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Mechanics of a Tennis Racket KAU Department of Physics and Electrical engineering Examiner: Igor Buchberger and Jürgen Fuchs Mechanics of a Tennis Racket By: Tom Linder Introduction Tennis is a worldwide sport that everyone knows of. It has been around for more than a century (1873). Despite of its age, there hasn’t been much of a development of the equipment, or to be more precise, the rackets. There is a lack of examination of the mechanics behind the racket. Initially the rackets were smaller and had a smaller surface that could hit the ball. Prince (a tennis racket brand) launched a new racket with a larger head, which is the standard today. The research in this paper will focus on the mechanics of the rackets, and see by how manipulating variables changes the properties. It is based on the work made by H. Brody. Center of Percussion Imagine a racket that is hanging freely in the air, and a tennis ball with momentum ΔP that strikes the racket. If it hits the center of mass the racket will get a momentum ΔP/M, where M is the mass of the racket. There will be no rotation in this case. If the ball would hit the racket at a distance b from the center of mass, both translation and rotation would occur. The angular momentum must be conserved. (1) (2) Let’s say that the racket translates to the right. If the ball hit the racket above the center of mass, a clockwise rotation would occur. At an instant of time, providing that the racket is sufficiently long, one point would still be at rest because that point rotates at the same speed as the racket translates, but at opposite directions (See fig 1). Fig. 1. The rod in the picture represents a simplified version of a racket. It can be easier visualized that when a ball strikes the rod, one point of the rod can get the same velocity as the translation of the rod, but at opposite directions. To write it mathematically: (3) where “a” is the distance from the c.o.m to the point at rest. Thus: (4) This can be rewritten as: (5) One can see that “a” would still be at rest no matter how hard the ball hit. Point A is defined as the point which is distance “a” from the center of mass. It is also called the center of percussion conjugate point. Equation (5) implies that he/she would experience no force if the racket was held at point A. The radius of gyration can then be defined as: (6) Point “B” is called the center of percussion, which is the distance “b” from the center of mass. The point on the handle where it might be good for a player to be holding a racket depends on where a player usually hits the ball, or in other words, where the center of percussion usually is. Therefore, it is of some interest to find the distance a+b. This distance can be found through an experiment in which the racket is used as a pendulum that oscillates around point B, as shown in fig 2. Fig. 2. Again, the rod is a simpler shape then a tennis racket, and used only to illustrate how the experiment can be done. It swings around point of percussion B. If the racket is dropped from any angle, the only force in action that will cause it to swing is the force of gravity. ( ) (7) is the moment of inertia with respect to point B. It can easily be calculated through Steiners theorem. (8) If one considers small oscillations, equation (7) can be simplified to: (9) Solving this differential equation gives: ( ( )) (10) By using Eq. (6) this can be rewritten as: (11) ( ) which can be rewritten a final time by substitution using Eq. (6) again as: ( ( ) ( ) (12) Now an expression for a+b can be obtained. (13) So a good placement for the hand would be a+b distance from the center of percussion, since there is no force on the racket to overcome when returning the ball. Values of the center of percussion can be seen in table 1. The ball should hit as close to the center of the head as possible, because the player then has a larger margin of error to get a decent hit if they were to miss the center of percussion. If a racket is held at the handle where it usually is held, the center of percussion is 5-6 cm below the center of the head. So there are two choices. Either the player holds the racket closer to the head, which would shift the center of percussion, or make alterations to the racket. The alterations that can be made would be either to increase the weight of the tip of the racket, or by increasing the size of the head. Racket Davis TAD Spaulding Smasher Prince Length of racket [cm] 68,9 67,0 68,6 Period of oscillation 1,26 1,28 1,30 around head [s] a+b [cm] 39,4 40,7 42 Distance between center of head and 54,6 54,0 51,1 butt end [cm] Distance between center of percussion 15,2 13,3 9,1 conjugate point and butt end [cm] Table 1. Shows some actual values of the center of percussion that H. Brody measure . The center of percussion is not necessarily the best place for the ball to hit. While the net force felt by the players hand is zero, a rotational force is still experienced. This force tend to bend open the fingers that are holding the racket. Also twisting motion could be a problem, which is especially common with beginner players. This can be analyzed further by measurements of the moment of inertia. Imagine that the racket is hanging by a wire that is connected to the pivot point that wishes to be examined, i.e. the handle. It is then put in a pendulum motion. The moment of Inertia can be calculated by using (14) where M is the mass, g is the gravity constant, d is the distance from the center of mass and T is the period, which can easily be measured with a stopwatch. Values of the polar moment of inertia can be seen in table 2. The polar moment influences the twisting motion. Racket Mass Maximum head Polar moment of [kg] width [m] inertia [kg ] Head Master 0,365 0,238 0,00117 Prince Pro 110 0,357 0,275 0,00155 Weed Aluminum 0,354 0,316 0,00224 Table 2. Show polar moment of inertia for different rackets. The values were calculated by H. Brod To decrease the twisting motion, the most efficient way is to make the racket wider, since the inertia is proportional to the radius squared. Another way is to make the racket heavier. Companies have done both to increase the moment of inertia. But larger moment of inertia is not always better. It depends on the play style. Bounce of the strings When the ball hits the racket off center, the strings will bounce the ball back but with a certain angle. If it hits at the center, and the spin of the ball and gravity are ignored, the incident angle should be the same as the angle of deflection. If a ball with velocity perpendicular to the surface of the head hits the racket, but off center, the ball will be deflected towards the center. The angle is proportional to (y/L)*(z/L), where y is the distance the ball goes into the strings of the racket, and z is the length from the center, and L is the total length of the strings when it is not elongated. fig. 3. The top part of the figure illustrates the initial state of the string, and the lower after it is hit by a ball that has incoming velocity perpendicular to the strings. “z” is the displacement from the center and “y” is the penetration depth of the ball. If the spring constant of the strings and the momentum of the ball are known, the greatest angle that the ball can deflect with can be calculated. An assumption that the strings vibrates as a simple harmonic is made. ∫ (15) Only do the integral over half a period, otherwise the answer would be zero, since the total momentum is conserved. (16) Since we assumed simple harmonic force, the following is also true: ( ) (17) The amplitude A can also be obtained if the integral is calculated over half a period: ∫ ( ) (18) Simply solve for A to get the following result: ( ) ( ) (19) “k” is a linear function that depends on T/L and the number of strings that is in contact with the ball. T is the string tension. A relation for the angle of error can be obtained: ( ) √ (20) From this the observation can be made that the angular error decreases when L increases, that is if the head is larger. Also it decreases when the tension is increased, but not as drastically as L. Another interesting observation is that the faster the ball travels, the larger the error is. In other words, it is harder to accurately return fast balls compared to slow ones. Not just because there is less time to react, but also if you hit the ball wrong the penalty will be even more severe. Tension of the strings and COR A factor that plays a part in the rebound speed of the ball is the tension of the strings and a variable called coefficient of restitution (COR).
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