## Derivation of Formulas

### The handwaving that goes with the math:

Derivation of a formula for finding the new balance point resulting from adding weight, such as by lead tape at the tip.

The math involved in the derivations is high school algebra. The conservation principles applied are first year physics.

The impact of a ball with a racquet is a case of “eccentric impact,” so both rotation (turning about the axis of rotation, at the middle of the hand) and translation (movement of the axis of rotation backwards or forwards) result from the collision. In other words, the hand will be both pulled and twisted when the ball hits the racquet. These resultant forces from impact should be minimized.

1. Sweet Spot (center of percussion) is the point along the racquet’s length where an impact produces no yank (zero Impulse Reaction) at the axis of rotation (the middle of the hand). In some modern handbooks and textbooks, the center of percussion, which was prominent in earlier books, has been given hardly any consideration, so modern-day students may have a blind spot in their education. See, for example, the total omission of this important concept in the CRC Handbook of Mechanical Engineering (Frank Kreith, Ed.)(CRC Press 1998), and compare Marks’ Handbook (4th Ed. 1941) p. 221, as well as the current edition. The center of percussion is that point on the racquet face at which an impact will produce zero Impulse Reaction at the axis of rotation — in other words, where the only resultant force from impact is a Torque. Those with baseball experience know what the Sweet Spot means in practice: a hit higher on the bat than the sweet spot stings the hands and is weak, but a hit on the Sweet Spot is smooth and strong. The distance of the center of percussion from the axis of rotation (the hand) is denoted by the letter q.

$q = \frac{I}{Mr}$

(I is swingweight in RDC units of kg·cm², M is racquet mass in kilograms, and r is the distance in centimeters from the axis of rotation to the center of gravity (balance point); q is in centimeters, measured from the axis of rotation to the center of percussion. The axis of rotation is 7 cm from the handle end on the forehand, and 5 cm on the serve. So you see that the Sweet Spot can shift according to where you hold the racquet.)

2. Moment. The moment of a force is its magnitude (in Newtons) multiplied by the distance of the point of application of the force from the axis of rotation. Weight is a force. In the metric system, which we use on this site, force is measured in units called Newtons; in the English system, force is measured in pounds (1 Newton = 0.225 lb). Weight is the product of an object’s mass and the acceleration due to gravity. We can idealize all of the mass of the racquet as being concentrated at the balance point (also known as the mass center, or center of gravity), so the balance point is the point of application of the downward force. Acceleration of the mass center downward due to gravity results in a force acting at the balance point. So the formula for Moment is simple: weight times lever arm. Working in the metric system, as we do here, start with the total mass of the racquet in kilograms (1 ounce = 0.02835 kg), multiply that by gravitational acceleration (9.81 m/s²) to get force in Newtons, then multiply the Newtons by the distance in meters from the axis of rotation to the balance point (one inch = 2.54 centimeters, or 0.0254 meters). The axis of rotation on the forehand is 7 cm from the handle end. Moment is a measure of how heavy the racquet feels when you hold it parallel to the ground. A head-heavy racquet will have a high Moment, which is bad not only because it tires the arm but also because it aggravates the twist on the arm from shots off the centerline of the racquet.

Torque and Impulse Reaction are the two resultant forces from an eccentric impact, such as the collision of a tennis ball with a racquet face.

3. Torque. Torque is the twist backward at the axis of rotation, resulting from impact with the ball. Derivation of a formula for Torque requires application of the principle known as Conservation of Angular Momentum. Unlike energy, momentum is conserved in any collision. Momentum is of two types: linear (straight line) and angular (rotating about an axis of rotation). If we pick any point (such as the axis of rotation 7 cm from the handle end), and sum up all of the angular momenta before impact, this sum will remain the same after impact even though the angular momenta of the ball and racquet have changed.

During the impact, there is also a force involved which is external to the racquet/ball system. This external force is a torque, or turning force, contributed by the player at the axis of rotation. But this is a non-impulsive force, therefore it is disregarded. Because the angular momentum contributed by this external force is small due to the short time that any angular acceleration of the racquet has to work, we can omit this external angular impulse from the conservation of angular momentum equation. Note on this assumption.

The angular momentum of the racquet is its swingweight ($$I$$) times its angular velocity ($$w$$); the angular momentum of the ball is its mass ($$b$$) times its velocity ($$s$$), times the distance ($$d$$) from the axis of rotation to the point of impact. From the equation arising from this conservation principle, derive an expression for the resultant angular impulse due to the racquet and ball alone, regardless of external forces. Note on this assumption.

The expression for the angular impulse due to the impact of the ball with the racquet is I(w1 - w2). From this, get the change in angular velocity of the racquet during the impact (w1 - w2)(in radians per second). See the derivation. Multiplying the change in angular velocity by the distance (r) from the axis of rotation to the mass center, or balance point, will give the difference in linear velocities, before and after impact, of the racquet’s mass center. Then dividing the difference in linear velocities by the time of impact will give the deceleration of the racquet’s mass center. By Newton’s Second Law, the deceleration of a mass is a force, so the result of multiplying a mass by a deceleration is a negative force in Newtons. Multiplying this force by its lever arm (r) gives the Torque resulting from impact. This Torque is a twist on the hand, backward, with the axis of rotation between the middle and ring fingers.

4. Torsion, or Longitudinal Torque. The screwdriver twist about the centerline of the racquet handle is called Longitudinal Torque, or Torsion. This is the cross product of two vectors: Torque and Moment. The cross product (also known as vector product) of two vectors A and B is AB sin q , where q is the angle between A and B. We assume for the tables and the use of this evaluation criterion that A and B are perpendicular to each other (Torque is perpendicular to the ground, and Moment is parallel to the ground and therefore perpendicular to Torque) so the angle is 90 degrees and the sine of it is unity, and therefore the cross product is simply Moment times Torque.

5. Impulse Reaction. Derivation of a formula for Impulse Reaction uses Conservation of Linear Momentum. Here we are considering linear, not angular, momenta with respect to only a certain direction, and not about an axis of rotation. The system momenta before impact, plus any external linear impulse (player push or pull on the axis of rotation, multiplied by the time it operates), equals the system momenta after impact — all along the direction from the player to the net. The player force is the Impulse Reaction applied by the player at the axis of rotation to restrain the axis from being yanked in this eccentric impact, and this Impulse Reaction operates for the dwell time of the ball on the racquet. Unlike Torque, Impulse Reaction is an impulsive force, so it must be included in the conservation equation. From this conservation principle we can write an equation and then solve for Impulse Reaction. See the derivation. Note that when Impulse Reaction is zero, the impact is at the Sweet Spot (center of percussion). The rankings under Impulse Reaction (positive = good, negative = bad) turn out to be the same as the rankings for Impact Force. So Sweet Spot location directly affects the amount of Impact Force.

6. Derivation of a formula for Shock is difficult but possible. The formula for kinetic energy is $$\frac{1}{2}Mv^2$$, and Shock is the change in kinetic energy from immediately before to immediately after impact, $$\frac{1}{2}M({v_1}^2 - {v_2}^2)$$ so finding the mass center linear velocities before and after will give a formula for Shock. Apply Conservation of Angular Momentum, as with Torque, to get the angular impulse and the difference in angular velocities ($$w_1 - w_2$$) of the racquet before and after impact, then multiply ($$w_1 - w_2$$) by the distance from the axis of rotation to the mass center (r) to get ($$v_1 - v_2$$), which is one factor in ($${v_1}^2 - {v_2}^2$$). The tough part is getting the other factor, ($$v_1 + v_2$$). First, convert ($$w_1 - w_2$$) to a difference in linear velocity of the impact point ($$p_1 - p_2$$) by multiplying the angular velocity difference by the distance (d) from the axis of rotation to the impact point. Knowing this linear velocity difference of the impact point ($$p_1 - p_2$$), apply the coefficient of restitution formula: the difference in velocities of the impact point and ball after impact is equal to their velocity difference before impact, times the coefficient of restitution (which is a measure of the elasticity of the ball/racquet system). Solve for the linear velocity of the impact point before impact (p1). Knowing that the linear velocities of the impact point and of the mass center are proportional in rotation and equal in translation, find the linear velocity of the mass center before impact (v1) by multiplying by the ratio r/d. Note on this step. Since ($$v_1 - v_2$$) is known, and so is (v1), v2 can be found. So now ($$v_1 + v_2$$) is known and the formula follows. Although the formula now looks hairy, that’s all there is to it. See Derivation of Shock Formula.

7. Work is equal to the kinetic energy of the racquet mass center just before impact, which is ½ Mv1², given a certain ball speed before and after. This kinetic energy of the racquet is the result of the effort, or Work, put in by the player. The less Work needed to produce a certain ball speed, the better. A Work formula requires application of Conservation of Angular Momentum, as with Torque. See the derivation. Also, as in the Shock derivation, the coefficient of restitution gives the key to finding the linear velocity of the mass center (v1) before impact, and once that is known, a formula for Work follows by the formula for kinetic energy.

8. Knowing the formula for Work, it’s an easy step to a formula for Shoulder Pull. Work is kinetic energy, whose formula is:

${½} Mv^2$

(M is mass, v is linear velocity of the mass center.)

Shoulder Pull is centripetal force (equal and opposite to the “centrifugal” force acting on the racquet), and the formula for centripetal (center seeking) force is:

$\frac{Mv^2}{R}$

(R is the distance from the shoulder to the racquet’s mass center, which is 0.01 r + 0.61 meters). The distance from the shoulder to the axis of rotation is assumed to be 0.61 meters. It is the centripetal force due to the racquet alone that is of interest, so forget the arm mass.

Dividing the Work formula by R and multiplying by 2, therefore, gives a formula for Shoulder Pull due to the racquet, in Newtons of force. One Newton = 0.225 pounds. See the Shoulder Pull derivation.

9. Shoulder Crunch is the change in the centrifugal force acting on the racquet. You will recall that the Shoulder Pull due to the racquet’s rotation was equal and opposite to the centrifugal force. Impact changes the speed of rotation, and therefore the centrifugal force. The change in centrifugal force is expressed as

${Mv_1}^2 \div R - {Mv_2}^2 \div R$

(Following the definition of centripetal force.) (M = racquet Mass, in kilograms; v1 is the linear velocity of the racquet mass center before impact, in meters/s; v2 is the linear velocity of the racquet mass center after impact; R is the distance from the racquet mass center [balance point] to the shoulder, in meters).

The excess centripetal force, which continues to act regardless of the impact lessening the centrifugal force it was countering, is the Shoulder Crunch. This is easily derived knowing Shock. Shock, or the change in the racquet’s kinetic energy due to impact, is ½ Mv1² - ½ Mv2² so by multiplying Shock by 2/R we get Shoulder Crunch. R is the distance from the shoulder to the racquet’s balance point (R = 0.01 r + 0.61 meters), so Shoulder Crunch = (2/(0.01 r + 0.61))(Shock).

10. Elbow Crunch is the excess of centripetal over centrifugal force due to impact, this excess pulling force acting at the elbow as what might be viewed as a muscle spasm. The same analysis as with Shoulder Crunch applies, except instead of using R = 0.01 r + 0.61 meters, it’s the distance from the elbow to the balance point: 0.01 r + 0.36 meters. Just multiply Shock by 2/(0.01 r + 0.36). Elbow Crunch turns out to be more severe than Shoulder Crunch due to the smaller distance to the mass center from the axis of rotation.

11. Wrist Crunch. See the foregoing discussions of Elbow Crunch and Shoulder Crunch. The R for Wrist Crunch is 0.01*r + 0.08 meters. Wrist Crunch is even more severe.

12. Tip Speed is the velocity of the racquet tip just before impact. From the Work, we can find v1, the linear velocity of the racquet’s mass center (balance point) just before impact. It will be the square root of [(2/M) * Work], because we know that the kinetic energy of the racquet on impact, i.e. the Work, = ½ Mv1². Then multiply v1 by the ratio of: the distance (which we will call e) from the axis of rotation to the tip, to the distance (r) from the axis to the mass center, or balance point. The distance e is found by subtracting the distance from the butt to the axis of rotation from the racquet’s length, in centimeters. The formula is:

$Tip\ Speed = \left(\frac{2}{M} \times Work \times \frac{e}{r}^2\right)^\frac{1}{2}$

13. Impact Force is the slowdown of the racquet on impact (Impact Impulse), divided by the time in which it occurs (the dwell time). Impulse is the change in momentum, M(v1 - v2). From the formula for Work, we know v1, the velocity of the mass center (balance point) just before impact. Getting v2 requires application of the coefficient of restitution formula, converting mass center velocities (v) to impact point velocities (p) by multiplying by the ratio of d/r. Having (p1 - p2), we convert back by multiplying by r/d, to get (v1 - v2), then reduce and multiply by the racquet mass M to get the Impact Impulse, in kg·m/s. It turns out that Impact Impulse correlates closely with distance from the tip to the center of percussion. Dividing now by the dwell time, we get the Impact Force, in kg·m/s². This is so because of Newton’s Second Law, which says that a mass (kg) times an acceleration (m/s²) is a force. The Impact Force multiplied by its lever arm gives Torque, exactly. Impact Force correlates exactly with Impulse Reaction, and the closer the sweet spot is to the tip, the less the Impact Force.

14. The new balance point resulting from adding weight can be found by the simple formula

$s = d \times \frac{m}{M + m}$

s d = shift in balance point from the original balance point, in cm = distance from the original balance point to the added weight = amount of added weight = original weight of racquet, before the weight (m) was added

Or in words: The shift in balance point ($$s$$) is equal to the product of the distance to the added weight ($$d$$), and the ratio of the added weight ($$m$$) to the new total racquet weight ($$M + m$$). Here’s the derivation:

The moment about the original balance point added by the new weight is the product of the added weight ($$m$$) and its lever arm ($$d$$), or ($$m \times d$$). After this weight has been added, the racquet will now have a new balance point, or mass center. The moment of the new mass center about the original balance point is the product of the new total racquet weight ($$M + m$$) and the lever arm ($$s$$) of the new mass center. This lever arm ($$s$$) represents the balance point shift, and the moment about the original balance point is ((M + m) × s). The two moments are equal, as they are both the result of the same weight addition, just looked at in different ways, so ((M + m) × s) = m × d and by simple algebra the formula follows.