A PHYSICIST looks at tennis in a special way. True, it is the same game with the same rules for everyone. But it must also obey the laws of nature.

The advice below is given from the point of view of a scientist trying to tell you how to take advantage of those laws to win more points and, as a player or spectator, to enjoy the game more.

Getting Strung Out

Tighter strings mean less power; looser strings mean more power.

The physics of this is clear and easy to understand. Then why do so many people think just the opposite. They see many of the top players get tremendous power from the rackets, which are strung tightly. Bjorn Borg strings his racket at 78 pounds, so they conclude this is why his balls have great pace on them.

Nothing could be further from the truth. It is because Borg and the other top players can hit the ball so hard that they can afford to string their rackets tightly to gain other advantages; they sacrifice some of their power in doing so.

Why Loose Strings Give More Power

By design, a tennis ball does not store and return energy efficiently. A ball dropped from a height of 100 inches onto a hard surface rebounds only to a height of about 55 inches. That is the official specification for the manufacture of a tennis ball. This means that about 45 percent of the energy that the ball had has been lost.

Strings, on the other hand, are designed to return between 90 and 95 percent of the energy that is fed into them. To give the ball the maximum energy (that is, the highest speed) when it is hit, the strings, not the ball, must store the energy by deflecting. They return almost all the energy they store when they deform, while the ball only returns about half the energy when it is hit and deforms.

If the strings have a lower tension, they will deflect more (that is, store more energy), and the ball will deform less (that is, dissipate less energy). The larger the string-plane deformation, the less the ball will deform. Try dropping the ball on the ground (equivalent to infinitely tight strings) and then from the same height onto a racket which is supported around the head, not by the handle alone. Compare the rebound heights: The ball will bounce back considerably higher in the second case -- to 80 percent of its original height.

Don't Dwell on It

The looser the strings (actually the greater the string plane deformation for a given force), the longer the ball will reside on the strings (the greater dwell time). The dwell time 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.

This last statement may seem contrary to common sense, if you picture a high-speed tennis ball sinking into the strings more deeply than a gently hit ball, and therefore conclude that dwell time should be greater for the ball that is hit hardest. It is nevertheless correct, since the strings get stiffer non-linearly the more they deform; this leads to shorter dwell times for harder hits.

How does dwell time affect your game? The actual time of contact for a normal shot with a normal string tension is about 4 or 5 thousandths of a second. A longer dwell time corresponds to a longer "carry distance" -- the distance the ball and the racket move together while the two are in contact. An excessively long dwell time can lead to loss of control if the racket head changes its angle rlative to the court while the ball is being carried. A longer dwell time also means that the shock of the ball being hit is spread over a longer time; and the magnitude of the force at any given time is therefore reduced.

One of the stock phrases many teaching professionals and tennis books use is: "For better control, keep the ball on the racket as long as possible." Such statements also appear in ads for tennis balls and in the patent literature. It is clear that all one need do is reduce the tension in the strings to ensure that the ball will spend more time on the racket. And yet, as we have seen, loose strings decrease rather than increase a player's ability to control the ball. Clearly, something is wrong with the advice being given.

What teachers should say is, "Swing your racket so that it appears you are maximizing the time that the ball spends on the racket."

Stringing Material

Between 50 and 75 pounds of tension, nylon string requires 12 pounds of additional tension to elongate it an extra 1 percent. Gut string needs only 6 pounds to do the same. What is important is the additional tension needed. When a given force is applied, gut strings will stretch more than nylon strings of the same gauge, provided that each set is strung to the same tension.

Because of the greater elasticity of gut, it will deform more and tend to cup the ball. The extra power gut gives you is a function of how you hit the ball. If you play a soft game, the advantage of gut is reduced. If, on the other hand, you really smash the ball, then you will appreciate the extra power that gut provides. In addition, the greater flexibility of gut may give you more control over the ball, because you will get more linear response from the strings as you vary your swing and the ball speed.

{That is, the rate of stretch per additional unit of force is much more consistent for gut string than for nylon, as the illustration shows. The more that the curve approaches a straight line, the more predictable a feel the string will have for the player.}

Sweet Spots & Sour Vibes

When you hit a shot and it really feels good, you claim that you have hit the "sweet spot." But can this feeling be quantified? Do some rackets have a sweet spot that is sweeter than others? Or is the size of the sweet spot the only relevant consideration?

A technical paper published in 1981 in the American Journal of Physics pointed out that there are actually three sweet spots in a racket, each of which measures a different physical characteristic of the tennis racket.

The three spots are defined as those places where, when you hit the ball, you experience one of three things:

The initial shock to your hand is at a minimum. This is the Center of Percussion (COP).

The uncomfortable vibration that your hand and arm feel is at a minimum. This is the node of the first harmonic (NODE).

The ball rebounds from the strings with maximum speed or power. This is the area with the maximum coefficient of restitution (COR).

In general, these three spots are found at separate locations on the face of the racket. Each spot is a defined region rather than an actual point. Players can determine the location of each of these spots on their own at home or in an ordinary workshop.

Sweet Spot No. 1 (COP)

The COP is defined relative to where you hold the racket. To find it, hold the racket between your index finger and thumb at the handle where you normally put the base of your index finger (about 4 inches from the butt end). Allow the racket to swing as a pendulum with your fingers as the pivot and your hand and arm motionless. Place your fingers so that the racket swings with its face open, as if it were hitting a low ball.

Then measure the time (in seconds) it takes for a complete back-and-forth swing. Better still, allow the racket to swing back and forth from the starting point five times, record the total time elapsed and divide this number by five. This will give you a more accurate value for the time of a single swing (T/swg).

The distance in inches from where you are holding the racket handle to the minimum initial-shock sweet spot is found by multiplying the time per swing squared (T/swg X T/swg) by 9.77:

Location of COP in inches = 9.77 X (T/swg) X (T/swg).

For example, if the time of one complete swing is 1.3 seconds, then the distance from your fingers to Sweet Spot 1 is 16.5 inches. (9.77 X 1.3 X 1.3)

Sweet Spot No. 2 (The NODE)

When a racket hits the ball, it deforms from the impact; then it begins to vibrate or oscillate for a short period of time. It can vibrate in several different ways. If the handle is firmly held, the racket will oscillate at either 1) about 20 to 30 cycles per second (with a motion like a diving board after the diver has jumped) or 2) about 100 to 150 cycles per second -- the "first harmonic" or next higher frequency at which the object will vibrate. (The stiffer the frame, the higher the racket's frequency of vibration for a given mode of oscillation.)

However, there is one impact point on the racket face (the node) where this vibration is not produced. This is Sweet Spot No. 2. The farther from this spot that the ball hits, the larger the amount of oscillation.

To locate it, tape a common index card loosely to the handle of your racket (or use a rubber band) and hold the racket between thumb and index finger about 5 or 6 inches from the butt end.

Hit the racket face at various places with a hand-held ball or butt end of antoher racket. When you hit the strings near the tip or throat, the index card will buzz loudly. When you hit the node, all you will hear is the ping of the strings.

Sweet Spot No. 3 (the COR)

On most rackets, you will get more power when the ball strikes the strings closer to the throat than in the center of the head. This means that as you move the impact point closer to the throat, you will get more ball speed. (A region of even higher power is accessible on a racket in which the shaft is made shorter and the head larger by extending the head toward the handle, as in the "oversized" rackets.)

If the head of the racket is firmly clamped in place, the point of maximum power will be at the center of the strung area because the strings are "softest" there. But if, on the other hand, it is the handle that is firmly clamped (which is a more reasonable way to simulate a hand-held racket), the point of maximum power is closer to the throat. This is so because the racket is flexible and the energy fed into racket deformation cannot be returned to the ball. The closer to the throat the ball hits, the greater is the effective stiffness of the frame and the less energy is lost in racket deformation.

This may seem wrong, since you were just told that loose strings give more power. Why, therefore, shouldn't a more flexible racket give more power? It has to do with the timing. A ball spends about 5 milliseconds (thousandths of a second) in contact with the strings before it is propelled away by the strings snapping back. The racket, however, takes about 15 milliseconds to return to its undistorted position. By that time, the ball is long gone and cannot benefit from any energy the racket absorbed in deformation.

The coefficient of restitution (COR) is defined as the ratio of the rebound speed of the ball to the incident speed of the ball. Maximum power is obtained when the COR is at a maximum. To locate Sweet Spot No. 3 (the area of maximum COR), clamp the racket handle in a vise with the face up. Drop a tennis ball from a height of a foot or more above the strings and measure how high it rebounds. Repeat the process at various points on the racket face. The spot that gives the maximum rebound height is Sweet Spot No. 3.

Finally, remember:

You can't beat the laws of nature. But you can use those laws to beat an opponent.