Golf Swing Physics
Guest article by Rod White
the previous sections we explained the principle of
the golf swing, how the unfolding of the club from the cocked position
rotational energy to be transferred from the arms and the body to the
that this unfolding can be passive requiring no effort from the golfer.
Now that we understand what is happening, let’s look
at a more realistic model of the golf swing, in the interest of
clarifying technique. In the actual
the golfer is applying torque, throughout the swing, to the inner arm
of the double pendulum -- by using muscles in the torso to turn the
improved model is a similarly-driven double
pendulum with a few extra features that allow us to investigate aspects
technique. It will give us an opportunity to see what sort
the golfer can do with the swing to improve his distance -- or screw
things up altogether.
The improved model can vary:
moment we will focus on the aspects of technique that have the largest
effect on the effectiveness of the golf swing – we will look at the
technological factors later.
- Torque applied to the inner arm of the pendulum, to
model the work done by the golfer via the torso and shoulders
- Torque applied to the outer arm of the pendulum, to
model work done by the golfers hands
- Wrist cock angle
- Release timing, to model the golfer releasing the
club early or late during the first phase of the downswing
- Arm mass
- Arm length
- Club shaft length
- Club head mass
- Coefficient of restitution for the club-ball collision
The animation shows the swing of a moderately good amateur golfer with
a sound golf swing.
the first part of the downswing,
the golfer holds the club
in a cocked
position and accelerates the shoulders and torso. Initially some
positive wrist torque is required to stop the club from being pulled
into the golfers neck (the hub). Remember the passive, steadily
rotating model on the
previous page? There, a string (providing negative torque) was needed
to keep the
club from swinging outward. Here, during the initial build up of speed,
some sort of "brace" (providing positive torque) is needed prevent the
from being pulled inward. The positive torque
required to brace the club falls rapidly as the club accelerates. When
positive ‘bracing’ torque falls to zero, the club can be allowed to
swing out -- ending the first phase.
phase of the downswing occurs as the club swings out.
If the golfer lets
the club swing out when the bracing torque falls to zero, then this is
described as a swing with a natural
the golfer holds the club in the cocked position for a short
while longer, this is described as a late release. If the golfer
club early, the club will swing in towards the neck for a small moment
swing out. We won’t
look at the effect
of release timing because to a good approximation release timing has no
During the second phase the golfer continues to turn his body and
no torque is applied via the hands – they are no more than a hinge
during this phase.
This model will be the starting point for
all our future calculations. The full numerical model includes a number
of variable factors, as indicated in the list above. The animation
solution for the swing we have just described.
pure coincidence (perhaps), the golf swing can be executed with the
natural release. This is not necessarily true for all stick-and-ball
sports, not even if the stick is a golf club. Consider:
- With baseball swings, the natural swing time is
much shorter because the bat is shorter, and the base-baller must
restrain the bat using negative
wrist torque to stop it from swinging out early.
- With professional long-drive
golfers, the shaft length is longer (up to 50
inches), the natural swing time is
much longer, and a swing with a natural release is anatomically
impossible. Professional long drivers must use positive wrist torque
(forcing the club out) to complete the swing.
- The normal golf swing
does not require positive or negative wrist torque during the second
phase of the downswing; therefore the hands are passive, and the golf
stroke can be more
accurate with fewer muscles involved.
the energy transferred?
We have discussed the golf swing in terms of the conservation of energy
and momentum, and showed that the energy is transferred to
the club as the swing unfolds, but what actually happens – where are
the forces that make this happen?
The figure shows a ‘stroboscopic’ view of the golf swing. Have a
close look at the direction of the clubhead midway through the swing –
this is indicated approximately by the red arrow. Now look where the
move at the same time – the blue arrow: in a different
Obviously the hands and clubhead cannot continue to move in different
directions, they are restrained by the fixed length of the shaft. The
of the club and hands results in a large tension in the shaft.
The tension pulls against the club head causing it to accelerate, and
against the hands causing them to decelerate. It is the
differing directions of the hands and club that are ultimately
responsible for the energy transfer.
In a professional golfer's swing, the tension peaks above
500 N (50 kg
equivalent, or over 100 pounds). During this phase of the
swing the rate at which
energy is transferred to the club peaks at about 5 kW (or
Now we’ll take a look
at the factors that affect the effectiveness of the swing. The two big
are the wrist cock angle and the wrist torque. Since greater wrist cock
increases the divergence of the trajectories of the hands and the
clubhead, we can
expect greater wrist cock (smaller wrist-cock angle) to improve the
swing. It is less
obvious whether wrist torque -- usually described in golf as "hand
action" -- helps or hurts. Let's look in more detail at the effects of
wrist cock first.
figure to the left shows the
clubhead speed versus downswing angle (the angle between the arms and where they were at the beginning of the downswing), for
three different wrist cock
angles (the angle between the arms and the club shaft). Note again
that the wrist-cock angle is
between the arms and the club, so a smaller angle corresponds to
cock, or greater "lag" as it is often called. If there were no wrist cock at all, the angle would be 180 degrees.
As expected, increasing the amount of wrist-cock (reducing the angle between
the arms and shaft) increases the efficiency of the swing. The
key point is that the peak speeds all occur at a very similar
angle, showing that the swing timing is almost unchanged.
The golfer expends the same effort for all three swings, yet we see a
10% increase in head speed resulting in a 10%
increase in distance – say 20 m for a 200 m drive --
with no extra effort.
chart at the right plots the driving distance versus wrist cock angle,
other aspects of the swing remain the same and that the ball is hit at
the peak head velocity. The
increase in distance that occurs with a decrease in wrist-cock angle
between 110 and 70 degrees is about 20 m – say 5m for each 10
Note again – the distance is gained with no extra effort from
the golfer – the difference is purely one of technique.
confirmation of this point from DaveT: In December 2009, I was playing
in a foursome about my own age. We were all within a year or two of 70,
and in relatively good shape for our age. Two of us had roughly a 90º
wrist cock at the start of the downswing; the other two had almost no
wrist cock at all. Throughout the round, it was telling that the two
with wrist cock were roughly equal length; so were the two without
wrist cock, but typically 30-50 meters back.
For those who are interested, I've estimated the distance from the
clubhead speed using a formula from Cochrane and Stobbs.
D = 3.75 x
speed – 25m
where the ball speed is in meters per second.
look at the effect of positive wrist torque during the second phase of
the swing. The use of the hands is a very frequent flaw with
amateur golfers. It is not uncommon to see the hands spread far apart
(like a baseball grip), or the right hand adjusted so the thumb is
behind the shaft through impact, or the right forefinger is set down
the shaft. All this is done in the hope of pushing the head faster
through the impact zone.
In fact it has the opposite effect. In this figure, the wrist torque is
expressed as a percentage of the shoulder torque. For the model I’ve
chosen, 10% corresponds to 1 kg.m of wrist torque in the
model. This is
a very large torque, but probably
typical for male beginners who have yet to learn to let the club swing
The graph shows that positive wrist torque causes the club to unfold
early, and therefore
causes the clubhead speed to peak early, and with a lower
velocity. Common symptoms include a pronounced swishing sound
that peaks before impact, drop-kicked shots (club ricochets off the
ground before impact), shots with a high trajectory, and often problems
with big high fades or slices. Researchers who have
tracked the swing speed for golfers with a range of handicaps find that
with low single-figure handicaps or better come close to hitting the
the peak clubhead speed. For most golfers, the club is decelerating
chart to the right plots the approximate driving distance versus wrist
torque with almost all other parameters kept the same. Remember that
wrist torque has two effects on clubhead speed. It (a) peaks at a lower
clubhead speed and (b) peaks earlier in the downswing.
Even if we assume that the ball is hit at the peak head velocity (blue
curve), the difference between a beginners swing (10% wrist torque)
and a swing with no wrist torque is about 20 m in distance.
beginner will take the same backswing as a low handicap golfer and lose
distance indicated by the red curve – nearly 40 m!
- The blue curve
assumes that the golfer changes his swing so impact still occurs at the
peak. We shorten or lengthen the swing so that impact will occur at
maximum clubhead speed. This golfer is then only bitten by (a) above.
- The red curve
assumes that the golfer simply makes the same length swing no matter
what the wrist torque. This golfer is then bitten by both (a) and (b).
Negative wrist torque also costs distance because the clubhead speed
peaks after impact (i.e., impact is at the black line in the curve
This is a very tough
lesson, yet all of us have experienced the occasion when we relax, try
hit a ball too hard, and hit the best drives of our lives. Learn to
relax, to shorten
your grip, and not to
use your hands.
trouble believing that you do not need to use wrist torque to have an
effective golf swing. But to prove a point,
some stunt golfers use drivers with a section of rubber tube or dog
replacing part of the shaft. They still hit the golf ball a long way --
much the same distance as with a proper shaft. With such a flexible
there is no way that wrist torque can have any effect.
Another good example is the trebuchet shown in a couple of
the videos on the next page, a medieval siege machine
used to fling rocks into or over castle walls. From the physics point
of view, the trebuchet is an upside-down golfer; the raised weight
represents the torque applied through the shoulders, the long wooden
beam represents the golfer's arms, and the rope sling represents the
shaft of the golf club.
Remember that almost all the energy transfer to the club is due to
tension in the shaft; the shaft does not need to be stiff, because the
vast majority of the force it transmits is along the length of the
shaft. If the club had a perfectly
flexible shaft -- like the rope sling of a trebuchet -- then there is
no way to apply wrist torque to get any action from the clubhead. Yet
the trebuchet was a very effective siege weapon. It was the best, most
powerful catapult in warfare for centuries until it was replaced by
cannons, demonstrating that "wrist torque" isn't all that important.
See the videos on
the next page for visual demonstrations that a trebuchet can
sling things a long way with no "wrist torque" at all.
Summary for Technique
by the golfer builds up kinetic energy in
the torso, shoulders, and arms. This is then transferred via tension in
shaft as the club and arms unfold away from the golfer’s body.
the negative effect of wrist torque and the positive effect of wrist
cock, account for most of the 70 m difference between the
the scratch golfer.
the greater the fold (wrist cock) the
more efficient the transfer of energy from the body to the club.
bad: the greater the wrist torque (use of the hands) the
the club unfolds and the less energy is transferred to the club.
These effects are also
not what the beginner golfer expects. This perhaps explains why a good
golf swing is so hard to learn.
Another factor making a good swing hard to learn is that it is mentally
difficult to hold onto the club firmly while not holding the wrists
firmly. It is curious that most people, when asked to throw a golf
far as possible, would swing the club around their shoulders without
using wrist torque, and
this is exactly the action required for a good swing. Swinging a club
loosely around your shoulders as if you were about to throw it will
help to train your brain to not use your hands. I have also found it
helpful to visualise throwing the club through the impact zone. In fact
a full vigorous swing around your shoulders like a baseball swing,
including hip and shoulder movement, captures all of the important
parts of the swing.
One of the benefits of the overlap grip is that it keeps
the combined length of the hands short and the right hand weak
(for a right-handed golfer). This enables the golfers to grip the club
firmly, but limits the ability to apply wrist torque.
have not given the full equations for the
numerical model here. As I indicated in the introduction, they are too
complicated to yield any insights directly – at least not for me. Also,
have to be solved numerically because there is no analytic solution. If
want to experiment with the equations, the full version can be found in
4 of Jorgensen’s book.
the analysis here I neglected gravity,
assumed constant shoulder torque, and assumed constant wrist torque
second phase of the downswing. This allows the equations to be
integrated analytically and reduces the number of dynamic variables in
numerical integration from 4 to 2. For
further information see the paper by Pickering and Vickers, or the
EPAPS document associated with my paper – it can be found quickly if
EPAPS, golf swing. To do the integrations you will need a moderately
integration algorithm. Applications like Mathcad, Maple, and
very good integration routines.
Physics of Golf 2nd
Ed", Springer Verlag, New York, 1994)
M. Pickering and G. T. Vickers, “On The
Double Pendulum Model Of The Golf Swing”, Sport.
Eng., 2, 161-172 (1999)
D R White, "On
the efficiency of the golf swing",
Am. J. Phys. 74,
pp. 1088-1094 (2006)
Last modified - May 14, 2013