Originally Posted by kaiser
So the pendulum action you are referring to is a rotation of the wrist around the axis going through the forarm, and not around the axis perpendicular to that going through the wrist (i.e. from thumb to little finger)?
Because if the latter, the pendulum action would be in the plane of the swing path which is mostly in the horizontal plane, and more so when the player meets the ball higher in the bounce. This means gravity, and hence g, would only play a small role in the pendulum action and g would need to be replaced by a, the accelleration of the racket arm executed by the player.
If the pendulum action you're referring to is a rotation around the forearm axis, as in the modern 'windscreen-wiper' stroke, it would be perpendicular to the plane of the swing path and therefore mostly in the vertical plane. Then gravity does play a role, but not exclusively. When you rotate your forearm to bring the rackethead up during the take-back, you store energy in the forearm by twisting the radius and ulna bones relative to each other. When you then bring arm foreward during the stroke, the radius and ulna are forced back into their original position, releasing this stored energy into a rotation of the wrist and, hence, the racket. This rotational force therefore also affects the pendulum action around the wrist, as well as gravity, creating an angular accelleration on top of g.
Shouldn't this also be accounted for in your formula? How would this affect your conclusions?
This has nothing to do with windshield wiping.
If you look at the slo-mo vids I posted earlier in this thread of Del Potro and Gonzalez hitting forehands, you can see that the racquethead typically drops from about 7 ft high to about 3 feet high before rising again toward the ball. During the portion of the stroke where the racquethead is dropping those 4 feet, the player does not need to exert much force to accelerate the racquet. Rather, it is gravity plus centripetal acceleration that accelerate the racquet (it matters little that the plane of the swing is not vertical, as long as there is a significant vertical component to the motion). In other words, the motion of the racquet still obeys the physics of a mechanical double pendulum even though the plane of the swing is only partly in the vertical plane.
Only after the racquethead reaches the bottom of the stroke does the player need to apply significant force to keep the racquet moving. This is because the racquet naturally accelerates during the downward part of the stroke (which is mainly gravity powered). But on the upward part of the stroke, the player must apply force to counter the deceleration caused by gravity. I believe that your body naturally senses the inflection point at the bottom of the swing when the motion of the pendulum sweep of your arm switches from acceleration to deceleration, triggering you to begin applying pressure on the handle to keep the racquet moving.
As Corners has mentioned earlier, Rod Cross published an article in 2009 in the American Journal of Physics, showing that the swing of a baseball bat can be modeled as a double pendulum, and that the player must actually apply a reverse-direction couple just before the moment of impact, otherwise the bat pivots too fast from the wrist, causing timing errors. In other words, MgR/I for the baseball bat was too high.