Applying Physics to Golf

7/29/2019 Applying Physics to Golf

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Introduction

I am a physicist and a keen golfer. A few years ago I was under doctors

orders to stay away from golf for at least 3 months in order to recover from

a shoulder injury. So instead of playing, I trawled through golf books and

papers to figure out what actually happens in the golf swing. After many

months, I had a good pile of paper describing a lot of things about golf, but

nothing simple to tell me how the swing really works. Even the best of the

classic books like Cochrane and Stobbs, Daish, and Jorgensen fail on this

point. Once I was back playing, the problem continued to nag.

The underlying problem with golf-swing physics is the complexity of the

math. The equations explain everything and nothing. Sure it is possible to

solve the equations, produce animations, and carry out numerical

experiments, which is what most researchers have done, including me.

However, the equations are so complex that it is not possible to look at them

and say, Ah ha! I understand.

Eventually I found the simplification that provides the insight, and this

presentation is the result.

The presentation builds the whole of the golf stroke including the swing and

the club-ball collision. The key points of technique identified here are simple

but so counterintuitive that they are hard to believe and execute when youare standing over a ball. The physics explanation helped me to defeat my

intuition. Im in my 50s now and 30 years of bad wiring in the brain is hard

to change, but over a period of two years I have gained 30-50 m distance on

my drives, and improved the accuracy on all of my shots, all with less effort.

Hopefully these pages will help you to do the same.

A careful reading of the presentation will take about an hour, and for most

people its probably necessary to view the animations to really understand

how the swing works. If you dont have an hour, you can try reading thetechnique section by itself, but you may find some of the advice hard to

believe.


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The golf swing is a combination of two physical processes: most obviously,

a collision between the club and ball, and perhaps less obviously, the

unfolding of a simple machine called a double-pendulum. Although simple,

the double pendulum has very complicated behavior. In recent years, it has

received a lot of attention because is a simple example of a systemexhibiting chaotic behavior (which may explain some of my golf shots). The

double pendulum also turns up in a lot of different places because it is an

efficient way of transferring energy from a heavy slow-moving object to a

light fast-moving object:

Baseball, tennis, hockey, and wood chopping are obvious examples. Football (kicking), javelin, and discus are not so obvious examples. Old war machines: Trebuchet (like a catapult), sling, atlatl or spear

thrower (also known as the woomera in Australia).

Builders and shipping cranes (especially on ships where it causestrouble), and the threshing flail.

Outline of the Article

1. The collisiono 2-ball collision, as a model of the club-ball collisiono 3-ball collision, a simple analogy of combined collision and

double pendulum

2. The swingo The double pendulumo Combined collision and double pendulum

3. The good and bad of techniqueo A driven double pendulumo How is energy transferred?o The good the effect of wrist cock (lag)o The bad the effect of wrist torque (using the hands)o Video examples of good golf swingso Video examples of trebuchet swings

4. Golf Technologyo Club-head masso Shaft lengtho Coefficient of restitutiono Shaft mass
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References:

1. Theodore Jorgensen, The Physics of Golf (Springer Verlag, New York,2nd ed, 1994)

2. C. B. Daish, The Physics of Ball Games Parts I and II (EnglishUniversities Press, London, 1972)

3. Alistair Cochran and John Stobbs, Search for the Perfect Swing(Triumph Books, Chicago, 1968)


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Golf Swing Physics

1. The Collision

Guest article by Rod White -- December 2008

This section is in two parts.

The first subsection, on the two-ball collision, provides a simplified

explanation of what happens in the collision between the clubhead and ball.

The second subsection, on the three-ball collision, considers what happenswhen two two-ball collisions are combined. This may seem like a bit of a

detour, but hang in there; it leads to an explanation of why the club head isso much heavier than the ball. More importantly, it captures the essential

features of the compromise required to maximize the overall energy transfer

in the golf stroke.Two-Ball Collision

Let us look at a moving ball colliding with another ball that is initially

stationary; our aim is to get as much kinetic energy as possible into thesecond ball.

In an efficient collision the first ball stops. This is the criterion for 100%

efficiency. Any movement of the first ball after the collision representskinetic energy that has not been transferred to the second ball.

Case 1: small ball collides with large ball

This slide shows an animation of the collision. After the collision, the large

ball moves slowly, and the little ball has recoiled and is moving slowlybackwards. As evident from the movement of the small ball after the

collision, not all of the kinetic energy of the small ball has been transferred

to the large ball. The collision is not 100% efficient.

Additional note: In these animations it is assumed that the collisions areideal, there is no energy lost through compression of the balls (i.e., the


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coefficient of restitution = 1.0)

Math note

To a physicist, this page illustrates perfectly the laws of conservation of

energy and momentum. Let's calculate the energy and momentum for eachball, before and after the collision.

Before the collision:

Kinetic energy: T = M1V12

Momentum: L = M1V1

After the Collision:

Kinetic energy: T = M1U12 + M2U22

Momentum: L = M1U1 + M2U2

Where V is the velocity before collision, U is the velocity after the collision,and M is the mass of the balls.

The conservation laws say the the "before" should be equal to the "after",

for both kinetic energy and momentum. So we now have two equations,Tbefore=Tafter and Lbefore=Lafter. Since we know everything except the after-

velocities U1 and U2 , we should be able to solve the two equations for thetwo unknowns -- and we will below.

Case 2: large ball collides with small ball

This time the animation has the larger ball colliding with the small ball.Again the collision is not 100% efficient, and afterwards, the large ball

continues to move forward: it retains some of the kinetic energy.

Note that the little ball moves faster after the collision than the big ball didbefore the collision. We will come back to this point in a few slides.


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Math note

Above we saw how conservation of energy and momentum give us twoequations in two unknowns. So it is now just algebra to calculate the

velocities after the collision in terms of the initial velocity of ball 1 and the

masses of the two balls. If we solve the equations, we get...

Velocity of first ball after the collision is

U1 = V1

M1 - M2

M1 + M2

Velocity of second ball after the collision is

U2 = V1

2 M1

M1 + M2= V1

2

1 + M2 /M1

Note: You have probably seen the equation for ball speed after collision withthe clubhead, cited by many references, as:

Vball = Vhead

1 + COR

1 + Mball /

Mhead

Let's look at the two-ball collision as a golf shot, with ball 1 as the clubhead

and ball 2 as the golf ball. Our model so far has no energy loss, so the CoR

is 1.0. Given this value of CoR and assignment of masses, the familiarequation for ball speed is exactly the same as the equation for U2, the speedafter collision of ball 2. Now we know where the equation for ball speed

comes from.

Case 3: two equal sized balls in collision

This time the two balls have the same mass, and the colliding ball comes toa dead stop with the collision. The collision is now 100% efficient.


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(M1 + M2)2

Three-Ball Collision

Now, lets look in detail at the three-ball collision. It might look like a bit of a

detour, but we will see shortly that two successive 2-ball collisions illustratesan important compromise that occurs in the golf swing.

Remember, when we ran the animation of the large ball colliding with the

small ball, how the small ball moved faster than the large ball? Although itsnot easy to tell, the small ball now moves faster than before. The

intermediate ball makes the transfer of energy from the large ball to thesmall ball more efficient.

This graphplots the efficiency for the two separate 2-ball collision that occur within the3-ball collision; first M1 with M2, and then M2 with M3. In the graph I have

assumed that the big ball has a mass of 1 kg, and the small ball has a massof 50 g. The mass of the intermediate ball can take any value as indicated

on the horizontal axis of the graph.


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The right hand dotted curve shows the efficiency of the collision between M1and M2, with the mass ofM1 equal to 1 kg. As expected from our

observations with the 2-ball collisions, the energy transfer is most efficientwhen the intermediate mass is also 1 kg.

The left hand dotted curve shows the efficiency of the collision between M2and M3, with the mass ofM3 equal to 50 g. It is most efficient when theintermediate mass is 50 g.

After the first collision, the intermediate mass carries a given percentage ofthe energy, and then this fraction of the energy is reduced by a further

percentage in the second collision. The efficiency of the overall three-ballcollision is therefore found by multiplying the efficiency the two collisions, as

given by the two dotted curves. The result is the solid red curve. Clearly, thebest overall efficiency is obtained when the intermediate mass is in between

the other two. If we read the mass off the graph we can see it is a littlegreater than 200 g.

Math note

The velocity of the third ball is given by:

U3 =

4 M1 M3

(M1 + M2)(M2 + M3)V1

Differentiation with respect to M2, and setting the result to zero gives the

condition for the maximum energy transfer:

In other words, the optimum intermediate mass is the geometric mean ofthe other two.

To improve the efficiency even further we could introduce a fourth ball, or an

infinite sequence of balls, with the masses graded between the large and

small masses. Now the efficiency approaches 100%. Practical examples ofthis type of exponentially graded coupling include the bullwhip, horn

loudspeakers, and graded electromagnetic absorbers used in stealthtechnology.

Golf Swing Physics

2. The swing


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Double-Pendulum Swing

Now we move on the model of the swing itself. Conventionally, physicistsmodel it as a double pendulum: one pendulum tacked onto the end of

another.

In this animation we show a double

pendulum composed of a large blue mass representing the arms of the

golfer, and a smaller red mass representing the clubhead. There is also athin string connecting the clubhead to the hub.

Initially the system rotates at a constant speed, and the string stops the

clubhead from swinging out. No energy is put into the system for this model;the system is rotating to begin with, and continues to rotate under its own

inertia. Since it is held in the same shape by the string, the moment ofinertia stays the same so the speed stays the same. An essential feature of

the animations is that no external forces or torques are applied to thesystem, and the connection between the two arms of the pendulum is simply

a hinge.

Now watch what happens when the string is removed. As the red massswings out, the blue mass slows down, and the red mass accelerates. When

the system is fully extended (near the point where the head would hit thegolf ball) the blue mass comes to a dead stop.

For the physicist, this is the magic moment! The blue mass coming to a dead


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stop means that all of the kinetic energy that was in the blue mass has been

transferred to the red mass (the club head); the swinging process is 100%efficient.

Most importantly, there were no external forces applied to make this happen

the system rearranged itself under its own inertia. If this was a golf swing,the golfer would not have to make any effort to make it happen.

In this example, I chose the value of the red mass so the blue ball would

come to a dead stop. In general this does not happen as we will see withthe next few animations.

The second animation shows thedouble pendulum again, but with a heavy clubhead mass -- heavier than

before and therefore heavier than optimum.

This time, the blue mass slows down more quickly than before, and whenthe system is fully extended the blue mass actually goes backwards (this

tells us that some of the kinetic energy remains in the blue mass).

This animation is similar to the two-ball collision when a small mass collides

with the big mass and recoils.


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The third animation shows the double

pendulum again, but with a light clubhead mass. The distribution of mass iscloser to a real golf swing than in the previous examples. This time, the blue

mass slows down but not as much as before, and when the system is fullyextended, the blue mass is still moving forwards. Again some kinetic energy

remains in the blue mass.

This animation is similar to the two-ball collision when a large mass collideswith the little mass and keeps moving.

Now although the motion of the double pendulum is complicated, we can

simplify the problem by considering the initial motion when the string is inplace, and later when it is fully extended. If we think about these twosnapshots, and ignore the complicated movements as the red ball swings

out, we can see a strong similarity between the two-ball collision and thegolf swing as it unfolds. Once again we see the Goldilocks principle at work.

The red mass representing the club head can be too big, too small or justright. We wont show you the curve just yet, but it is very similar to the

efficiency curve for the 2-ball collision.

The double pendulum is much more complicated than the simple collision, sothe condition for a 100% efficient swing is much more complicated that the

condition for a 100% efficient collision (which was M1 = M2). We give theequation for the optimum clubhead mass in the math note below.

In addition to the mass of the blue ball representing the arms (M1), thecondition depends on the length of the two arms of the pendulum

representing the golfer's arm and the shaft of the club (L1 and L2), and theangle between the arm and the shaft (, which is the wrist cock angle).


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The equation also tells us how to reduce the optimum clubhead mass and

increase the efficiency of the swing:

Lengthen the shaft, and/or Increase the wrist-cock angle.

In fact the whole golf stroke: the unfolding of the swing, and the club-ballcollision looks a lot like a three-ball collision.

The equations of motion for the system are still complicated, but if we

restrict ourselves to thinking only about the energy and momentum at these

two points (the point the string is released, and the point of full extension),the system can be understood relatively easily.

Remember that the energy and momentum are the same throughout theswing and through impact. The momentum must be the same; that'sphysics. The energy is the same because we have not yet considered energy

losses in the collision (that is, the COR is 1.00 for our model).

Math note

Here is the basis of themathematical model I am using. The diagram shows how Ive defined the

various angles, shaft lengths, etc.

We are only considering the instantaneous motion of the system at the time

of two snapshots:


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a. The moment just before the string is released. At this moment,everything is rotating; nothing is flying out from the centre of rotation-- though it will the instant after the string is removed.

b. The moment of impact, calculated to correspond to "complete release"-- a straight line from the golfer's center of rotation down the armsand the club to the ball.

By picking these two snapshots, and having the system move under its owninertia, the system is very much simpler to analyze. Mainly, there are noradial terms to consider; only tangential terms involving angular velocities.

Only two things you really need to note here, both of them at the momentthe string is released:

The downswing angle , which measures how far the arms move, and

The wrist cock angle , which measures the angle between the armsand the shaft of the club.

These are the simplified equations for the swing. (We still have not included

the collision with the ball at this point.) They might still look horrific, butthey are much simpler than the full equations, and they can be solved

algebraically to give a simple expression for the condition for 100%

efficiency.

To read the equations you need to know that terms of the form mass

x length2 measure a quantity called the moment of inertia. Moment ofinertia measures the resistance to rotational forces (torques) in the sameway that mass is a measure of resistance to a translational force. Everybody

knows "F=ma". Well the rotational analogue is "torque = moment ofinertia x angular acceleration".

The Greek symbols with dots over them are angular velocities.

Kinetic energies have theform:

moment of inertia x (angularvelocity)2

Momenta have the form: moment of inertia x angular velocity

The subscripts on the angles, i and f, indicate initial and final velocity.

First, we require the kinetic energy and angular momentum to be the samefor the two snapshots of the swing.


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Note that the angular velocity of the two masses, beta-dot and alpha-dot, isthe same in the first snapshot.

As we did with the equivalent equations for the collision, we can solve these

two equations to calculate the final angular velocities for the two masses.This is how the swing efficiency curve was calculated. However, it is more

interesting to find out the condition for a 100% efficient swing. Ideally, wewant the blue mass to come to a dead stop at impact, and this leads to a

further simplification, as given by the second set of equations.

This pair of equations is much simpler and has one non-trivial solution:

m1L12 + m2R

2 = m2 (L1 + L2)2

The bottom line of this analysis is that the swing is 100% efficient when themoment of inertia of the system in the first snap shot equals the moment of

inertia of the club in the second snap shot.

In words, this last equation reads, the moment of inertia of the initial system

(with the string in place) equals the moment of inertia of the club when fullyextended. This is analogous to the condition that the M1 = M2 in the two-ball collision.

Most importantly, this observation tells us that the energy transfer occursbecause of the changing moment of inertia of the club as it swing out.

We can now calculate the optimum clubhead mass for this passive model bysubstituting the value for R, which is given by the cosine law (from diagram

a above):

R2 = L12 + L2

2 - 2L1L2cos

Which then gives us the required condition for optimum clubhead mass:

m2 =m1 L1


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2[1 + cos()] L2

Ice skater analogy of the golf swing

The most important picture to keep in mind is the transfer of energy withunfolding.

Think of this as the reverse of the ice skater effect. The ice skater initially

has the arms extended and works to move them as fast as possible. Theskater then pulls the arms in close to the body. This causes the body speed

up so that the skater spins fast. It is important to note that the handsactually slow down in this process, but we only watch the body. The body is

spinning faster, true. But, because the hands' radius of rotation is shorterthan before, the velocity of the hands is lower than before.

In the golf swing, the opposite occurs. The golfer initially holds the club close

to the body using wrist cock, and works very hard to build up the kineticenergy in the body and arms. The golfer then allows the club to swing awayfrom the body, so the body and arms slow down and the club speeds up. I

emphasise the term allows, because it can be an entirely passive process;the golfer does not have to make the club swing out it happens naturally.

In summary...

The ice skater:o First builds speed in arms,o then folds the arms close to the body,o to speed up the body

The golfer:o First builds speed in body,o then unfolds the club from the body,o to speed up the club.

Model golf stroke = pendulum + collision

We are now in a position to understand the basic principles underlying thegolf stroke. There are two parts in the golf stroke: the swing and the

collision. This graph is idealised we discuss the non-ideal bits shortly.


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Part 2: the collision

between the club and ball is exactly as we described for the two-ballcollision. The efficiency is represented by the left hand of the two dotted

curves, and the maximum efficiency occurs when the two masses are equal(golf ball mass is close to 46 g).

Part 1: the swing is where energy is transferred from the golfer's arms and

torso to the clubhead. This is represented by the right hand dotted curve.The right hand dotted curve for the swing efficiency is not exactly the same

shape as that for the collision but very close. The exact position of the curve

depends on a variety of factors that we havent considered in detail yet -arm mass, arm length, shaft length, wrist-cock angle, and coefficient of

restitution.

The maximum efficiency in this case for the swing alone requires the clubhead mass to be about 1 kg (according to the complex equation given in the

math notes). As with the 3-ball collision, the overall efficiency is given bythe product of the two separate efficiency curves.

We can now see why the clubhead is so much heavier than the ball. To

maximise the efficiency of the total golf stroke, it must be midway between

the optimum for the swing (1 kg) and the optimum for the collision (46 g).The right hand dotted curve is not exactly the same shape as thecorresponding curve in the three-ball collision but very similar.

Anything we can do to move the swing curve further left improves theefficiency of the golf stroke. The only two factors included in this model that

are within the golfers control are:

The wrist-cock angle (make smaller). That is why many of the longest


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hitters have a very acute lag angle well into the downswing. (We shallsee this in the videos later.)

Shaft length (make longer). But shaft length is limited by the rules,and in many cases further limited by the individual golfer's ability to

control a long club. Long drivers routinely use long shafts, but they too

are limited by their competition rules.

The model we have developed so far suggests that the optimum club headmass is about 150 g. In practice, real club-head masses are a little heavier.

This is because, unlike our model (which was entirely passive), in a real golfswing, the golfer is constantly working to build up energy and momentum.

Work done early in the swing is transferred efficiently, just as our modelsuggests. Work done late in the swing, when the wrists are un-cocked, is

transferred less efficiently. The overall efficiency is therefore less than ourmodel predicts in effect the average wrist cock angle is greater than we

have assumed. There are other factors too:

Energy lost due to the compression of the ball as described by thecoefficient of restitution means that the swing is less efficient, andthe clubhead mass must be heavier to give the maximum ball speed.

Our muscles also produce greater torques when they are movingslowly and this favours a slightly heavier mass.

Finally,the optimum mass changes with the value of m1 thatrepresents the mass of the golfers arms, and our value of 1 kg on a

0.66 m pendulum arm may not be representative of the moment ofinertia of a typical golfer.

Golf Swing Physics

3. Technique


In the previous sections we explained the principle of the golf swing, how

the unfolding of the club from the cocked position causes rotational energyto be transferred from the arms and the body to the club, and that this

unfolding can be passive requiring no effort from the golfer. Now that we

understand what is happening, lets look more closely at a more

realistic model of the golf swing, in the interest of clarifying

technique. In the actual golf swing, the golfer is applying torque,

throughout the swing, to the inner arm of the double pendulum -- by using


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muscles in the torso to turn the shoulders. The improved model is a

similarly-driven double pendulum with a few extra features that allow us to

investigate aspects of technique. It will give us an opportunity to see what

sort of things the golfer can do with the swing to improve his distance -- or

screw things up altogether.

A Driven Double Pendulum

The improved model can vary:

Torque applied to the inner arm of the pendulum, to model the workdone by the golfer via the torso and shoulders

Torque applied to the outer arm of the pendulum, to model work doneby 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

For the 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.


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The animation shows the swing of a moderately good amateur golfer with a

sound golf swing.

During the first part of the downswing, the golfer holds the club in a cocked

position and accelerates the shoulders and torso. Initially some positive wristtorque 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 club from

being pulled inward. The positive torque required to brace the club falls

rapidly as the club accelerates. When the positive bracing torque falls to

zero, the club can be allowed to swing out -- ending the first phase.

The second 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 release. If the golfer holds the club in

the cocked position for a short while longer, this is described as a late

release. If the golfer releases the club early, the club will swing in towards

the neck for a small moment and then swing out. We wont look at the effect

of release timing because to a good approximation release timing has no

effect.

During the second phase the golfer continues to turn his body and arms, but

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 shows the solution for the swing we have just

described.

Important note:

By 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


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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 to50 inches), the natural swing time is much longer, and a swing with anatural release is anatomically impossible. Professional long drivers

must use positive wrist torque (forcing the club out) to complete theswing.

The normal golf swing does not require positive or negative wristtorque during the second phase of the downswing; therefore the hands

are passive, and the golf stroke can be more accurate with fewermuscles involved.

How is the energy transferred?

We have discussed the golf swing in terms of the conservation of energy andmomentum, 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


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the direction of the clubhead midway through the swing this is indicated

approximately by the red arrow. Now look where the hands move at the

same time the blue arrow: in a different direction!

Obviously the hands and clubhead cannot continue to move in differentdirections, they are restrained by the fixed length of the shaft. The diverging

directions 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 pulls

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 almost 7horsepower).

Now well take a look at the factors that affect the effectiveness of the

swing. The two big factors 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.

Wrist Cock Angle


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The figure to the left shows

the clubhead speed versus downswing angle (the angle between the arms

and the body) for three different wrist cock angles.

As expected, increasing the amount of wrist-cock (reduced 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 downswing 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.


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The chart at the right

plots the driving distance versus wrist cock angle, assuming all 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

degrees wrist cock. Note again that the wrist-cock angle is measured

between the arms and the club, so a smaller angle corresponds to greater

wrist cock, or greater lag as it is often called.

Note again the distance is gained with no extra effort from the golfer the

difference is purely one of technique.

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 ball speed 25m

where the ball speed is in meters per second.

Anecdotal confirmation of this point from DaveT: In December 2009, I wasplaying 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 90

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


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30-50 meters back.

Wrist Torque

Now we 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 Ive 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 by itself.

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


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range of handicaps find that only golfers with low single-figure handicaps or

better come close to hitting the ball at the peak clubhead speed. For most

golfers, the club is decelerating through impact.

The 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.

The blue curve assumes that the golfer changes his swing so impactstill occurs at the peak. We shorten or lengthen the swing so that

impact will occur at maximum clubhead speed. This golfer is then onlybitten by (a) above.

The red curve assumes that the golfer simply makes the same lengthswing no matter what the wrist torque. This golfer is then bitten by

both (a) and (b). Negative wrist torque also costs distance because theclubhead speed peaks after impact (i.e., impact is at the black line in

the curve above).

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 swingwith no wrist torque is about 20 m in distance. More typically the beginner

will take the same backswing as a low handicap golfer and lose the distance

indicated by the red curve nearly 40 m!

This is a very tough lesson, yet all of us have experienced the occasion when


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we relax, try not to 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.

Many people have 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 usedrivers with a section of rubber tube or dog chain replacing part of the shaft.

They still hit the golf ball a long way -- in fact, much the same distance as

with a proper shaft. With such a flexible shaft, 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 theshoulders, 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 gunpowder-powered

cannons, demonstrating that "wrist torque" isn't all that important.

See the videos on the next pagefor visual demonstrations that a trebuchet

can sling things a long way with no "wrist torque" at all.

Summary for Technique

Work done by the golfer builds up kinetic energy in the torso, shoulders, and

arms. This is then transferred via tension in the shaft as the club and armsunfold away from the golfers body.

The good: the greater the fold (wrist cock) the more efficient thetransfer of energy from the body to the club.
http://www.tutelman.com/golf/clubs/golfSwingPhysics3b.php?ref=#trebhttp://www.tutelman.com/golf/clubs/golfSwingPhysics3b.php?ref=#trebhttp://www.tutelman.com/golf/clubs/golfSwingPhysics3b.php?ref=#treb


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The bad: the greater the wrist torque (use of the hands) the earlierthe club unfolds and the less energy is transferred to the club.

These two effects, the negative effect of wrist torque and the

positive effect of wrist cock, account for most of the 70 m difference

between the beginner and the scratch golfer.

These effects are also counterintuitive 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 club as 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 wristtorque.

Math note

I have not given the full equations for the numerical model here. As I

indicated in the introduction, they are too complicated to yield any insightsdirectly at least not for me. Also, they have to be solved numerically

because there is no analytic solution. If you want to experiment with theequations, the full version can be found in Appendix 4 of Jorgensens book.

For the analysis here I neglected gravity, assumed constant shoulder torque,and assumed constant wrist torque during the second phase of the

downswing. This allows the equations to be partially integrated analyticallyand reduces the number of dynamic variables in the numerical integration

from 4 to 2. For further information see the paper by Pickering and Vickers,or the supplementary EPAPS document associated with my paper it can be


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found quickly if you Google EPAPS, golf swing. To do the integrations youwill need a moderately good numerical integration algorithm. Applications

like Mathcad, Maple, and Mathematica have very good integration routines.

References:

Theodore Jorgensen, "The Physics of Golf 2nd Ed", Springer Verlag, New

York, 1994)

W. M. Pickering and G. T. Vickers, On The Double Pendulum Model Of TheGolf 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)

Golf Swing Physics

3+. Video Examples


In this section well look at slow motion videos of some professional golfswings.

The real golf swing is more like a triple pendulum with the beam betweenthe neck and the shoulder forming the third arm of the pendulum. While

watching the videos, remember the essence of a good golf swing is thereverse of the ice skater effect the transfer of energy with unfolding. Thetotal power produced by the golfer (about 2.5 kW for a professional) comes

from the biggest muscles in the body the upper legs and torso. Theproblem is how to transfer this power, from the muscles, through the

shoulders, arms, club, and eventually, to the ball.

The backswing

consists of a large shoulder rotation against the hips to pretension themuscles of the torso.

minimises the moment of inertia of the arms and the club by foldingthe left arm against the shoulders and cocking the wrists. In all casesthe down swing begins with the left arm folded almost parallel with the

shoulders, and the club typically cocked to 90 degrees.

The downswing movement


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leads with the large muscles of the legs and torso rotating and buildingmomentum in the hips, with

the whole folded system -- the upper body, shoulders, arms and club -- moving almost as one.

Once the rotation is firmly established, the arms swing out transferring energy from the body to the arms,

then the club swings out, transferring energy to the club.

Tiger Woods

Tiger's swing is the swing closest to the textbook swing of the golfers here.

Things to note:

The long extended backswing to create a large shoulder turn andtension in the muscles of the torso.

The system stays "folded" through the early part of the downswing. Excellent retention of wrist angle well into the downswing. Hands slow noticeably as he approaches impact. This is not

"deceleration" in the golf instructor's [negative] sense; it is caused bythe tension in the shaft exceeding the shoulder torque, and is evidence

of the transfer of energy from the arms to the clubhead.

Hand, wrist, and forearm strength are not a factor here -- except to beable to stay relaxed with the large shaft tension trying to yank the clubfrom his hands. And the same comment applies to the other videos as

well.

Sergio Garcia

Sergio is not as tall as many pro golfers, so to hit the ball as far as he doeshe must have a more efficient swing. The distinguishing feature here is the

large wristcock midway through the down swing. Things to note:

Even more shoulder turn than Tiger. As he starts the downswing, he actually increases his wrist cock angle.

This "early release" of the club means that he's not using sufficientpositive wrist torque to maintain even a 90 angle.

Midway through the swing, the wrist cock is probably well under 60,so that work done during this part of the swing is transferred moreefficiently to the club when the wrists unfold.

That is where his exceptional clubhead speed comes from.


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Freddy Couples

Fred has one of the most admired swings in the game, superficially an easyswing, but hes actually doing a lot of work very early. The important thing

to note here is how Fred starts his downswing. At the top, his whole body is

coiled, including his hips. His first move is a big rotation of the hips.Everything above the hips turns, but only because the hips are turning.

Everything above the hips -- torso, shoulders, arms, and club -- are almostfrozen, and just move with the hips. That is what we mean by "staying

folded". The rest is pure swing -- no hands.

Of course Fred, like Tiger and Sergio, keeps his wrists cocked until very latein the downswing.

Jamie Sadlowski

Jamie is not a household name for most golf fans -- unless they are fans oflong driving. Jamie, at only 165 pounds, has been the world champion of

this sport for the past two years (2008 and 2009) with record drivesexceeding 400 yards. How does he get the clubhead speed to accomplish

this against much bigger, more-muscled competitors?

Jamie is an incredible athlete with extreme flexibility, and none of us couldhope to reproduce this swing. But because the downswing is so long, it

shows very clearly the ideal of staying folded.

As a long driver, all of his actions exceed those of PGA tour golfers: an

incredible shoulder turn and even more incredible wrist cock. He also has theadvantage of a long shaft on the driver (see next section on technology).

Many people have trouble believing that you do not need to use wrist torque

to have an effective golf swing. But there are some good examples fromoutside golf illustrating this point.

Perhaps the best example is the trebuchet, a medieval siege machine used

to fling rocks a long way. YouTube has many videos of trebuchets. Let's seewhat we can learn from them:

Let's start with a good overview of a working trebuchet from DGseward.

The raised weight represents the torque applied through theshoulders,

the long wooden beam represents the golfers arm,


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and the rope sling represents the shaft of the golf club.Think about it; if the club had a perfectly flexible shaft -- like the rope slingof 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 wasreplaced by gunpowder-powered cannons, demonstrating that "wrist torque"

isn't all that important.

Let's turn DGseward's video upside-down to show the similarity to the golfswing. It should look more like a real golf swing this way. The armsaccelerate from the turning "shoulders" (the counterweight), and the club

(the rope sling) lags until it is whipped outward by the tension in the rope.

Trebuchet competitions remain something of a sport today. Games include

distance and target competitions, with "punkin chunkin" being an Americanchampionship throwing large pumpkins for maximum distance. Here's a

pretty basic "punkin chunkin" trebuchet.

Finally, if you want to experiment with a trebuchet yourself, here's a sciencekit you could build.

Golf Swing Physics

4. Technology


Golf has always been a game of technology, with a long history of

improvements to golf balls and clubs. However it is easy to be sceptical ofthe impact technology has had surely coaching and swings have improved

too.

On this page we look at some of the effect of some of the recenttechnological changes on golf driving distances.


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The pink curve on the graph shows the mean driving distance for PGA tour

professionals for every year between 1980 and 2008. The two curves eitherside show the average plus and minus 1 standard deviation (a measure of

the variation amongst the golfers). The dots show the average drivingdistance for the longest driver of the year.

The graphs show a consistent increase of about 40m with much of that

occurring between 1995 and 2005. How much of that gain is due totechnology?

Clubhead Mass

As we should expect

from the simple model we developed earlier, the driving distances are moreor less independent of club head mass. As the club-head mass increases, theswing efficiency increases (more kinetic energy transferred the clubhead),


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and the collision efficiency falls.

The chart shows that there is a maximum between 170 and 180 g, but it is avery broad peak. That is, you have to get a long way from the peak before it

makes a lot of difference. Any club head mass between 130 g and 240 g

yields the same distance within about 4 m.

The optimal mass not only gives the greatest distance, it also means thatthe same club will do for almost all golfers (which is good news for the club

manufacturers).

The optimum mass indicated in the figure is still a little short of the 195 g to205 g typical of modern drivers. There are a couple of reasons for this.

Firstly, muscles work better at slower speeds, favouring a slightly heavierclub and slower swing. Secondly, the driven pendulum model used to

produce the graph assumes a constant shoulder torque. In reality it will takea short while for the shoulder torque to build so that more of the work is

done at the end of the swing where it is not transferred efficiently, and thelower efficiency results in a heavier optimum mass. Finally, the value I chose

for the effective mass of the golfers arms is a nominal value that had beenused previously by other researchers, a 20% increase in arm mass is

sufficient to account for the difference.

Shaft Length

The effective shaft length usedin the graph is defined by the point where the arms and the club hinge, and

can be a couple of inches less than the full length of the club.

We have already mentioned the increase in efficiency that accompanies

increased shaft length. In fact, increasing the shaft length increases driving


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distances through two effects:

1. An increased downswing angle which means that more work is doneby the golfer, and

2. The improved efficiency of the swing a greater fraction of the workdone by the golfer is transferred to the club head.

The total work done by the golfer is downswing angle x shoulder torque, andclubs with a longer shaft take a longer time to swing out, and results in agreater down-swing angle, and hence greater work done by the golfer.However, a natural release swing (with no wrist torque) becomes

anatomically impossible with long shafts. For the average male golfer, most

of the modern standard length drivers (45-46) are a little long, althoughthis depends a lot on the golfer's grip and the location of the centre of thehinge at the wrists.

A number of professional golfers have observed that their driving accuracyimproves with shorter shafts. This is probably due to a reduction in the use

of the hands.

Coefficient of Restitution

By far the largest contribution

to increased driving distances has come about from the improved Coefficientof Restitution (CoR) of the club-ball collision. At the time of the introduction

of the solid core ball, the CoR of most balls was in the low 0.7 range. Asrubbers improved, the CoR improved markedly. During this time,manufacturers had to compromise between a soft skin (better spin control)

and distance (with a harder, longer wearing skin). Nowadays it is possible toachieve the maximum CoR permitted under the rules while still having a soft


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skin.

The rules have two tests on balls, one based on ball velocity after a welldefined impact with a solid steel object moving at a prescribed speed. This

limits the CoR of the ball (in collision with solid steel) to about 0.79. The

second limits the driving distance with a specified driving machine.

With the development of metal drivers, it became possible to build large thinclub faces with a resonant frequency low enough that they would flex and

store energy during impact. This "trampoline effect" reduces thecompressive forces on the ball, and therefore reduces the energy dissipated

in the collision. As a consequence of the trampoline effect, the CoR for thecombined club-ball collision has risen to about 0.83 -- where it is now limited

by rule.

This means, since the 1970s the improved CoR has resulted in a 20mincrease in driving distances for the average golfer.

Shaft Material

Another of the major technologicaladvances has been the change from stainless steel shafts to graphite shafts..

The effect is due to an improvement in the swing efficiency.

The club-ball collision time for a typical drive is below 0.0005 seconds.During this time, the kinetic energy must be carried from all parts of the

clubhead to the ball. You can envisage this happening as a compressionwave in the metal clubhead carrying the energy forward. The same is true

for the first few inches of the shaft. However, most of the kinetic energy inthe shaft cannot be carried into the head and to the ball because the speed

of the wave, carrying the energy from the shaft, is too slow. So the greatmajority of the shaft's mass is not involved in momentum transfer to the


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ball. This means that all of the work done by the golfer to put kinetic energyinto the shaft is wasted -- it does not participate in energy transfer to the

ball -- and reducing the mass of the shaft will make the stroke moreefficient.

Reducing the shaft mass from about 130 g (steel shafts) to about 60 g inmodern lightweight shafts has resulted in an increased driving distance ofabout 10m.

Note: contrary to common claims, the flex of a shaft has no effect on theclub head speed, and there is little energy stored in the flex of the shaft that

can be recovered at impact this is all advertising BS. The shaft flex doeshowever affect the loft of the club at impact. The huge (50 kg for a

professional) tension in the shaft pulls at the heel of the clubhead. However,the centre of mass of the club head is a few inches away from the heel, near

the centre of the head, and with a high swing speed the centre of masstends to line up with the shaft. The shaft therefore bends towards the target

and toe down a few centimeters. This effect provides club manufacturerswith a way of expanding the range of lofts on their clubs. A flexible shaft will

enhance loft by as much as 5 degrees, while a stiff shaft only about 2degrees. Tip-stiff shafts also reduce this effect, and tip-flexible shafts

emphasize it. Golfers with a slow swing speed require drivers with a high loft(14-15 degrees say) to achieve the greatest distance, whereas golfers with a

high swing speed achieve optimum distances with a lower loft (as low as 9

degrees).

Summary

So there you have it! The great distances produced by good golfers using

modern equipment are due to both technique and technology. The importantfactors we have discussed are:

Technique:

Use the transfer of momentum and rotational energy from your turning bodyand arms to the clubhead and then to the ball. Don't try to impose your will

on the club, but rather allow the energy to flow to it. (This is not some new-age proverb. As we saw, it is classical physics.) Particular techniques to useinclude:

Start the downswing with the club (and even the arms) "folded" closeto the body.


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Keep them folded there as late as you can into the downswing. Thatmeans the first part of the downswing is just rotation, with nothingfrom the hips up moving relative to any other part.

Do not apply wrist torque to swing the club out or forward into theball. You may think you are increasing the clubhead speed with "hand

action", but the physics shows that it does exactly the opposite.

Focusing on the mechanical aspects of technique, as we describe it above, isoften unhelpful when you are playing. Our brains work better if we can

establish an emotional connection to the correct swing. When yourepracticing at the range, practice by first swinging the club about your

shoulders as though you are going to throw the club dont use your hands:

remember how it feels, both the rhythm and the feel in your muscles. Nowwhen you stand over the practice ball, remember the rhythm and feelings as

you throw the club through the impact zone. When you hit a good shot,pause and enjoy it, and remember the feeling. Learn that feeling so you can

recall it during your pre-shot routine on the course.

Technology:

Clubhead speed is maximized by using the lightest, longest shaft youcan control. But keep in mind the modern driver shafts are too long forthe average golfer to swing without use of the hands, which means a

tendency to lose control under mental pressure. Like some pros, youmay find that backing off on the length and making sure of a

consistent center-face impact is a better strategy for both average

distance and control. Modern drivers with the larger heads and flexible face have a higher

Coefficient of Restitution than older drivers, and are a little more

tolerant of off center hits. Update your driver if you have not alreadydone so. If you already have a 460 cc driver, there is probably little to

be gained.

Dont be tempted by drivers with a low loft. The optimum loft formodern drivers with a low centre of mass and low spin rate, is a higher

than for the older drivers. Few golfers need a loft below 10.5 degreesthese days. Also, as your swing improves with less use of the hands,

the less the club will be ahead of the hands at impact, and the lowerthe trajectory.

Don't worry excessively about clubhead weight. Choose it so you swingthe club comfortably and consistently. Any commercially

Applying Physics to Golf


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Now let's start to apply some of the physics we just learned. We'll look at

what happens during the golf swing, during impact, and during the ball'sflight. Nowhere near everything, of course. We will focus on some issues

which are important for clubfitting, especially:

Where does the power for the swing come from? The answer willprobably surprise you.

What happens during impact, and how does it turn clubhead designand path into launch conditions? This is the bottom line, turning a

swing into a golf shot.

What determines the ball's flight?How do the launch conditions (ballspeed, launch angle, and spin) translate into distance and direction.

Once we have these fundamentals under our belts, let's look criticallyat a few mythsthat have perpetuated themselves in golf lore.

What powers the swing?

Many golfers, especially in the United States, grew up with sports in which a

ball is thrown or hit. Baseball, American football, basketball, or hockey are

usually mastered -- well, at least played sort of competently -- before golf isattempted. And those sports generally involve powering the ball with the

hands and arms. Yes, the body can play a part in adding to the power. Butthat part is generally getting the whole mass of the body moving in the

direction you want the ball to travel.

The golf swing is different, not just in degree but in principle. Golfers who

grew up hitting things with a bat or a hockey stick have developed swinghabits that are counterproductive in golf. (Well, the very best at those sportsmay have inorporated the important elements of a golf swing. But the

average Sunday athlete has not.)

Most of the power in a golf swing comes from centrifugal force, generated bythe muscles that rotate the body through the swing. Before explaining it

further, let's look at the physics of the golf swing. After we see the forces at

work in a good golf swing, we'll go back and see what sort of bad habitsmost golfers carry over from their other sports. Finally, since you're probably

going to be skeptical about this -- it really is counterintuitive, and you shouldbe skeptical -- we'll review how we know this to be true.

The double pendulum

When an engineer sets about analyzing a real-world system -- like a golfswing -- he creates a physical "model" of the system. This is a set of

elements that are simple enough to yield to calculations, yet complex
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enough to represent what is actually going on. Finding the right model -- the

right balance between simplicity and complexity -- is the first and often thehardest step in engineering analysis.

The simplest model that makes any sense at all for

the golf swing is a double pendulum. The twomembers of the pendulum are:

The golfer's shoulders and arms, taken as asingle rigid unit. That's the green triangle in

the diagram. We'll call that "the triangle" inthe discussion that follows.

The golf club, also taken as a single rigid unit.The triangle is hinged to the golfer's body (the tan

elements in the diagram) so it can turn. Similarly,the golf club is hinged to the other end of the

triangle.

This is a very simple model, having only two

moving elements hinged together. To see justhow simple, let's re-draw it the way an engineer

would: as a collection of free, hinged bodies. Nowwe can see why the model is a double pendulum;

it is a black pendulum (the club) hanging fromthe end of a green pendulum (representing thetriangle). While the diagram looks different fromthe golfer above, it works exactly the same when

it comes to physics.

Given the simplicity of the model, it's pretty amazing how close it can get tothe actual measured performance of a golfer's swing. True, there are a lot of

nuances of the swing that it doesn't capture. But experience has shown it is

rich enough to explain where the clubhead speed comes from in a goodswing.


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Let's look at the next question about using the

model. We have two hinges, and we can applya torque at each of those hinges. Those two

torques -- plus gravity -- are the only forces in

this model that will cause the golfer to swingthe club.

So the engineering model has to say what kind

of torque:

The body applies to the shoulders to turnthe triangle.

The hands and wrists apply to the club touncock it and bring it to impact.

It turns out that the torque the body applies tothe triangle is considerable, but a good swing

applies almost no torque to the grip of the clubs by the hands. Yet morethan half of the clubhead speed comes from the club turning about the

hands at the bottom of the swing -- much more than could be explainedsimply by shoulder turn. What is creating that very strong rotation of the

club about the hands, if the hands are not being used to supply a "hit" force?

The answer is centrifugal force.

Remember that a body in motionwants to keep moving in a

straight line. But the golfer ispulling the club around in a

circle. According to Newton, the

club wants to fly outward fromthe circle; the force that is trying

to pull it out straight with thearms is centrifugal force. That

centrifugal force is generated by

pulling the club in a circle aroundthe shoulder hinge, and the forcewants to pull the club straightout along a radius from thathinge.

How big is that centrifugal force?

Let's look again at the formula:
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m v2F = ------

rThe mass m is a property of the golf club, and the radius r is a combination

of the extended arms and the wrist-cock angle. The more acute the wrist-

cock angle, the closer the club is to the shoulder hinge -- and thus thesmaller the radius. As for velocity v, it increases as the body torqueaccelerates the triangle.

So, what does the golfer have to do to get maximum centrifugal force inorder to get maximum clubhead speed? His job is to "hold the lag" -- keep

the club cocked at a right angle to the arms -- until fairly late in thedownswing. This keeps from releasing the club until v is nearly as large as

it's going to get, which allows a large F to accelerate the club outward anddownward just before impact. This, and not torque applied by the hands, is

the way to reliable high clubhead speed.

(It is worth noting there is criticism of centrifugal force as the mechanism ofthe golf swing. More on thathere.)

If you spent years swinging a baseball bat before you started golf, this

probably defies your understanding of power. A baseball slugger has strongforearms and wrists, the better to whip the bat through the ball. Yes, body

rotation is also important, but the hands are active, while for this model of

the golf swing the hands are passive.

On top of that, you never hear this from the pros. Instead, you hear stuffabout clearing the hips, or keeping your right elbow "tucked" instead of

"flying". If this is true, why don't the pros teach it?

So you're probably skeptical that this is how the golf swing works. And youshould be! But it really does work this way. First, let me address why thepros don't teach it; then I'll spend some time on how we know it's true.

OK, why don't the pros teach it?

1. They don't know it! Very few teaching pros or TV commentators havea clue about physics. Terms like "centrifugal force" and "moment ofinertia" are buzzwords they throw around without a clue what they

really mean. Please don't get me started on this; it's a pet peeve and Icould go on all day.

2. Actually, there are a few pros who do understand, but they are indeedfew and far between -- and often skeptical.Jack Nicklaus writesabout
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an incident in 1972, when an expert in golf physics told him about this.

He wasn't at all sure he believed it...

"But his theory seems to explain a shot I hit at the par-3 fifteenth inthe second round at Firestone. The choice of club lay between a two-

iron and a three-iron, and I decided to go with an easy two-iron.Coming into the ball I was deliberately 'soft' with my hands. I've never

hit a better two-iron in my life! The ball finished over the green.

"Maybe this explains what happens on those good drives where I have

a 'soft' feeling in my hands through the ball... My hands merely wentalong for the ride."

3. In a sense, they do teach it -- without knowing or believing it. The twoexamples above are not contradictory to the way the swing works:

o "Clear the hips" creates body rotation with the large muscles,which causes rotation of "the triangle". Anything that increasestorque on the triangle will contribute to power.

o Tucking the right elbow has little to do with power. That move'spurpose is to control the swing plane. When done right, itassures that the ball will go in the direction of the target -- and

none of the analysis above deals with accuracy at all, just power.

Now let's look at the reasoning why we should believe it works this way.

Jorgensen's study

Theodore Jorgensenset about to find a physical model that would match the

behavior of a golfer with a good, "classic" swing. Here is how he went aboutit:

He found a golfer with the sort of swing he needed. He outfitted the golfer with reflective dots on his joints, as well as on

his golf club's grip, shaft, and head.

He took a sequence of strobe pictures of the swing, with the reflectivedots indicating exactly where all the important parts of body and clubwere at every moment. At this point, he had a completely

instrumented swing, and could compute velocities and accelerations of

club parts, body parts, wrist cock angles, etc.

He started his mathematical modeling with a simple double pendulum,and fiddled with the torques until the model gave the same swing asthe golfer did.
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He couldn't do it with the simple doublependulum, so he added complexity alittle at a time until he had an exact

match between the mathematical modeland the golfer's swing.

So Jorgensen's model isn't quite a simple

double pendulum. The figure shows thechanges he had to make to the simple model in

order to get it to behave exactly like the realgolfer. He has to insert a right-angle "stop" so

that the wrist-cock never exceeds 90. And he

had to put a little "sway" into the golfer -- asmall forward motion of the shoulder hinge

during the downswing.

But the important change he did not have tomake was to add any wrist torque to release

the club at the bottom of the swing. That is accomplished completely bycentrifugal force. In fact, once he had a mathematical model that behaved

like the golf swing, he ran some "what if" analyses to see whetherapplication of wrist torque could add to power. He found that there is a

critical time about 70-100 milliseconds before impact (where the arms are60 back from vertical) where torque changes from hurting clubhead speed

to helping it. That is, any uncocking wrist torque before the critical time willreduce clubhead speed at impact. You can indeed increase clubhead speed a

bit by applying wrist torque, but only if you can do it for just the last 70milliseconds before impact, and not before. It takes a very well-coordinatedathlete to get away with this.

Interestingly, Jorgensen found that that the same critical time works the

other way as well. If you use negative torque (that is, use strength in thewrist to prevent uncocking) early in the swing and then release it 100

milliseconds before impact, you will increase the clubhead speed. In fact,you'll get as much increase in clubhead speed as that well-coordinated

athlete would have gotten by a late application of positive torque. And it's

much easier to hold off release than to apply a release-aiding torque atexactly the right time.

So Jorgensen's study confirms the notion that power in a golf swing --clubhead speed -- is a product of centrifugal force and not wrist torque. He

adds a lot of detail, but nothing that denies that basic truth.


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Muscle energy

If classical scientific-method physics (Jorgensen's approach) doesn't satisfyyou, how about biology and physiology?

Coming into impact, a golf club's kinetic energy is based on its mass andspeed. It gets there from zero kinetic energy during the time of the

downswing, less than half a second. This implies that the muscles have toput out a certain amount of power for half a second. Physiologists know how

much power a muscle can provide for a short burst (say, half a second).

When this fairly simple calculation is cranked through, the answer is over 30

pounds of muscle mass needed to impart that energy to the golf club. This ismuscle that is engaged in generating motion, and does not include muscle

used to stabilize the body in the golf swing posture. The 30-pound numberhas come up consistently in quite a few separate studies aimed at this

question.

There isn't anywhere near that much muscle in the forearms, hands, andwrists, so they can't be the major driving force of the swing. You need the

big muscles -- the legs, thighs, torso, and shoulders -- to create that muchpower. That verifies that the clubhead's energy comes from body rotation,

not hand torque. But it doesn't unambiguously point to centrifugal force asthe enabler.

But we should be able to compute the clubhead speed that would result if we

only used body rotation and not centrifugal force. Without any velocity atimpact from uncocking the wrist, just from body rotation, we get only abouttwo thirds the clubhead speed that a good swing actually accomplishes. So

we need centrifugal force because:

We know the bulk of the power comes from body rotation. We know that body rotation without wrist-uncock velocity gives a third

less clubhead speed. In order for body rotation to generate wrist-uncock velocity, we need

centrifugal force -- because the small muscles in the hands andforearms can't generate that much power.

Trebuchet

Still skeptical? Don't believe physics or biology? How about history...

A few years ago (probably 2004), I was watching a show about Siege


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Engines on The History Channel, and had a "Eureka" moment. They were

talking about the Trebuchet, a rock-hurling device that was invented about1200AD. It replaced the catapult over only a few decades, because it had

more range for a smaller and lighter device. (Still big and heavy to be sure,but definitely more efficient than what came before it.)

My Eureka was because, watching it,

I saw an upside-down golf swing.The principle of a double pendulumdriven by centrifugal force was rightthere, and history has proven it very

effective. For a description of how a

trebuchet works, see the page andthe animation I clipped from TheTrebuchet Store. (They sell

trebuchet kits and the like, in case

you find this stuff interesting for itsown sake, not just what it teaches

about the golf swing.) In short, theinner arm of the pendulum (corresponding to the triangle) is a rigid, pivoting

structure, but the outer arm (corresponding to the golf club) is literally astring. You couldn't apply "wrist torque" to it if you wanted to -- it must

operate by centrifugal force.

Now one of the most interesting thing about this design is that it has neverbeen significantly improved upon! It has been around for eight hundred

years, and it is still the most efficient catapult known. Of course, catapultsare no longer used for sieges; cannons and gunpowder took over a few

hundred years after the trebuchet's introduction. But:

For those few hundred years, the trebuchet remained king of the siegeengines.

Even today, there are catapult-engineering contests; they call them"punkin chunkin'" and obviously they hurl pumpkins -- and bowlingballs and other large objects up to and including major kitchen

appliances. But the design that still dominates this "sport" -- even inour engineering-knowledgeable age -- is the trebuchet.

Since the equations of motion for the trebuchet are basically the same asthe zero-wrist-torque golf swing, we can rest assured that the centrifugally-

driven golf swing is very effective indeed.

Hitters and swingers
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I'd like to take this opportunity to state very specifically what I mean later in

these notes by the terms hitter and swinger. Most clubfitters and manyinstructors make this distinction, but it tends to be intuitive and imprecise. I

believe that:

A swinger is a golfer who depends exclusively on centrifugal force forclubhead speed, and adds no wrist torque during the downswing

except that needed to hold a 90 wrist cock.

A hitter is a golfer who depends to some extent on torque applied tothe club's grip via the hands and the wrists.

Of course, there are few pure swingers and no pure hitters. But, comparing

two golfers, we now have a way to say which one is more of a hitter and

which more of a swinger. And, in fact, we can tell from this whether a golfer

isprimarilya hitter or a swinger.

Estimating Slice and Hook

Dave Tutelman - November 5, 2008

This work was instigated by an email conversation with James Smith. James

is a golf coach and clubfitter in Texas (Craftsmith Enterprises), and wanted

to know if there was a rule of thumb to estimate the amount of curve on a

deliberate fade or draw. The rule in this article may or may not be useful forteaching or course management -- or it may not. But it is physically correct

and may be useful to estimate errors due to misfit clubs.

We know a few things

about what happens

when a ball is struck with

the clubface not square to

the movement path of the

clubhead (see the figure).

1. The ball takes off ina direction betweenthe clubface

direction and theclubhead path.

2. The ball's direction is closer to the clubface direction than the clubhead
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path, in a proportion usually between 80:20 and 90:10.

3. The ball has a sidespin that increases proportionally to the ballclubhead speed and roughly proportionally to the difference angle.

4. This spin produces a draw or a fade -- or, if the spin is high enough, ahook or a slice.

Relating the difference angle to the amount of curvature (slice or hook) is a

complex aerodynamics problem. Frank Schmidberger and I have developed

a program (TrajectoWare Drive, which is available for download at no

charge), that traces the ball's entire trajectory, including the amount of hook

and slice. But you are not going to have a computer with you on the course,

and that approach would not be legal anyway. So is there some rule of

thumb that we can use?

This was James' challenge to me. After we went back and forth a bit to

better define the problem, I was able to come up with a pretty good

approximation. The way I did it was to do a large number of runs on

TrajectoWare Drive, then look for a simple equation that fits the data. The

equation was not based on any knowledge of physics -- just any simple

equation that was close enough to the data to give useful results. In other

words, if I could fit a simple linear equation to within 5% of the actua

Applying Physics to Golf

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