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The Complete Idiot's Guide to Choosing a Tennis Racquet

Twelve Objective Racquet Performance Criteria

What Causes Tennis Elbow?

Is a Lightweight Racquet a Good Idea?

Should Your Racquet Be Head-Heavy or Head-Light?

Are Big Head Racquets Better?

Are Extra-Long Racquets Better?

Does a Two-Handed Shot Have Any Advantages?

The Effect of String Damping Gadgets

Benchmark Conditions for Meaningful Racquet Comparisons

The Complete Idiot's Guide to Choosing a Tennis Racquet

by Wilmot McCutchen (www.RacquetResearch.com)

An idiot wouldn't be reading this. Instead, he would swing a racquet a few times in the pro shop, and say: "Wow, it's so light! I'll take it!" After a few weeks of violent banging, he would drop out of the game with tennis elbow.

Tennis elbow should be your main concern. Light, stiff, head-heavy racquets are bad for tennis elbow, so avoid them. Simple physics (which I will not bore you with, here) is clear on this, and so are the big hitters on the men's tour.

First, the bottom line: heavy and head-light is best. Best for performance, best for avoiding injury.

Pete Sampras uses a 14 oz. racquet that has an even balance, Andre Agassi uses a 13.2 oz. racquet that is 5/8 inch (5 points) head-light, and Mark Philippoussis uses a 13.5 oz. racquet that is 3/4 inch (6 points) head-light. These guys, from their outstanding performance, obviously know something about what works in top echelon tennis. What they use is no heavier than the old wood racquets, and even children used to be able to swing them.

Most of the best sellers today, however, are about 4 ounces lighter. Why won't the racquet manufacturers offer the same racquet that these pros use? If it's not the same, why mislead people when you know -- or any reasonable person should know -- that it is deceptive to have the pro out there with the same paint job as the light racquet? Exhibit "A": Andre Agassi plays with what is purportedly a Head Ti Radical, but his racquet weighs 2.5 ounces more.

If you are inclined to buy a granny stick, consider this: if you were in a car accident, which would you rather be driving, a compact or a truck? We all know that the light car will get crushed. The collision of a racquet and a ball is the same thing: a heavy racquet will keep going on impact, crushing the ball more for better pace and spin.

Light racquet partisans argue that because you can swing the light racquet faster, it will hit harder than a heavy racquet. OK, granted that if you have the time and energy to execute a long violent stroke, you can swing the light racquet faster and get greater head velocity on impact. Three problems with that: (1) a violent stroke is harder to control; (2) when you are stretching for a shot, you don't have time to execute a long stroke, so velocity will be small and because racquet weight is small also, your shot will be weak; and (3) the light, fast racquet will slow down a lot on impact, stressing the arm. Momentum (mass times velocity) and not force (mass times acceleration) or energy (1/2 mass times velocity squared) is what counts in a collision. Oops, sorry -- a little scary physics there, but the point is crucial for understanding the principle.

What you want is a racquet that will give you the most ball speed for the least effort (Power), on a shot that goes in (Control), and which will not stress your elbow or shoulder (Elbow Safety and Shoulder Safety). What you don't want is to put in a lot of effort on a wild shot that wrecks your arm.

But what if you put most of the mass in the head, making the racquet head-heavy? Wouldn't you then have a light racquet that hits hard? The light, head-heavy racquet will have a high swingweight, which is good for pace and spin. Swingweight is the inertia (resistance to change in motion) of the racquet as it rotates, and what this means in practice is that it's harder to whip, but once you get the racquet rotating (e.g. on the wrist snap of the serve) it will want to keep rotating when it meets the ball and will crush through, mashing the ball against the strings for better spin and pace. That's the advantage of the Hammer and the extra-longs. But in combination with light weight, there are these drawbacks on closer scrutiny: (1) a light and head-heavy racquet is bad for the elbow and shoulder, for technical reasons explained elsewhere; (2) it feels heavy and sluggish to position for volleys and returns; (3) the power comes from your effort, not the racquet, and you have to work a lot harder to get a certain ball speed than with a heavy and head-light racquet. Tactical tip: stretch the Hammer player wide with heat to the forehand -- so they don't have time to execute a long stroke -- and watch them slap it in the net.

Although the trend for years has been in the wrong direction, toward light and head-heavy racquets, there are some excellent oldies still available. But as time goes on, they get discontinued. So do like Pete Sampras (who uses the legendary St. Vincent ProStaff 6.0 85, long out of production): don't buy just one, but stock up when they go on closeout. And remember to restring often, even before you break a string, because strings quickly lose their bounce.

© 1999 Wilmot McCutchen, All Rights Reserved.

Twelve Objective Racquet Performance Criteria

An Introduction to Scientific Evaluation Concepts

Here's an interesting question: why don't the touring pros use the same racquets that their sponsors sell to the public? The pros heavily customize to add mass and swingweight. Why is such a fix necessary?

It has been estimated that half of all players over 30 suffer from tennis elbow. So it's important to have good information because the consequences of a bad racquet choice are worse than the waste of $300. But how can a player make an informed choice? Racquet ads are not very informative, and often deliberately misleading. Playtests depend on the personal opinion of the testers, and like all subjective tests are suspect for obvious reasons.

What's needed is a set of numbers, some objective performance criteria -- quantifiable and meaningful terms in place of hype and subjective playtests. There are twelve quantifiable performance criteria:

1. Sweet spot (center of percussion) location relative to the racquet tip, given a certain axis of rotation at the hand in the stroke (see formula for sweet spot location). The distance of the center of percussion from the racquet tip correlates closely with the impulse from impact, so the closer to the tip, the smoother the followthrough.

2. Moment, a measure of how heavy the racquet feels to hold up parallel to the ground (not merely the weight of the racquet, but this weight multiplied by its lever arm). Low Moment is good for easy positioning on volleys and returns.

3. Torque resulting from impact, this torque being a twist of the hand about the axis of rotation of the racquet in the stroke, which is 7 cm from the handle end on the forehand (First Benchmark Condition), and 5 cm on the serve (Second Benchmark Condition. This is not the screwdriver twist of the handle (Torsion, or Longitudinal Torque), but the bending back of the racquet. (See formula for Torque, see derivation of formula for Torque);

4. Torsion, or Longitudinal Torque, is the screwdriver twist about the longitudinal axis (i.e. the centerline, the imaginary line running through the center of the handle and the strings) of the racquet, this twist resulting from impacts on the centerline. This criterion is the cross product of Moment and Torque, and is not independently evaluated.

5. Impulse Reaction, a push (positive) or pull (negative) on the axis of rotation (the hand) resulting from impact (see formula for Impulse Reaction, also see derivation of formula for Impulse Reaction). Impacts above the center of percussion (Sweet Spot) result in a pull on the hand; below is a push.

6. Shock from impact, the energy load on the racquet which results in vibration and determines Shoulder Crunch and Elbow Crunch, (see formula for Shock, see derivation of formula for Shock). Shock is the change in the racquet's kinetic energy on impact.

7. Work, the player effort necessary to achieve a given ball speed by the stroke (see formula for Work, see derivation of formula for Work). Work measures the energy efficiency of the racquet, so high Work is bad.

8. Shoulder Pull from the stroke, the centripetal force at the shoulder to counter the centrifugal force of the racquet when it is swung (see formula for Shoulder Pull, see derivation of formula for Shoulder Pull).

9. Shoulder Crunch due to impact, the excess centripetal force acting to contract the shoulder muscles suddenly when the centrifugal force of the racquet drops on impact (see derivation of formula for Shoulder Crunch).

10. Elbow Crunch due to impact, the excess centripetal force acting to contract the elbow muscles suddenly when the centrifugal force of the racquet drops on impact (see derivation of formula for Elbow Crunch).

11. Impact Impulse is the change in the racquet's momentum on impact. It should be low, so that the followthrough is smooth. (See derivation of formula for Impact Impulse).

12. Polar Moment is the rotational inertia (resistance to twisting about the axis running up the handle and through the head). It should be high to provide stability on off-center impacts. Also, the handle size should be large to help resist this twisting.

1. The Sweet Spot is known to cognoscenti as the "center of percussion." This is a point (not an area) where one of the resultant forces from impact (Impulse Reaction) is reduced to zero, which is good for accuracy. Impacts above the center of percussion will produce a yank on the hand toward the net, causing shots to be weak and sail long; impacts below the center of percussion will cause a push on the hand, which in turn causes the racquet to jerk forward during impact and produces a faster shot, perhaps faster than intended.

Generally speaking, the higher the sweet spot on the racquet face (i.e. the longer the distance (q) from the hand to the sweet spot) the better. However, because racquets are of different lengths, it is misleading to compare q values directly, because q is the distance of the center of percussion from the axis of rotation, and for a long racquet a high q may still mean a low sweet spot. So what counts is the distance of the sweet spot from the tip. The formula for finding the center of percussion on any racquet is

q = I / Mr

[q is the distance in centimeters from the axis of rotation to the center of percussion; M is racquet mass in kilograms; r is the distance in centimeters from the axis of rotation to the mass center, or balance point, of the racquet; and I is racquet swingweight about the axis of rotation (also known as moment of inertia or rotational inertia or Second Moment) in the units measured on the Babolat Racquet Diagnostic Center (kg.cm2). The axis of rotation is 7 cm from the handle end on the forehand, 5 cm on the serve, and the Babolat RDC measures swingweight about an axis 10 cm from the handle end, so the RDC swingweight must be converted using the Parallel Axis Theorem].

There has been some confusion in the use of the term "sweet spot," with some calling it a large area on the racquet face instead of a point along the racquet's length. The center of percussion -- which is the real sweet spot -- is not an area but a point. Another alias for the center of percussion is "center of oscillation." The sweet area on the string bed is where the racquet's bounce is maximized.

The sweet spot is interesting to know, but it does not figure prominently in the racquet rankings. It is not a force or an energy load resulting from the stroke. It turns out that the closer the sweet spot is to the tip, the smoother the followthrough on the stroke.

2. Moment is the turning force twisting the racquet head down when you hold the racquet parallel to the ground. Moment is an important performance criterion, and is measured in units of torque (twisting force), not weight, because leverage, as well as weight, counts. Moment is a twist downward as the mass center (balance point) of the racquet is drawn by gravity, and it's key how far away this mass center is from the hand. High Moment is bad because it makes the racquet head heavy to hold up and sluggish to position.

Moment should be especially important for juniors and ladies. A light racquet having a balance point far from the hand may have a larger Moment than a heavy racquet with a head-light balance, so merely knowing the weight of a racquet is not enough, and may be misleading.

It is deplorable that ignorant consumers are being enticed to buy dangerous racquets by misleading sales ploys like inviting them to pick up the racquet by the wrong end to see how light it is. This "pick up appeal" pitch is causing tennis to lose players and popularity at an alarming rate. What counts is Moment, not weight. Moment is the racquet's weight times its lever arm, which is the distance to the balance point from the axis of rotation (this axis is at the middle of the hand, 7 cm from the handle end). Weight in the metric system is the mass of the racquet, in kilograms, times the acceleration due to gravity (9.81 meters/s2), and the distance to the balance point is in meters. The unit of measure of torque is the Newton.meter (1 Newton.meter = 0.7375 foot pounds = the same force you would feel holding a 3.6 ounce weight at the end of a meter stick). The lower the Moment, the better. Moment figures range from 0.648 to 0.967 Newton.meters.

Moment is key for two reasons: (1) a racquet with a high Moment is bad because it is hard to hold up and to position for volleys and returns, especially for juniors and ladies; and (2) Moment multiplied by Torque gives the Longitudinal Torque, or Torsion, which is the screwdriver twist about the racquet's handle centerline resulting from impacts, even impacts on the centerline. High Longitudinal Torque is bad for tennis elbow. See the discussion below on tennis elbow. See Moment rankings for 276 racquets.

Torque and Impulse Reaction are the two resultant forces from an "eccentric impact," such as the impact of a racquet with a ball. This is called an "eccentric impact" because the two mass centers do not move along the same line. As we mentioned above, when you hit the Sweet Spot, Impulse Reaction is zero. But there is still some Torque.

3. Torque, which is a resultant twisting force about the axis of rotation (this axis being perpendicular to the ground and on groundstrokes), tends to twist the racquet back on impact with the ball. This winds up a catapult in the wrist, which flings the racquet forward after impact. The stronger this catapulting force, the worse the whipsawing stress cycle resulting from the stroke, and thus the worse for tennis elbow -- so high Torque is bad. See the Torque rankings for 276 racquets.

Some loss of energy could be expected from the conversion of Torque into the subsequent forward catapulting force, due to absorption of Torque in the bending of the racquet frame and stretching of the muscles, so it will be difficult to quantify the catapult effect. Note, however, that a stiff racquet will not absorb as much of this bending force, and therefore a stiff racquet is a risk factor for tennis elbow.

The pros prefer the slim, more flexible racquets, which absorb Torque in frame bending and thereby reduce the catapulting force that flings the racquet forward after impact. The "widebody revolution" (of the late 80's) never caught on among the pros, for good reason. Stiff racquets may be good for power, but they are bad for tennis elbow.

Note that Torque is a different twisting force from Moment, although both are measured in the same torque unit (Newton.meters or, in the English system, foot-pounds). The difference is in the axis of the twist. Moment is about an axis of rotation parallel to the ground, while Torque is about the axis determined by the path of the ball and the attitude of the racquet face (for ground strokes, an axis perpendicular to the ground). For all racquets, in both benchmark conditions, we assume an impact on the centerline with the racquet face perpendicular to the ground.

4. Longitudinal Torque (Torsion) is the twisting force around an axis running up the handle. As we assume a centerline hit in our calculations under both benchmark conditions, the twist from an off-centerline hit is not evaluated. See the discussion below on whether weighting at 9 and 3 o'clock on the head is good. Longitudinal Torque is simply the cross product (vector product) of Moment and Torque. That means you multiply Torque by Moment and then multiply by the sine of the angle between these two vectors. For groundstrokes (assuming the axis of rotation is perpendicular to the ground and the racquet is parallel to the ground), the angle is 90 degrees so the sine is 1. This cross product produces a twist in the racquet handle -- even for a dead-center hit. For the serve, Longitudinal Torque is disregarded because Moment is practically zero when the racquet is pointing straight up.

Both high Moment and high Torque contribute to high Longitudinal Torque. For a right-handed forehand, Longitudinal Torque would be a twist in the clockwise direction. This twist winds up the racquet to release in a sudden handle twist in the opposite direction (counterclockwise) once the ball leaves. The magnitude of this second twist depends on racquet stiffness (stiff is bad).

5. Impulse Reaction is another resultant force from impact, along with Torque. Impulse Reaction is a translational force (straight ahead push or pull) on the axis of rotation (at the hand), measured in Newtons (1 Newton = 0.2248 pounds, or 3.6 ounces, of force). Impulse Reaction is the pull or push that a player must exert to counter the resultant pulling or pushing on the axis of rotation to keep the shot on target, so it is actually equal and opposite to the resultant translational force from impact. For impacts above the Sweet Spot (center of percussion), Impulse Reaction is a pull from the player to counter the yank on the hand (negative value). For impacts below the center of percussion, Impulse Reaction is a push against the hand (positive value). Right on the center of percussion (sweet spot), there is no Impulse Reaction at all and the shot is smooth and accurate. Our directional convention is that positive is toward the net, so a pull is a negative Impulse Reaction. Close to zero is best, and positive is better than negative because positive Impulse Reaction adds more speed to the racquet during the impact, while negative Impulse Reaction tends to bring the head under the ball, causing sailing by undesired slice. In the Rankings under Impulse Reaction, the following adjustment was made for purposes of rankings: negative Impulse Reactions were given their full value as a difference from zero, but positive values were halved because of the beneficial power effect offsetting the loss in accuracy, then changed to negative values and sorted by difference from zero. If you don't accept that method, the raw evaluation data are there for you to apply a different ranking method.

6. Shock loading of the racquet and arm results from a sudden change in the racquet's kinetic energy on impact. In other words, how much the racquet slows down when it slams into the ball. Shock is measured in joules (the same metric unit as work, heat and energy). Although the term "shock" has no generally accepted definition in engineering, for our purposes we will call Shock the difference between the initial and final kinetic energy of the racquet. See note.

Before impact, you put energy into the racquet to get it up to speed for the collision, and during the impact you put in a little more energy to aim the shot. After the ball leaves, the racquet mass center (balance point) moves at a slower speed, and this means a loss of its kinetic energy (kinetic energy = 1/2 mass times velocity squared). The ball gets some of this lost energy (the same for all racquets under the two benchmark conditions), and the rest becomes internal energy, wasted in bending the frame, which becomes vibration. Vibration, especially where the frame is stiff and light, can stealthily sap the elbow.

If the frame is stiff and light, frame vibration will not be damped by the frame but will have to be dumped into the arm holding on to the racquet. Don't place any reliance on string buttons to save your arm. Damping gadgets on the strings are too small to do much besides reduce residual string vibration, which is a minor annoyance, and damping gadgets in the frame must be expected to handle an energy load of the magnitude determined by the design of the racquet.

Better than damping is prevention of Shock by proper weight distribution in the racquet (head-light and heavy overall). The most effective vibration damper is a large particulate handle end weight. Also effective at vibration damping is the Pro Kennex Kinetic system, which has particulate weights in the racquet head. The high performance of the Kinetic racquets in the evaluations does not take into account this additional damping benefit. High Shock is bad also because it means high Elbow Crunch and Shoulder Crunch.

7. Work is the energy required to produce a certain ball speed with the racquet. High Work is bad because the player has to swing harder to get the same result. Work quantifies a racquet's power: the less work the player has to put in, the more powerful the racquet. Of course, a player may put in lots of effort and get lots of ball speed, especially with high swingweight racquets, but the power comes from the player, not the racquet. In the evaluations, string tension, frame stiffness, and ball bounce are comprised in a standard bounce or elasticity (coefficient of restitution) of the racquet/ball system (0.85), so these factors -- which anecdotally are said to add power -- are not used independently for evaluations. We assume that all racquets are strung such that they have the same bounce. Work, like Shock, is measured in terms of joules of energy, and the formula for Work (and kinetic energy) is 1/2 Mv2 (M is racquet mass in kilograms and v is the linear velocity of the racquet mass center (balance point) just before impact, in meters per second). The Work done during the impact is not counted, because studies show that it does not add materially to the speed of the ball. It turns out that head-heavy racquets require a lot more Work, which is bad. They are also hard on the elbow and shoulder, which is worse. Head-light and heavy racquets are the most powerful (lowest Work).

8. Shoulder Pull is the force (in the metric unit of Newtons, a Newton being about a quarter of a pound) exerted by the shoulder muscles in opposing the centrifugal force acting on the racquet as it moves around the shoulder in the swing done by the player's Work. This counteracting force is called a "centripetal" force because it acts toward the axis of rotation (here the shoulder socket); Shoulder Pull is equal and opposite to the centrifugal force while the racquet is getting up to speed for the impact. The formula for centripetal force is Mv2/R (where M is racquet mass in kilograms, v is the linear velocity of the mass center, in meters/second, and R is the distance from the racquet mass center to the axis of rotation, here the shoulder). Note that, in rotation, the mass center linear velocity (v) decreases the closer to the hand the balance point is located, so head-light balance means low Shoulder Pull. The variable v is squared in the formula for centripetal force, so a light racquet having a head-heavy balance may still have a large Shoulder Pull, despite its light weight, due to its distant mass center and consequent high mass center velocity in rotation. That's bad.

9. Shoulder Crunch is the change in the centrifugal force acting on the racquet, a change that occurs due to impact slowing the racquet down. The shoulder muscles that were countering this centrifugal force with an equal and opposite centripetal force (Shoulder Pull, see above), and which were in equilibrium before impact, suddenly are released from some of their load but continue to contract as if it were still there, because the slowdown is so sudden. This is effectively a muscle spasm.

This excess centripetal force is Shoulder Crunch, and is measured in units of force (Newtons in the metric system, pounds in the English system, 1 Newton = 0.225 lb.). Once the Shock is known, Shoulder Crunch follows by a simple calculation: Shoulder Crunch = (2/R)(Shock) (where R = distance of the racquet's mass center from the shoulder). See the derivation. Please note that it is only the Shoulder Crunch due to the racquet that is calculated here -- there is also mass in the arm swinging the racquet, and it slows down too, but for purposes of racquet comparisons we assume that it is the same arm for every racquet, as if one player were trying out all of them.

10. Elbow Crunch is the excess centripetal force acting at the elbow, an excess that occurs because on impact the racquet slows down, so its centrifugal force drops. The centripetal force of the muscles attaching to the elbow and the centrifugal force of the racquet in its swing had been balanced before the impact, but the sudden slowdown creates what is effectively a muscle spasm. The muscle continues to contract as if it still had a full load, so it suddenly shortens and yanks on the tendons that attach it to the elbow. This yank (Elbow Crunch) is a cyclic stress which, repeated over time, may be a contributing cause to tissue failure. Like Shoulder Crunch, it is calculated by multiplying Shock by 2/R' (where R' is the distance of the racquet's mass center from the elbow). See the derivation of Elbow Crunch. Elbow Crunch is larger than Shoulder Crunch because the elbow is closer than the shoulder to the mass center of the racquet, so R' is smaller than R. Elbow Crunch is measured in units of force (Newtons, 1 Newton = 0.225 lb. or 3.6 oz.).

11. Impact Impulse is the change in the racquet's momentum on impact. For a smooth followthrough, this should be low. See the derivation of a formula for Impact Impulse. Impact Impulse correlates closely with the sweet spot: a high sweet spot (center of percussion) means a smooth stroke. Presently, this criterion is not used in the evaluations. Perhaps it should be included as a factor in the macro criterion known as Control.

12. Polar Moment is the racquet's rotational inertia about its longitudinal axis: its resistance to a screwdriver twist. This should be high. Shoulder weighting, such as by Wilson's Perimeter Weighting System, or by lead tape at 9 and 3 on the racquet head, increases Polar Moment. So does larger racquet head size. Measurement of Polar Moment so far is not available, so it is not used in the evaluations done on this site.

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What Causes Tennis Elbow?

Many professional researchers are still looking for an answer. Damage to the tendon attaching the extensor carpi radialis brevis (ECRB) muscle to the elbow is the cause of the pain, but the cause of this cause is a mystery. However, it is fairly certain that this type of damage is the result of repetitive stresses, such as hitting a tennis ball.

Producing causes of tennis elbow may include the following mechanisms, which are offered here for comment and further investigation:

(1) Elbow Crunch is a sudden shortening of the ECRB due to impact (explained at greater length above under Elbow Crunch). This effectively is a muscle spasm that stresses the tendons.

(2) On impact, the resultant Torque twists the racquet head back, while Moment is dragging the head down, and the hand is holding the racquet steady. The resultant twist of the handle (Torsion, or Longitudinal Torque) is clockwise for a right-handed forehand. This twist winds up a catapult. When the ball leaves the racquet, the catapulting force is counterclockwise for the right-handed forehand. The two opposite screwdriver twists in a short time give a severe stress cycle to the extensor carpi radialis brevis muscle that attaches the middle of the hand to the elbow, even for a dead-center hit.

(3) The back-and-forth catapulting stress cycle of Torque from impact twisting the racquet back, followed by catapulting the racquet forward when the ball leaves, aggravates the handle twist cycle mechanism discussed above under (2). The extensor carpi radialis brevis muscle is anchored at the elbow and at the metacarpal (hand) bone of the middle finger, on the index finger side. The resultant Torque from impact is a twist backward that tends to yank this muscle as the middle finder is extended. On impact, this muscle is either straining (on the backhand) or slack (on the forehand). On the backhand, the first twist yanks this straining muscle, further stressing the tissues attaching it to the elbow. Then the muscle suddenly loses resistance but continues to work against the combined stress, so it suddenly shortens after impact, giving an even more severe yank to the elbow (cf. the discussion below on Elbow Crunch). For the forehand, the muscle is slack on impact, so the catapulting stress cycle cracks the muscle like a whip, stressing the points of attachment at the wrist and elbow. Elbow straps help because they damp the whip effect.

(4) Shock becomes internal energy, which expresses itself as frame vibration, and this vibration is transmitted to the arm holding on to the racquet unless it is damped somehow. (The correct term is damped, not "dampened.") In the old wood racquets, vibration disappeared quickly because it was damped by the flex of the wood, but the new stiffer and lighter frames do a poor job of damping, so they efficiently transfer the subtle shaking to the arm. Undamped high frequency frame vibration can stealthily sabotage the elbow, so the price of power may be pain. Vibration of the frame shakes the extensor carpi radialis brevis muscle that attaches the middle of the hand to the elbow. This causes cyclic stressing of the tendons at the lateral epicondyle, where the fat half of this long teardrop-shaped muscle attaches. Cyclic stressing is how you break a coathanger by bending it back and forth. Eventually, with enough stress cycles, fatigue can cause tissues to snap, even without any tremendous force.

What you don't want if you are concerned about the risk of tennis elbow is a stiff, high-Torque, high-Moment, high-Shock racquet. That means a light, head-heavy racquet. Poor stroking technique is frequently accused, conveniently diverting scrutiny from racquet design, but, as the calculations on this site prove, risk factors for tennis elbow include: (1) light racquet weight and (2) head-heavy balance.

Stiff frames are also bad. In the racquet evaluation criteria, what is good for minimizing elbow damage is low Shock, low Elbow Crunch, low Torque, and low Moment.

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Is a Lightweight Racquet a Good Idea?

No, a lightweight racquet is a dumb idea, as pro customizers attest. Weight is not bad. You need weight to return a "heavy" ball (lots of pace and spin). Wimpy racquets can't put much pace on the ball if you don't have time to develop a long stroke, such as when you are stretched wide. Pete Sampras uses a racquet that is 14 oz. and evenly balanced, and when he is going for a putaway, he chokes down so the swingweight is even higher. Agassi uses one that is 13.2 ounces and 5/8 inch head-light. Mark Philippoussis uses one that is 13.5 ounces and is 3/4 inch head light. Lest you think that these heroic sticks are as unwieldy as the sword of Goliath, remember that the lightest wood racquet was 13 ounces. Ladies and children used them.

Maybe, in the short space that you have to execute your stroke, you might swing the wimpy racquet a little faster -- but swing speed is not the whole story. Momentum, not energy, and not force, is what counts in a collision, and in computing momentum the racquet's mass is just as important as its speed (momentum = mass times velocity). Readers with baseball experience know what happens when you try to hit a hardball home run with a softball (i.e. lightweight) bat. A softball bat cannot hit a hardball very far because it doesn't bring enough mass to the collision, and therefore its momentum on impact is low.

Aside from the foregoing performance considerations, there is the even more important question of safety. What doth it profit a man to win a match, but hurt his elbow? Light racquets are bad for tennis elbow.

Most racquet customers and their stringers know little, and care less, about the difference between weight, Moment, and swingweight. They don't have game enough to tell the difference in practice, and theory is a closed book to them. "Pick up appeal" (how light the frame is when you pick it up in the pro shop) is the predominant criterion (after cosmetics) for the ignorant. An epidemic of elbow and other arm injuries has been the result. Tennis is losing players at an alarming rate, and slowly declining in popularity. It's all because of the fundamental mistake of amateurs regarding racquet weight, a mistake that some racquet salesmen apparently have chosen to exploit for their short-term profit.

The touring pros know better. They add weight when they customize their racquets. A more massive (heavier) racquet will crush majestically through the ball instead of bouncing off, which makes it more comfortable on impact and more accurate. See the Official Rules of the ATP Tour regarding racquets. This little secret vexes the sponsors that pay them lots of money to pretend to play with granny sticks, so you won't hear much about it. See page 8 of the June 1996 issue of Stringer's Assistant (published by the US Racquet Stringer's Association) for some data on pro customized racquets.

The Effect of the Sweet Spot

The "center of percussion" is the real "sweet spot." Every so often, a shot is smooth and accurate when the impact is at just the right spot, and that is because the resultant push (positive) or pull (negative) on the hand from impact (Impulse Reaction) has been reduced to zero. Proper weight distribution, using a heavy handle end counterweight, can raise the center of percussion significantly, and the higher the Sweet Spot, the better.

But a racquet with a relatively high center of percussion (such as the appropriately named Hammer) is not necessarily good. Even though it may have some extra punch due to a high positive Impulse Reaction for impacts at the center of the strings, if its balance is head-heavy, that power comes at a painful price. Its Torque and Shock will be high, even with a high sweet spot, and so will the Work, Shoulder Pull, Shoulder Crunch, and Elbow Crunch. Hammer-type racquets bring up the rear of the rankings. What you want is high sweet spot together with head-light balance. Both can be achieved with a handle end counterweight. Some customizers squirt silicone into the handle, but much more weight, at a more distal location, is better.

The center of percussion is a point along the racquet's length; it is not a "spot" having an area. Do not be misled by deceptive advertising suggesting that some manufacturer has succeeded in expanding the sweet spot from a point to an area, so that you can get sweet spot performance nearly everywhere on the racquet face. Another misconception is that there is no shock if the impact is at the center of percussion. One resultant force from impact (Impulse Reaction) is reduced to zero, but the other (Torque) still exists, and there is still some Shock, Work, Shoulder Pull, Shoulder Crunch, and Elbow Crunch. Even when you hit the sweet spot, you feel something.

An interesting fact is that the impulse (impact force) on the racquet upon impact (Impact Impulse, see derivation) is that the higher the center of percussion, the lower the Impact Impulse -- they correlate closely.

The area concept of the sweet spot has to do only with mapping where the coefficient of restitution (a measure of the bounce, or elasticity of the collision) exceeds a certain arbitrary value. It is a plot of elasticity, with the more elastic region being inside the sweet area. Manufacturers who claim a large sweet area claim to have succeeded in making their racquets bouncier, or more elastic. The upside of more elasticity (bounce) is less Shock, Work, Shoulder Crunch and Elbow Crunch. String tension as well as frame geometry affects elasticity. In the benchmark conditions used for evaluation on this site, the elasticity is assumed to be the same (0.85) for all racquets at the point of impact, which is assumed to be at 16 centimeters from the tip for all racquets because this is the center of the head of a standard-length (27") racquet, which most of us were trained on.

The Effect of String Tension

The variables affected in the formulas by string tension are dwell time (t) and coefficient of restitution (c). Dwell time (t) is the length of time the ball stays on the strings. Coefficient of restitution (c) is the measure of the elasticity of the collision between the ball and the racquet (high c means more elastic, livelier bounce).

Longer dwell time (high t) means lower Torque and Impulse Reaction on impact, which means better accuracy. Dwell time decreases with increasing string tension, which is bad. And, after a point, as string tension increases, coefficient of restitution goes down. Lower coefficient of restitution (low c) means higher Shock, Work, Shoulder Pull, Elbow Crunch, and Shoulder Crunch.

The conventional thinking is that loose strings give more "power" (presumably, this means higher c) and tight strings give better "control" (presumably, higher t and therefore less resultant forces from impact). Dr. Jack Groppel, in his interesting book, High Tech Tennis (available from the USPTA bookstore), has shown that these variables are not affected as presumed. See pp. 25-27. Although one might think that the ball would stay on the strings longer when the tension is higher (because it is flattened more), dwell time decreases with increasing string tension, which is not good for accuracy. And the relationship between string tension and coefficient of restitution (c, or racquet bounce) is not linear, especially with midsized racquets, where there is a pronounced peak of bounciness at 60 pounds for gut and at 50 pounds for nylon. For oversized racquets, it is true up to 70 pounds of tension that lower tensions mean higher bounce for both gut and nylon. There seems to be no consensus among the pros. Borg strung as tight as possible (85 lb.) on specially fortified racquets with two extra plies, while McEnroe preferred very low tension (44 lb).

Anecdotal evidence suggests that there is truth in the rule that tight strings give better control, but it isn't for the reason that dwell time increases. With a loose racquet, especially a big head racquet, an off-center hit will deform the string bed more severely than it would a tight, small-head racquet (like Pete Sampras uses), and therefore there will be less certainty as to the path of the rebounding ball. Thanks to Greg Raven for pointing this out. And Ronald Yepp points out that with a tight racquet, the ball is flattened more, so topspin is easier to produce. This would be particularly true where the head is small. Pete Sampras is a case in point: he can generate amazing topspin on his second serve using his heavy, small-head, tightly strung (75 lb.) racquet. Bjorn Borg is another. Spin gives greater control, and greater spin is possible with tight strings.

Recommendation: for power, use a midsized racquet strung with natural gut at 60 pounds because this tension gives the maximum bounce. If control is your main concern, and your stroke puts a lot of top on the ball, string very tight and use small-head racquets. Restring often because string tension decreases quickly, as Crawford Lindsey has shown.

The Effect of Frame Stiffness

Many correspondents have asked why flex data is not included in the evaluations. The answer is that flex, or frame stiffness, is comprised in the uniform coefficient of restitution (0.85) stipulated in the benchmark conditions. Other factors comprised in the coefficient of restitution are string type, string tension, ball bounce, and bounce-boosting innovations such as Power Holes, etc. The assumption was made that this mix of factors produced a coefficient of restitution that was the same for all racquets tested. When better data becomes available as to the effect of stiffness on racquet bounce, with the other factors controlled, then a more exact understanding may be possible. For now, we must plod along with a rough but reasonable assumption that all racquets are strung such that they have the same bounce.

Frame stiffness is measured on the Babolat RDC as flex numbers, which represent deflection under a 25 kilogram load. High flex numbers mean stiff frames. A stiff frame would have the effect of flattening the ball more (if used with tight strings and a small head), thus making topspin easier to impart, and it would allow the strings to do their job better because the frame would not be deformed so much on impact. The downside of stiff frames is that they do a poor job of damping frame vibration, and therefore may be a risk factor for tennis elbow in racquets having high Shock.

The Effect of Handle Size

Bigger is better. A large handle size gives more area to apply friction and a wider radius to apply the frictional force in order to resist racquet twisting about its longitudinal axis (Torsion, or Longitudinal Torque) on off-center hits. Handle size is the circumference (distance around). It can be increased by adding an overgrip (e.g. Tournagrip ® or moleskin), or by building out the handle under the grip with tape or shims. Bigger handles should also be better for preventing blisters.

The Effect of Heavier, Larger, or Softer Balls

To slow down the men's game, and thereby increase its entertainment value for the unsophisticated, the rulers of tennis want to change the balls.

Particularly troubling is the lack of prior consultation with the pros who will be using these balls. Which of them are in favor? True, it is rare to find any touring pros who might be called educated, so naturally they must expect their opinions to be ignored and their objections overruled. If they don't want to play, there are plenty of ambitious youngsters eager to take their place. However, even though there might be no reason for courtesy or compassion, wouldn't it be economically prudent to prolong the careers of the marquee players, and not to increase the already alarming rate of injuries among the recreational players?

The ITF has authorized a ball with a 15% greater diameter to be used "on an experimental basis." The intention is that the bigger ball will meet more air resistance, therefore play will be slower. Fluffing the nap (felt covering of the ball) will increase diameter and drag, but apparently the intention of the ITF is to require ball manufacturers to mold a larger rubber core. See the diameter test for the "slow ball" in Appendix 1 to the Rules of Tennis -- fluffed nap will not keep the ball from falling through the bottom hole of the testing apparatus. The larger diameter of the rubber core, even if the weight of rubber remains the same, will result in a higher rotational inertia for the ball.

That means a "heavy" ball because players will be able to impart a lot of angular momentum (spin). Angular momentum is the product of the rotational inertia and the rotation speed, and the higher rotational inertia permits a much "heavier" ball at the same spin rate. High angular momentum of the ball on impact will aggravate Torsion (screwdriver twist on the handle), causing more stress on the arm of the receiver.

Another problem with bigger balls: they will radically change the game in the same way that the "spaghetti string" racquet did, by giving junkballers an edge. The ITF banned (retroactively) the spaghetti strings which imparted such extreme spin. The same reasoning should ban these balls.

Yet another problem with bigger balls: if the same ball weight (57 grams) is to be maintained, the rubber of the bigger ball must be made thinner to stretch over the larger surface. Thinner rubber means that the air will leak out easier, and higher air pressure will be needed to maintain the same ball bounce. These balls will go flat faster. They will also be less bouncy in actual pro-level play because of higher hysteresis loss from more air being compressed. These will be soft balls.

Presently, for professional tournament play, a ball must bounce more than 53 inches and less than 58 inches when released from a height of 100 in. That means that the coefficient of restitution for the ball itself (apart from the racquet) is between 0.73 and 0.76. It should be noted that the 100 in. drop height does not approximate the speed of a pro serve, so for testing the hysteresis loss from the bigger ball this test would be inadequate. Using softer balls, having a bounce at the low end of this range (low c), means higher Shock, Shoulder Pull, Work, Shoulder Crunch, and Elbow Crunch for the players.

As you can see from the formulas, heavier balls (high b) means both higher resultant forces from impact (Torque and Impulse Reaction), and higher Shock, Shoulder Pull, Work, Shoulder Crunch, and Elbow Crunch. With heavy balls, the game becomes more painful and less accurate. See the formulas.

Club players can take a lesson here, especially those who play on clay, where the balls get heavier as play goes on. Change balls often to protect your arm. Tennis balls are a bargain, so leave them on the court.

Should Your Racquet Be Head-Heavy or Head-Light?

Head-light is better, no question. A head-light racquet (balance point closer to the hand than the midpoint of the racquet's length), has significantly lower Moment, resultant forces from impact (Torque and Impulse Reaction), Shock, Work, Shoulder Pull, Shoulder Crunch, and Elbow Crunch. And it can have high mass (M) and high swingweight (I), but low Moment with a handle end counterweight. That's good, remember.

In the formulas, the key variable is r (the mass center radius, or the distance from the axis of rotation to the balance point). Head-light balance means that r is small. When r is small, r2 will be tiny, and the key coefficient in the formulas, Mr2/I (which, as astute students will note, is equal to r / q because q = I / Mr) in the formulas for Torque and Shock, will be small, which is good. The linear velocity of the mass center is critical, and when the mass center is close to the hand (small r), its linear velocity (v) in rotation will be smaller than when it is distant. A distant mass center goes much faster in rotation -- remember the carousel at the playground? So head-light is the smart choice. Head-light (low r) with a high sweet spot (high q) is the really smart choice for reducing the risk of tennis elbow. That means a racquet with a large handle end weight (~5 ounces). This handle end weight customization produces significant improvement: check this.

An important additional benefit of head-light balance is that Moment is less, so the racquet is easier to position for volleys and returns, and is not so heavy to hold up all afternoon. Moreover, with a low Moment, the Longitudinal Torque from impact will be small, so the racquet will be easy on the elbow. Head-heavy racquets, on the other hand, increase the risk of tennis elbow because of their high Moment and high Torque (therefore high Longitudinal Torque), their high Elbow Crunch, and their high Shock.

The Effect of Mass and of Swingweight

More mass is definitely better. More swingweight (moment of inertia) is also definitely better. The touring pros, in customizing their racquets, add mass and increase swingweight, because they know from personal experience what really works. Their customized racquets bite on the ball more, so they are able to generate heavy spin on their forehands and serves. Pete Sampras' heavily customized Wilson Pro Staff 85 (a modification of the legendary St. Vincent ProStaff, which is no longer in production) weighs 14 ounces, about the same as the old woodies, but much heavier than the heaviest racquets marketed to the public these days.

Chain store customers, and even those who buy at pro shops, demand lighter racquets -- completely the opposite of the pros! A candid observer must find it somewhat incredible that even though the racquet makers pay the pros lots of money to display what appears to be the same racquet they are selling to consumers, in reality the racquet is not at all the same in weight or swingweight. Racquet manufacturers tout light weight as if it were something good, when in fact they should be putting a warning label on their racquets advising buyers that they are increasing their risk of disabling injury if they insist on banging away with a wimp stick. The heavier, the better. If 14 ounces sounds big to you, consider that even ladies and juniors used to play with wood racquets that weighed that much, and Don Budge won the Grand Slam with a 16 ounce racquet.

Mass (in kilograms, symbol M) is a measure of the racquet's "inertia," a word that means essentially its resistance to change. The change resisted by mass is change in linear (straight-line) velocity. More mass means that the racquet will not slow down so much on impact. A short, controlled swing of a heavyweight racquet can hit the ball harder than a frantic flail of a featherweight. Same as bats in baseball: if you want home run power, bring a heavy bat to the plate. Babe Ruth's bat weighed 52 ounces. You want a racquet that will hit through the ball, instead of bouncing off. Pete Sampras' second serve has an incredible amount of spin on it because his racquet can bite on the ball due to its high inertia on contact.

Mass is not the same thing as weight, but weight units such as ounces may readily be converted into mass units of kilograms, using the conversion factor of 1 ounce = 0.02835 kg.

Swingweight (symbol I) is another measure of the racquet's inertia, but it is a different kind -- rotational inertia, or a resistance to change in a racquet's angular velocity about an axis of rotation. A useful way to understand swingweight is as the energy storage potential of a racquet. Just like flywheels, racquets store up the player's effort. A racquet with a high swingweight (I in the formulas) requires more effort to swing, but will not lose much angular velocity on impact, and will snap through the ball more, biting for more spin, especially if the strings are tight and the head is small.

Angular velocity means how much of a circle is swept out every second, in radians per second. There are 2p radians in a circle so an object having an angular velocity of 6.28 radians/s will rotate at one revolution per second.

It is most important to realize that swingweight has meaning only with respect to a specified axis of rotation. A swingweight number where you don't know the axis of rotation is of no use. The published swingweight numbers of the USRSA (so far the best authority, but of dubious accuracy sometimes) come from measurements on their Babolat Racquet Diagnostic Center (RDC), which measures swingweight about an axis of rotation 10 cm from the handle end. In play, this is not the axis of rotation of the racquet, so you will have to use the Parallel Axis Theorem to convert the RDC swingweight figure to a value for I that can be used in the formulas. I have a dream: that an independent testing laboratory, under no pressure from advertizers, would use an accurate pendulum timing device to compile accurate swingweight data on strung, gripped racquets -- at least the new ones.

On the forehand, the axis of rotation is 7 cm from the handle end, which is between the ring and middle fingers. Hold a racquet and waggle it to see that this is true. On the serve, where most top players use a choked-down grip over the butt cap, the swingweight will be higher because the axis of rotation has moved to 5 cm from the handle end. In other words, swingweight will change depending on how low you grip the racquet. Notice when Pete Sampras is going for a forehand winner, he chokes way down over the butt cap to increase the swingweight.

The unit of measurement of swingweight in tennis is kg.cm2 (kilograms times centimeters squared), which is the unit for the swingweight measurements of the Babolat Racquet Diagnostic Center (RDC), presently the industry standard measuring device. Other scientific names for swingweight are moment of inertia, rotational inertia, and Second Moment. More swingweight is good for accuracy and comfort (low Torque, Shock, Shoulder Crunch, and Elbow Crunch).

A racquet with a high swingweight takes more effort to whip around the axis of rotation, but on impact that investment pays off in better speed and accuracy. "Maneuverable" is a term loosely used to describe a racquet with a low swingweight (although there seems to be some confusion between swingweight and Moment as this term is used). Maneuverability in a racquet, under this definition, is not good, because high swingweight is good.

Two racquets that weigh the same (have the same mass) may have very different swingweights because of the way this mass is distributed. More mass to the head of the racquet, such as by adding lead tape, will increase swingweight. Although more swingweight is good, head-heavy balance is bad for your arm, so lead head tape fixes one problem (higher I) but aggravates another (higher r). Increasing r means a higher Shock, Torque, Moment, Work, Shoulder Pull, Shoulder Crunch, and Elbow Crunch. Note that the effect of r is squared in the Shock and Torque formulas, but the effect of I is not. So lead tape to the head should be counterbalanced somehow with a tailweight.

Tailweighting is a better idea for adding mass and increasing swingweight because it has the added benefit of giving a head-light balance. The effect of a tailweight is increased where it is distant from the axis of rotation. A handle end weight of 5 ounces, extending 1 inch below the heel of the hand during play, will radically improve performance under all criteria. Here's proof.

The Effect of Shoulder Weighting

Adding weight to the 9 and 3 o'clock positions of the head (shoulder weighting) adds swingweight, but not as much as adding the same amount of weight to the tip. Shoulder weighting also increases the Polar Moment of the racquet, which may be necessary to counter the "heavy ball" encountered in top echelon tennis when the opponent hits with a lot of pace and topspin. The heavy ball tends to rotate the racquet, even when the impact is dead center, thus making it more difficult to put your own top on the ball and giving an uncomfortable screwdriver twist (Torsion). A racquet with a lot of rotational inertia about its longitudinal axis will not be pushed around so much by the impact and will be more dominant in play. Shoulder weighting should be offset by a tailweight at the handle end, so the balance is head-light.

Are Big Head Racquets Better?

No. The increased length of string to be stretched in a large racquet head gives a more pronounced give to the string bed, and therefore presumably a longer dwell time (t) and a more pronounced trampoline effect (higher coefficient of restitution c), both of which are good. But the trade-off is that accuracy on off-center hits may be worse because the string bed is more deformable, and therefore the path of the rebounding ball is less certain. Also, the ball is not flattened against the strings as much, so it tends to just roll down the face when you stroke for topspin. Pete Sampras plays with an extremely small head racquet (85 square inches in area). The wood racquets were even smaller (65 sq in), and the tubular metal racquets that Jimmy Connors used were smaller still, and had a head size like a squash racquet. These world number ones are persuasive authority against big heads.

Another downside of big heads is that, due to their large width, they have a large Polar Moment (rotational inertia about the axis running through the handle and head). That is especially true if there is a weighting system at 9 and 3 o'clock. The stress cycle of torques (the resultant angular impulse from impact and the countering catapulting twist forward after impact) will be more severe because the torques are larger.

The Longitudinal Torque for an off-centerline hit is defined as the product of the racquet's Polar Moment (I^), and its angular acceleration about the longitudinal axis (a): torque = I^a. What we would like to know is whether the angular impulse (change in angular momentum) is greater with a bigger head racquet. Angular momentum is torque multiplied by the time it operates. If what the proponents of big heads say is true -- that the higher Polar Moment (I^) makes the racquet stay in contact with the ball longer -- then the angular acceleration (a), although smaller, has more time to operate, and therefore the resultant angular momentum from impact may be greater. That means a greater whipsawing stress cycle. The consensus among physical therapists seems to be that big heads are a risk factor for tennis elbow. The pros who make their living winning tournaments do not favor them. Therefore, in view of the foregoing theoretical discussion, and the preferences of the pros for accuracy over power, the tentative conclusion must be that big head racquets are not better.

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Are Extra-Long Racquets Better?

Sometimes. If the impact point is at its standard distance from the hand (i.e. where it would be using a 27-inch long racquet), extra-long performance is worse on groundstrokes but better on the serve. Moment is increased by the extra length, which is bad for the reasons discussed above. There is no need to adjust your striking point on the serve because you do better with the impact point at the usual distance from the hand, even though that moves the impact point lower on the face than the center of the strings. Two-handers should definitely consider an upgrade to an extra-long.

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Does a Two-Handed Shot Have Any Advantages?

Yes. The axis of rotation on the two-handed shot is between the hands, which is higher up the handle than the axis on the one-handed shot. That's good, because decreasing r and d in the formulas improves performance under all criteria, and you get those desired decreases by shifting the axis of rotation up the handle. Many players use a two-handed backhand with good results, but few use the cross-handed forehand of Seles and Gambill. Pancho Segura's two-handed baseball style forehand was, in its day, considered the best shot in the game because it was so deceptive and powerful. A two-handed shot has the additional advantage of allowing a much heavier racquet to be used.

The Effect of String Damping Gadgets

Damping doo-das on the strings damp only residual string bed vibration, and do not really protect the arm by damping frame vibration. Adding more mass to the head in the form of a damping gadget is a bad idea because it increases r in the formulas and therefore worsens performance, so the damper should be light. Pete Sampras' string damper is just a cable grommet.

Benchmark Conditions for Meaningful Racquet Comparisons

It is helpful to establish some benchmark conditions so that abstract formulas can yield some meaningful numbers for racquet comparisons. Benchmark conditions stipulate values in the formulas for b, c, t, s1,and s2, and establish common impact point and axis of rotation locations for all racquets. With these stipulations, comparisons are possible using racquet measurements to determine M, r, d, and I in the formulas.

The First Benchmark Condition is a 70 mph return (s2) of a shot received at 20 mph (s1) (its speed parallel to the ground, after bouncing), with duration of impact on the strings (dwell time, t) 0.004 seconds, coefficient of restitution (c) 0.85, axis of rotation 7 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet). From the latest research figures on ball speed, the postulated speeds are at the heroic end of pro performance.

The Second Benchmark Condition is a 110 mph serve, with duration of impact (dwell time) 0.004 seconds, coefficient of restitution 0.85, axis of rotation 5 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet).

Under both benchmark conditions, the mass of the ball (b) is 0.057 kg (57 grams, or about 2 ounces), and string tension, string type, racquet stiffness, etc. are all comprised in the uniform coefficient of restitution (c = 0.85 for all racquets). This figure of 0.85 is typical for impacts at the center of the strings. See H. Brody, "The physics of tennis, III. The ball - racquet interaction" Am. J. Phys. 65, 981 (1997). With the foregoing stipulations fixing constants, the only variables remaining in the formulas are M, r, I, and d, which are calculated using the racquet measurements.

The axis of rotation used for the First Benchmark Condition is 7 cm from the handle end, where it would be on the forehand. For the Second Benchmark Condition, the axis of rotation is 2 cm lower (5 cm from the butt end) because top players typically choke down over the butt cap on the serve to increase swingweight. Michael Change, it appears, does not, so his axis of rotation on the serve would be the same as on the forehand (7 cm) and the swingweight, r and d in the calculations would have to be adjusted accordingly.

Dwell time is put at 0.004 seconds, which is a realistic figure based on available information, but it may vary slightly with string tension, frame flex, and other factors. More investigation is needed, using high speed video. Roland Sommer, the inventor of the kinetic system used in the ProKennex line, claims a much longer dwell time using the kinetic system (which is good) but the superior performance of the ProKennex kinetic racquets in the evaluations does not depend on this, but on their intelligent weighting scheme.

Only high school algebra is required to understand the formulas given for evaluating racquet performance, and the derivations of these formulas. The concepts from which the formulas are derived are first year physics and the simple related engineering regarding an "eccentric impact."

Where Did the Racquet Data Come From?

Racquet data are from measurements published in the June 1998 issue of the official magazine of the US Racquet Stringers Association, Stringer's Assistant, in the "Powerability Table". In January 1999, this magazine was renamed Racquet Tech. In the January 1999 issue were data on the new Spring 99 racquets, and in the May 1999 issue were more racquets. Altogether, the racquets evaluated presently total 276. All data are for strung and gripped racquets, measured on the Babolat Racquet Diagnostic Center by USRSA technicians.

The Four Measurements You Need to Know

The measurements to ask for in order to evaluate objectively a racquet's performance by means of the formulas are:

M = racquet mass, in kilograms (1 ounce = 0.02835 kg), after the racquet has been strung and gripped.

r = the mass center radius, which is the distance from the axis of rotation to the mass center (balance point). The axis of rotation is 7 cm from the handle end (First Benchmark Condition, anyway). In lieu of that, simply the distance of the strung and gripped racquet balance point from the butt end can be adapted for use in the formulas.

I = swingweight (moment of inertia) about the stipulated axis of rotation for the given benchmark condition, in kg.cm2, for the racquet when strung and gripped. Published measurements, and swingweight measured on the Babolat RDC, are with regard to an axis 10 cm from the handle end. Note that there is no such thing as an absolute swingweight -- swingweight only has meaning with respect to a specified axis of rotation, so swingweight numbers are useless unless you know where the measurement was taken. The axis of rotation for the First Benchmark Condition is 7 cm from the handle end, where it would be on the forehand. The swingweight on the serve, where there is a lower axis of rotation (5 cm from the handle end), as well as the swingweight for the forehand, may easily be found from the published swingweight measurement by using the Parallel Axis Theorem.

d = distance, in cm, from axis of rotation to point of impact (16 cm from the tip for all racquets under both benchmark conditions). This is easily derived from the overall racquet length. The axis of rotation for the First Benchmark Condition is 7 cm from the handle end and the impact point is 16 cm from the tip, so subtract 23 cm from the racquet's total length to find d for the First Benchmark Condition. Subtract 21 for the Second Benchmark Condition.

It is critical that a standard axis of rotation be agreed upon so that racquets may be compared objectively. We fix the axis of rotation for the First Benchmark Condition at 7 cm from the handle end, where it would be on the forehand. Cf. H. Brody, "The physics of tennis, III. The ball - racquet interaction" Am. J. Phys. 65, 981 (1997). Of course, the formulas still work whatever the axis of rotation might be (as in the case of the Second Benchmark Condition, where the axis is 2 cm lower).

Mass (M) may be found by weighing the racquet and converting ounces to kilograms (1 ounce = 28.35 grams or 0.02835 kilograms, so multiply your scale reading in ounces by 0.02835). The distance r may be found by experiment, balancing the racquet on a knife blade, then measuring the distance from that point to a point 7 cm from the handle end. To find d, subtract 23 cm from the length of the racquet under the First Benchmark Condition, and 21 cm for the Second Benchmark Condition. Measuring swingweight (I) requires a special device, such as the Babolat Racquet Diagnostic Center or the AccuSwing from Alpha Sports.