'Linear Golf' - some basic notions

A golf swing is a complex event. To get more readily an intuitive feeling for the basic forces at work it is better to strip the problem at hand to its most simple expression at least as a starting point. So we will look at two point masses connected with a slender rod as shown in Fig1. A force F is applied to mass m1 of which the motion is constrained without friction to a straight line AA'.

Any single force F acting at a point P can always be replaced by an equal and similarly directed force acting through any point O and a couple with moment τ =  F r sinΨ, as shown in Fig2. Visually it should be clear from Fig2 that the linear force F exerted on point mass m1 is hence equivalent to a force F acting through the center of mass O and a couple with a moment τ = F r sinΨ, as shown in Fig3.

For the laymen it should be of considerable help to really catch on with this simple basic scientific notion. It will definitely help for getting a feeling for what goes on in a real golf swing. Hence, there are really two basic motions to consider. A linear motion of the center of mass caused by F acting through O and a rotational motion caused by the torque having moment τ, around O, both caused by the single linear force F acting on m1 at P in Fig1,  see Figs3,3a,3b.

The force F is not altered by the motion of the ensemble but the moment τ is a function of the orientation of the rod with respect to the line AA'. Hence it varies continuously and can be either positive or negative. Since we are primarily concerned with the motion of m3 this has therefore a direct bearing on the possible resultant value for the velocity of m3. The magnitude of the moment is also determined by the mass distribution, hence the position r of the center of mass O re. to P.    

Now let's do some thought experiments. Starting position for the segment perpendicular to the line AA' and a constant force acting, along the line AA', on mass m1. The results for the ensuing motion is show in Fig4a. The resulting force acting on the distal mass m3 is shown vectorially as a black arrow. Notice the sign of the moment it causes. First positive and then negative when the segment drops below the line AA'. This indicates a braking torque.
    

Figs 4b shows the torque acting on the segment. Fig4c the respective kinetic energies of the two point masses m1 and m3 and the connecting rod mass m2. Fig 4d show the work done by the force on the moving segment. At t=0.2 sec the force is reduce to zero and hence from thereon the kinetic energy and the work done remain constant.  Fig 4e shows the kinetic energy of the distal mass m3 which sort of functions as the clubhead in our thought experiment.

The next experiment is to vary the duration of time that the force F is applied to the mass m1. Instead of 0.2 sec the force F is only applied for only half that time hence only 0.1 sec. The results are shown in Figs 5a,b,c,d,e. Notice the interesting fact that the moment acting on the segment remains now positive and hence  no braking as above but instead there is positive angular acceleration for the segment, through 'impact'.  
  

The interesting part of this thought experiment is to  compare the results.
    
-1- From Fig4c and Fig5c it is seen that there is substantial transfer of kinetic energy form mass 1 to mass 3 in experiment 2 and virtually none in experiment 1. Even if there are no linked segments, working with only one rigid segment, the kinetic energy re-distributes itself very much like in a kinetic chain.

-2- From Fig 4d and Fig 5d it is seen that we have an effort of only 92 Joule in experiment 2 as 232 Joule in experiment 1, hence 2.5 times less work done, or 250 % less effort employed. Comparing the maximum velocities, notice that there is however, surprisingly, little decrease in maximum velocity, only from 36.1 to 28.4 m/s, hence only a reduction of about 28 % .

-3-  In experiment 2, at t ≥ 0.1 sec,  the force F drops to zero, hence no further work is done. Consequentially the kinetic energy remains constant from thereon. Nevertheless it is now the positive inertial torque acting on m3 which takes charge and continues to accelerate the mass m3.

Fig 6 shows more systematically the relation between inputted effort (work done) and maximum velocity of mass 3. Notice that one can almost reduce the effort by 2 before there is any significant loss in maximum velocity for m3. It shows the useful contribution of the inertial torque if we give it a chance to act.

Conclusions

It is amusing to see that even stripping the golf swing down to a simple stick being pulled along that still interesting and rather useful notions scan be learned about the golf swing.

A release torque does not need centrifugal force or positive torquing with the wrists - simply pulling/pushing the golf club along can produce a release moment.

Again we find that an efficient release is one where one let go before impact, hence action followed by letting go, a balancing act between yin and yang.