Formula: Sweet Spot
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 kgf/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.)
| \(A_x\) | = | 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 |