Formula: 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.
\[ Moment = M \times \frac{r}{100} \times 9.81 \]
(Nm about axis of rotation)
So the formula for Moment is simple: weight × 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.
| \(A_x\) | = | 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 |