How String-to-Ball Friction Affects Spin
Crawford Lindsey, Tennis Warehouse, San Luis Obispo, CA, 93401
February 21, 2012
This research looks at the role of friction between the ball and the strings in spin generation. Several qualitative experiments were performed to see friction in action between string and ball and to observe the effects on spin.
Also, experimental procedures were developed to measure the friction between the string and ball for the ball's motion both parallel (cross strings) and perpendicular (main strings) to the length of the string.
The conclusion is that strings with maximum perpendicular string-to-ball friction and minimum string-to-string friction will produce the most spin.
String manufacturers have always marketed tennis strings according to their ability to grip the ball via sticky coatings, rough/grabby surfaces, or "biting" shapes. The underlying theory is that greater friction between the string and the ball will cause more spin.
This was universally accepted as true until it was demonstrated that for all impacts less than 50 degrees away from perpendicular, the strings will bite the ball to the maximum extent possible, no matter the texture, shape, or material of the string. This is important because when biting occurs, friction ceases. This result assumes that the stringbed is laterally rigid — i.e., the strings do not move sideways, or, if they do, they do not snap back into position. This was the situation in the pre-polyester string days. As we will see below, polyester changed everything.
Sliding, Biting, Rolling (no lateral string movement). The biting process is as follows: at impact the ball slides across the strings with backspin and friction slows and rotates the ball. At the ball's contact point with the strings, both its linear motion and direction of spin are the same. Friction reverses the rotation direction so that the linear motion is forward and the rotational motion is backward at the contact point. When friction causes the linear and rotational speeds relative to the strings to become equal and opposite, the contact point is then at rest, and thus, there is no longer any motion-resisting frictional force. If all strings were to produce this same result, then they would all produce the same spin.
Sliding, Biting, Rolling (with lateral string movement). But this conclusion only applies if the main strings do not slide sideways across the cross strings. If sliding occurs, which is a dominant effect of using polyester, then additional spin will be imparted by the sideways snap-back of the string. It used to be assumed that the main string movement was undesirable because the strings did not snap back and would get stuck out of position. This was a waste of elastic energy, as well as an annoyance to rearrange strings between points. But recent research has shown that main strings do move and snap back - how far, how fast, and with how much energy depending on the string's stiffness and the friction between the mains and crosses. A string that has a low coefficient of friction between strings will commence, sustain, and reverse its sideways slide more easily, optimizing the timing of the snap-back and losing less energy in the process. In this phase of spin generation, low inter-string friction is key.
But, string-to-ball friction is important here also. If there is greater string-ball friction, then the ball can laterally move the main string farther and faster, giving it more energy and time with which to snap-back. And here too, string-ball friction is important, because the string that grabs onto the ball with the greatest force will most efficiently increase the torque generated by the snap-back.
Measuring String-to-Ball Friction
When the ball moves across the string-bed it moves parallel to the cross strings and perpendicular to the mains. Thus, two experiments were devised to measure the friction in each direction. In each experiment, we will determine the coefficient of friction (COF) of each string.
1. Friction Parallel to the String. The equation for friction is F = μN, where F is the friction force, N is the normal force pushing the two surfaces together and acting perpendicular to those surfaces, and μ is the coefficient of friction (COF), which is the constant of proportionality between the friction force and the normal force. μ is the "stickiness" factor. It will depend on the molecular attraction between surfaces, the surface typographies, and the relative stiffness/hardness of the surfaces. These factors determine how readily two surfaces will slide across each other. μ will vary depending on the experimental setup and input parameters, but the relationship between the various materials tested should remain the same independent of setup and parameters.
To measure the string-to-ball COF parallel to the string, a string incline was created by hanging a 20 lb weight from two parallel strings spaced about 30 mm apart. The incline was set to 57 degrees. This steep angle was chosen to make sure that the ball would slide all the way down the incline. This is a necessary condition of the experiment. Gravity continuously accelerates the ball faster than friction can slow and spin it (especially because the only force pushing the string and ball together is the weight of the ball), so biting and rolling never occurs, and the ball slides all the way down the incline. Therefore friction acts throughout the trip down the incline so we can then use beginning and ending velocities and spins to determine the effects of friction in-between. At lower angles, depending on the COF, the ball might begin to roll for portions of its trip. For example, if the COF parallel to the string were 0.3, rolling would occur even at 40 degrees.
Fig.1 shows the geometry and some of the math used to arrive at the string-to-ball COF parallel to the string. Geometry and calculations used to obtain the string-to-ball COF parallel to the string.
In summary, the ball impacts, slides, bites and friction ceases. As the ball slides, it also pushes the main string in the direction of its motion. It will push the string farther and store more energy the more friction there is between the ball and the string and the less friction there is between the strings.
Upon snapback, the string begins to slide against the ball in the opposite direction, reactivating the friction force. At this point, the greater the friction between ball and string, the more the ball will be slowed parallel to the the string-bed, the spin will be increased, and the angle of rebound will be increased.
That being the case, the greatest spin should be generated from strings with high string-to-ball friction and low string-to-string friction. Do such strings exist? Yes. It is interesting that there is no predictable correlation between the string surface condition and the COF. This is true for both string-to-ball and for string-to-string friction. But it is true that the top performers in each category are shaped, textured or coated.
Thus, it would seem that the ultimate spin string (i.e., maximum spin generation) involves maximizing string-to-ball friction while minimizing inter-string friction. The data above demonstrates that this is indeed possible.