Tennis Matrix
Exploring the Physics Behind Tennis
Strings and Balls
Dead Spot
Dead spot: a spot near the tip where the ball doesn't bounce at all. At this spot, all of the energy of the ball is given to the racquet, and the racquet does not give any energy back to the ball because the mass of the racquet at that point is equal to the mass of the ball. The effective mass is the ratio of the force to the acceleration at that point.
Sweets spots
In racquet sports, the desired point of contact between the ball and the racquet is called the “sweet spot”. If a ball impacts at the sweet spot, the force transmitted to the hand is sufficiently small that the player is almost unaware that the impact has occured. If the ball impacts at a point well away from the sweet spot, the player will feel some jarring and vibration of the handle.
Forces on the hand come from three independent motions of the handle: rotation, translation and vibration. The vibrational component is absent when a ball strikes the vibration node. The rotational component, exerts a torque on the hand. A force is always exerted on the upper part of the hand, and a force in the opposite direction is always exerted on the lower part of the hand. In tennis, there are two sweet spots and one dead spot.
Vibration Node
The first two vibration modes of a freely suspended tennis racquet are shown below. A racquet behaves like a uniform beam since the centre of mass of a racquet is near the centre of the racquet. The fundamental mode has a frequency of about 100 Hz for a relatively flexible frame or about 180 Hz for a stiff frame. One node is near the centre of the strings, and the other node is in the handle.
F = ma
m = F/a
A good place to hit a ball when serving is near the dead spot. However, when returning a fast serve, the dead spot is the worst place to hit the ball. The best spot is nearer the throat of the racquet since that's where the ball bounces best.
Center of Percussion
The center of percussion (or center of the racquet face) provides the best "feel" on groundstrokes, and provides the most stability.
Consider a racquet that is freely suspended by a long length of string or balanced vertically on the end of its handle. If a ball impacts at the centre of mass (CM), the racquet will recoil at a speed V. All parts of the racquet will recoil at the same speed V. If the ball impacts at any other point on the strings, the racquet will recoil and it will also rotate about its CM. The whole racquet then moves away from the ball with a speed V1 due to the recoil , but the handle simultaneously moves towards the ball with speed V2 due to rotation of the racquet. If there is any point in the handle where V1 = V2, then that point will remain stationary and the rest of the racquet will rotate about that point as shown below.
The axis of rotation is called the conjugate point with respect to the impact point, and the impact point is called the centre of percussion (COP) for that particular axis of rotation. The axis and the COP form a pair of conjugate points. For an impact near the tip of the racquet, the axis of rotation is about half way between the end of the handle and the CM. For an impact near the throat of the racquet, the axis of rotation is beyond the end of the handle.
Impulse reaction - a push or a pull on the hand resulting from an impact. Impacts above the center of percussion will be defined as a pull (negative force) on the hand and below will be a push (positive force). A positive impulse reaction is considered to be better because it leads to less impact force. An impact at the center of percussion, leads to an impact reaction of zero. An impulse reaction is calculated in units of force, because it is a translational force just like a push or a pull on the axis of rotation, which in this case, is the hand. The unit of measurement of force in the metric system is the Newton (1 Newton = 0.2248 lb force). These forces, from the impact to the racquet, generate shock.
q= I / Mr
'Q' is the distance from one's hand to the center of percussion (which is in cm). 'M' is the racquet's mass in kg. 'R' is the distance from one's hand to the center of mass. 'I' is the racquet's swing weight about the hand, which is also known as the moment of inertia. Using this formula to solve for 'q' gives you the location of the center of percussion.
Tennis Balls
The ball loses about 45% of its energy when dropped on concrete, but it loses only 30% of its energy when dropped on the strings.
Why?
This is because the strings absorb some of the impact energy and then give almost all of that energy back to the ball. The amount of energy lost by the ball depends on its compression of the strings. The bigger the compression, the more energy is lost when the ball expands back to its original shape meaning that at high impact speeds, where the ball compresses more, the energy loss is even greater than a ball at low impact speed.