less power the player will get. Why do loose strings give more power than tighter
strings? Tennis balls do not store and return energy efficiently. For example, imagine
throwing a tennis ball from a height of 100 inches onto a hard floor. The tennis ball only
rebounds to a height of about 55 inches, a loss of about 45 percent of the initial energy
of the ball. Strings, however, are designed to return 92.5 percent of the energy that is fed
to them (Watts 84). To give the ball the maximum energy, the strings must store the
energy by deflecting. If the strings have a lower tension, they will deflect more and the
ball will deform less. So why not string all rackets loosely? By reducing the tension too
much, the speed of the ball will be inadequate and the strings will wear out too fast from
excessive rubbing. Moreover, by stringing a racket loosely, control must be sacrificed.
Reasons for loss of control because of loose stringing includes: making the speed of the
ball more dependent upon the pace of the opponent?s shot, changing the angle at which
the ball leaves the racket, and increasing the dwell time of the ball on the strings. This
allows the racket to twist or turn more while the ball is still in contact. The looser the
strings, the longer the ball will reside on the strings. The dwell time of the ball on the
strings should increase as the inverse of the square root of the tension. In addition, the
dwell time of the ball on the strings decreases the harder the ball is hit, because the
strings become effectively stiffer the more they are forced to deform (Brody 12).
When a player hits a shot and feels great, he or she has hit the sweet spot.
According to the American Journal of Physics, there are three sweet spots of a racket
(Bloom 4). Sweet spot number one is the initial shock to a players hand. To some this is
known as finding the node of the first harmonic (See figure 3). Sweet spot number two is
when that uncomfortable vibration that many players feel is also a minimum. Sweet spot
number three is when the ball rebounds from the strings with maximum speed and
power. When a racket is struck by a ball, the racket recoils to conserve momentum. If
the ball hits the racket at its center of mass, the racket recoil is pure translation and there
would be no rotation of the racket. Instead, if the ball hits in the center of the strung
area, the racket both translates and rotates. If the ball is not hit exactly at a sweet spot,
however, there will be an initial net force on the player?s hand. If a player hits the ball
closer to his or her hand than this sweet spot, the initial force will pouch on the palm of
his or her hand.
The oscillation amplitude of the racket depends on the point of impact for the
occurring vibrations. When a racket hits the ball, the racket deforms due to the impact
and then begins to oscillate for tenths of seconds (See attachment 4 &5). Since most
tennis players, like myself are not able to hit the ball at the second sweet spot every time,
manufacturers have attempted to reduce the vibrations with special vibration-damping
materials. Some say these small devices that fit on the strings are purely psychological.
Research, however, shows that the feedback from the racket is dramatically affected.
These small devices ?damp the vibrations of the strings that oscillate up to 500 to 600
cycles per second? (Randall). In doing this, they change the sound of the interaction
between the ball and the racket.
When a tennis player hits the ball off-center, the racket tends to twist and the shot
is more than likely to go out of bounds. The property of the racket to resist this change in
twisting is known as the roll moment of inertia. The quantity m(r squared) represents the
rotational inertia of the particle and is called its moment of inertia. It is calculated as the
mass of the object times the distance of that mass from the axis squared. If the moment
of inertia is made larger, the racket is less likely to twist and will gain stability along the
long axis (Brody 214) (See figure 2). The moment of inertia can be increased by adding
masses along the outside edge of the head. The Wilson?s Hammer System was created to
do just this. The theory behind the Hammer (another racket) is ?that it is head heavy,
providing more power due to an increased moment of inertia? (Brody 214). In addition
to the head?s weight, the moment can be increased by increasing head-width. Because
inertia depends on the factor m(r squared), increasing the width also increases the polar
moment significantly more than increasing the mass. The polar movement is the
property of an object to resist twisting. Increasing the head on the racket reduces the
likelihood that the racket will twist in the player?s hand after an off center hit.
Through the understanding of the motion of the ball, characteristics of swings,
and general anatomy of the racket, one can see how physics influences even the most
basic aspects of tennis. Even though people participating in the game of tennis are not
completely aware of the physics in each shot, they are still able to enjoy the game. A
person who is seriously interested in the game of tennis, however, can figure out a lot by
studying the various laws of physics and how they determine the course of the sport of
tennis. That was my father?s intention when challenging me to research the Radical Tour
260. I did eventually obtain the racket. Through research? No, the coach called and
suggested the racket to my parents. Researching racket science and characteristics of the
sport of tennis has brought much humor to my parents. Was it fate that determined that I
would one day be researching the physics of tennis, or is this all a big dangerous
conspiracy between my professors, coaches, and parents?
Works Cited
Barnaby, John M. Racket Work- the Key to Tennis, Allyn and Bacon. Boston, MA. 1969.
Bloom, Phil. ?Finding Sweet Spots.? Phil Bloom.
(14 March 1998).
Brody, Howard. ?The Moment of Inertia of a Tennis Racket? Physics Today. April,
1985; (p. 213-215).
Brody, Howard. Tennis Science for Tennis Players, University of Pennsylvania Press.
Philadelphia, PA. 1987.
Cantin, Eugene. Topspin to Better Tennis, World Publications. Mountain View, CA.
1977.
Randall, James. ?The Tennis Racket,? Newton at the Bat: the Science in Sports. ed.
Schier and Allman. 1984.
Watts and Bahilli. Keeping Your Eye on the Ball, University of Pennsylvania Press.
Philadelphia, PA. 1994.