height 5'10'' with arm 24'' and weight 176 pound. The various
swing parameter of this golfer are compared with another golfer of height 6'2''. During down
string let the wrist torque is zero and shoulder torque is 58 foot pound. The club head speed is
109 mph and driver is 45'' with normal shaft and head weights. In case of taller golfer the arms
are increased by 5.7% and thus it is equals to 25.4'' long. As a result of this club head speed
drops to 107 mph. Further if club length is increased to 47.6'' then it again help to restored the
2mph club head speed. Similarly the 5.7% scaling in height results in 11.7% strength scaling in
shoulder torque resulting in increased speed of 113 mph. Due to this reason the taller golfer tends
to hit farther as compare to shorter golfers.
Velocity of club head during golf swing
The factors such as torque at beginning of swing, force at impact and torque at grip
contributes to club head velocity. The force and torque act on club grip and thus motion of club
is given by equation:
F – mg = ma
T + r *F = R (Iw' + w'* Iw')
where F = force acted on grip
m = mass of club g = gravity a = acceleration of club grip
T = torque r= position vector from grip to point at which T and F act
R = rotational transformation matrix w = angular velocity
The acceleration of club head is given by a(ch)
a(ch) = AT + A [r X] F + (F/m) – AR (W' * I'w') + w * (w * I(ch)) – g
AT = torque A [r X] F + (F/m) = force
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AR (W' * I'w') = GYRO w * (w * I(ch)) = Cent. Acceleration
g = gravity
Effect of height and length of arms
Application of swing at correct stage can help to exert maximum force for achieving
perfect goal swing. As compare to wrists arms of golfers plays more important role in improving
swing. With long arms bigger arc are formed or leverage is increased. Among taller golfers arms
length is also higher which reduces the club head speed. Thus taller golfers must be able to
handle proportionally longer club (Cole and Grimshaw, 2016). A longer driver must provide
higher club head speed so that good impact can result in longer distance. To accelerate the club
small force is applied at hands because the same torque drives longer arm lever.
Hence it can be concluded that the club head speed which is reduced to long arms can be
regained by using a longer club. Another parameter which is affected by height is weight. Taller
players have usually higher weight and it will require more torque for rotating the extended arms
and torso. Due to this reason the club head speed is again reduced in equal amount as it was
increased by arm length. Though increased muscle must relate with higher muscular force but
change in weight does not affect the torque.
Among taller and shorter people if all other parameters are same then force increases with
increase in muscle cross sectional area which is directly proportional to square of height. Torque
at any joint is product of muscle strength and moment arm from pivot of joint to muscle
attachment. The distance between pivot and muscle attachment is longer and thus torque scales
with cube of height. As compare to short golfers, tall players have advantage of increased club
head speed which is provided by increased torque value. Thus it can be concluded that taller size
of players gives them a frame of more muscles and large lever arms at joints which is the reason
that they hit ball farther (Smith and et.al., 2017). The bigger or longer arc is defined as the arc
travelled by hands and can occur in the form of longer arms (bigger radius) and increased
shoulder turn (bigger angle).
Tangential velocity (v) = rw
In golf swing v = club head velocity
r = radius of lever from rotating point (club or arm)
w = rotational velocity
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Thus when taller golfer creates same rotational velocity w then on increasing v ball will hit
farther.
CONCLUSION
It can be concluded that by calculating and integrating maths in finding various forces,
trajectories and ratios, golfers can improve their swing and performance. The application and
knowledge of pendulum models or impact of arms can add to efficiency in their game.
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REFERENCES
Books and Journals
Balzerson, D., Banerjee, J. and McPhee, J., 2016. A three-dimensional forward dynamic model
of the golf swing optimized for ball carry distance. Sports Engineering. 19(4). pp.237-
250.
Cole, M.H. and Grimshaw, P.N., 2016. The biomechanics of the modern golf swing:
implications for lower back injuries. Sports Medicine,.46(3). pp.339-351.
Han, K.H., 2016. Lower body mechanics of golf swing and its association with maximum
clubhead speed in skilled golfers(Doctoral dissertation, Texas Woman's University).
Hume, P.A. and Keogh, J., 2018. Movement Analysis of the Golf Swing. Handbook of Human
Motion. pp.1755-1772.
Ivanov, A. and Javorova, J., 2017. Three–dimensional golf ball flight. Journal Tehnomus–New
Technologies and Products in Machine Manufacturing Technologies. pp.54-61.
Langdown, B.L. and et.al., 2019. Acute effects of different warm-up protocols on highly skilled
golfers’ drive performance. Journal of sports sciences. 37(6). pp.656-664.
Matsumoto, K. and et.al., 2016. The Influence of a Golf Club's Inertia on Shaft Movement
During the Golfer's Swing. Procedia engineering. 147. pp.360-365.
Morrison, A., McGrath, D. and Wallace, E.S., 2018. Analysis of the delivery plane in the golf
swing using principal components. Proceedings of the Institution of Mechanical
Engineers, Part P: Journal of Sports Engineering and Technology. 232(4). pp.295-304.
Smith, A.C. and et.al., 2017. Comparison of centre of gravity and centre of pressure patterns in
the golf swing. European journal of sport science. 17(2). pp.168-178.
Steven, G.P. and Celermajer, J., 2016. Simulating the Contact of Golf Club to Ball for Improved
Performance. In Applied Mechanics and Materials (Vol. 846. pp. 294-299). Trans Tech
Publications.
Zhang, M. and et.al., 2016. Wrist lagging angle impact on both golf driving distance and
accuracy. International Journal of Sports Science 6 (6).
Online
Mgrdichian, L., 2006. Physics reveals the key to a great golf swing. [Online]. Accessed through
<https://phys.org/news/2006-12-physics-reveals-key-great-golf.html>
Salzberg, S., 2013. The Physics of Golf: What's the Ideal Loft to Hit the Ball Farthest?. [Online].
Accessed through <https://www.forbes.com/sites/stevensalzberg/2013/04/29/the-physics-
of-golf-whats-the-ideal-loft-to-hit-the-ball-farthest/#f45e30b69268>
Tutelman, D., 2019. Flight of the Golf ball. [Online]. Accessed through
<https://www.tutelman.com/golf/design/swing3.php>
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