Vertical Jump

Ground Forces, Linear and Angular Momentum

On occasion one sees reference being made to a particular golf swing as using the ground forces efficiently. Sometimes it is carried one step further and allusions are made as if some magic ground force can be multiplied or energy can be sucked up from the earth. However nothing really magic about it all.  In an effort to show some light on ground reaction forces/torques let's analyze a simple vertical jump. The analysis has to be done slowly and in small steps since some like lcdv are easily overfed.


The rather particular feature of our interaction with mother earth is that it is generated in a rather curious round about manner. First the golfer generates some internal force/torque between body parts. This results in an external force/torque acting on the earth. The earth responds with an equal and opposite inertia reaction force/torque acting back on the golfer. This inertia reaction force/torque however then becomes the external force/torque acting on the golfer allowing him to swing his club with vigor.


Many scientists treat inertia reaction forces as being no force at all, vaguely disposing of it all by describing them as some side effect of inertia or referring to them sinply as being a fictitious force. Well, they will be very surprised to find out that it is precisely their fictitious or pseudo force/torque which is solely responsible for making it possible to swing their golf club. It seems as if many never learned to look beyond the usual simple text book spring forces and simply disposed of all inertia reaction forces/torques simply as being fictitious.  

In Fig1 we have a very simple model of a vertical jump in which m1 and m2 represent the upper and lower legs, whereas m3 represents the remaining of the body. The point mass m3 is restrained to have no lateral motion, hence moves only vertically. Fig2 shows the starting position for the vertical jump. If a torque of sufficient magnitude is applied at the knee joint, m3 will rise vertically and eventually the jumper will get temporarily free of the earth.

Fig3 shows the angular momentum of the three masses during the jump. Since the point mass m3 moves vertically along the y-axis it does not acquire any angular momentum relative to the origin. The two leg segments acquire some angular momentum as shown. Fig 4 shows the vertical linear momentum of the three segments during the jump. The two leg segments acquire some but the point mass m3 acquires plenty.  


biomechanic, you have moved seemingly to my side of the fence relative to conservation of angular momentum, but you still keep thinking that it is still perhaps different for linear momentum. You will notice however from Fig4 that exactly the same argument applies both for linear and angular momentum. There is nothing at the start, it is continuously created as motion is generated by forces/torques. Hence it is simply erroneous to believe that there is conservation of either linear or angular momentum for either a jump or a golf swing.

Figs 5 and 6 show respectively the horizontal and the vertical ground reaction force, whereas Fig7 gives a vector representation of the ground reaction force.  Notice from Fig 6 that the applied internal torque at the knee joint results in an external inertia vertical ground reaction force a bit higher that the static weight of the three masses, as indicated by the dotted horizontal line, and hence sufficient for a jump to occur.


From Figs 2, 6 and 7 it can be seen that the peak vertical force occurs a bit before maximum extension is reached. Close to maximum extension very rapid changes occur in both the horizontal and vertical ground reaction force. The jump starts with a vertical force of about 1000 N and just prior to maximum extension we have almost zero vertical force and maximum horizontal shear force. This is more readily visualized looking at Fig7.

Conclusion

A similar analysis could be done for a golfer. The main difference, besides being more complex, is that one has to deal more with torques instead of forces. There is nothing mysterious about ground forces/torques. They are brough into being by the golfer himself by generating internal forces/torques using his muscles. Hence it is the golfer which is solely at cause. The earth responds, following faithfully Newton's third law, by creating equal and opposing inertia ground reaction torque/forces.  There is no magic involved, no force to be found in the earth and somehow mysteriously multiplied. It is all just simply hard work by the golfer. Nothing comes for free.

mandrin