Formula: New Balance Point
New Balance Point distance (\(x\)) from original balance point, resulting from adding weight (\(m\)) at a distance (\(y\)) from the original balance point of a racquet having an original weight (\(M\)).
\[ x = y \times \frac{m}{M + m} \]
Or in words: The shift in balance point (\(x\)) is equal to the product of the distance to the added weight (\(y\)), 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 (\(y\)), or (\(m \times y\)). 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 (\(x\)) of the new mass center. This lever arm (\(x\)) represents the balance point shift, and the moment about the original balance point is ((\(M\) + \(m\)) × \(x\)). The two moments are equal, as they are both the result of the same weight addition, just looked at in different ways, so ((\(M\) + \(m\)) × \(x\)) = \(m\) × \(y\) and by simple algebra the formula follows.
| \(A_x\) | = | Impulse Reaction, the translational force acting at the axis of rotation due to impact, in Newtons. Note that when \(d\) = \(q\) (\(q\) is the distance from the axis of rotation to the center of percussion), the expression within the second set of parentheses becomes zero. |
| \(a\) | = | linear acceleration of the mass center, in m/s² |
| \(b\) | = | mass of the ball, in kg |
| \(c\) | = | coefficient of restitution of the racquet/ball system |
| \(d\) | = | distance from the axis of rotation to the impact point, in cm |
| \(e\) | = | the distance from the axis of rotation to the tip |
| \(F\) | = | force applied at mass center, in Newtons |
| \(I\) | = | moment of inertia (swing weight) of racquet, in kgf/cm² |
| \(I_5\) | = | moment of inertia (swing weight) of racquet at 5cm from the butt, in kgf/cm² |
| \(I_7\) | = | moment of inertia (swing weight) of racquet at 7cm from the butt, in kgf/cm² |
| \(I_{10}\) | = | moment of inertia (swing weight) of racquet at 10cm from the butt, in kgf/cm² |
| \(I_a\) | = | moment of inertia (swing weight) of racquet at distance \(a\) from the butt, in kgf/cm² |
| \(M\) | = | mass of the racquet, in kg |
| \(m\) | = | mass in kg |
| \(ω\) | = | angular velocity of racquet, in radians/s |
| \(p\) | = | linear velocity of impact point, in m/s |
| \(r\) | = | distance in cm from mass center (balance point) to axis used in the stroke |
| \(s\) | = | ball velocity, in m/s (positive is away from player) |
| \(s_1\) | = | velocity of ball before impact, in m/s |
| \(s_2\) | = | velocity of ball after impact, in m/s |
| \(T\) | = | torque at axis of rotation, in Nms |
| \(t\) | = | dwell time, or duration of impact, in seconds |
| \(v\) | = | linear velocity of the mass center, in m/s |
| \(v_1\) | = | linear velocity, just before impact, of racquet mass center, in meters/second |
| \(v_2\) | = | linear velocity, just after impact, of racquet mass center, in meters/second |