The Physics of Tennis Rasquet

The sweet spots

A tennis racquet, like a baseball or cricket bat, has two sweet spots. If a ball impacts at either of these spots, 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 spots, the impact can be quite painful. One of the sweet spots is the vibration node, located near the centre of the strings. The other sweet spot is called the centre of percussion (COP). These and some other significant spots on a racquet are shown in Fig. 1.

Fig 1

Contrary to popular opinion, the two sweet spots do not coincide with the point at which the ball rebounds with maximum speed, nor do they define the spots where the force on the hand is zero. Forces on the hand arise from three independent motions of the handle, namely rotation, translation and vibration. The vibrational component is absent when a ball strikes the vibration node. The rotational component, arising from recoil of the racquet head, exerts a torque on the hand, causing it rotate about an axis through the wrist. As a result, 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. The two forces are equal and opposite for an impact at the COP, with the result that there is then no net force on the hand or forearm. This means that there is no sudden jarring of the arm for an impact at the COP.

Vibration Node

The first two vibration modes of a freely suspended tennis racquet are shown in Fig. 2. A racquet behaves like a uniform beam in this respect, despite its round head, 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 140 Hz for a stiff frame. One node is near the centre of the strings, and the other node is in the handle. It is easy to hear this vibration if you hold the handle lightly at the node in the handle, with the handle near your ear, then strike the frame or strings. The vibration node on the strings is easily located using this technique. If you hold the handle firmly, the frame vibrations (but not the string vibrations) are strongly damped.

Fig 2

The next mode, for a uniform beam, has a frequency 2.75 times the fundamental frequency. It is not excited with any significant amplitude since the impact duration, T, of the ball on the strings is about 5 ms. The frequency spectrum of this pulse, approximately a half sine waveform, peaks at zero frequency and is zero at f = 1.5/T = 300 Hz (see Fig. 3), close to the second mode frequency. The impact will still excite string vibrations at about 500 Hz since the strings are not as strongly damped as the frame.

Fig 3

Centre of Percussion

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 in Fig. 4.

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.

Now consider a racquet that is suspended by a rod passing through a hole drilled through the handle so that the racquet can rotate freely about this axis when a ball strikes the strings. When a ball impacts on the strings, the handle will exert a force on the axis unless the ball impacts at the COP. Consequently, the COP is regarded as a second sweet spot since the force on the hand should be zero for an impact at the COP. However, the hand extends over a reasonable length of the handle, and every point in the handle will have a different centre of percussion on the strings.

Fig 4

It seems that no matter where the ball strikes the strings, there may well be a point under the hand where the force is zero, but there will always be other points where the force is not zero. In fact, that is exactly what is measured. The impact causes the racquet head to recoil, so the whole racquet rotates in the hand, exerting a force on the upper part of the hand and a force in the opposite direction on the lower part of the hand (see Fig. 5). The torque causes the hand to rotate about an axis through the wrist. However, if the impact is at the COP for rotation about the end of the handle, then the force on the upper part of the hand is equal and opposite the force on the lower part of the hand, so there is no net force on the hand or the forearm. Therefore, the forearm will not receive a sudden jolt if the ball impacts at this COP.

The COP can be located approximately by holding the end of the handle between your finger and thumb and throwing a ball onto the strings. If the handle jumps out of your hand, then you missed the COP. It is usually located about 5 cm away from the centre of the strings, as shown in Fig. 1. A more accurate measurement of the COP can be made with a piezo disk (extracted from a piezo buzzer) between the thumb and the handle to measure the force acting on the thumb.

The dead spot

Clamp the end of the handle on a table, using your hand to press on the handle, so the rest of the racquet hangs over the edge of the table. Then drop a ball onto the strings at various points. The ball will bounce best near the throat. There is a spot near the tip where the ball doesn't bounce at all. That's the dead spot. At the dead 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. The reason is that the effective mass of the racquet at that point is equal to the mass of the ball. The effective mass is the ratio of the force at that point to the acceleration at that point (F = ma so m = F/a). If a ball of mass m collides head-on with another ball of mass m at rest, then the incident ball stops dead and gives all its energy to the other ball.

Fig 5

Similarly, if a moving racquet strikes a stationary ball at the dead spot, then all the rotational energy of the racquet is given to the ball. The best place to hit a ball when serving is at the dead spot. However, when returning a fast serve, the dead spot is the worst place to hit the ball. The best spot is near the throat of the racquet since that's where the ball bounces best.

The ball

The rules of tennis specify that the ball must bounce to a height between 53 and 58 inches when dropped from a height of 100 inches onto a concrete slab. What happens in actual play is hard to predict, but a good test is to drop a ball onto the strings when the head is clamped (eg by placing the racquet on the floor and stepping on the handle near the head). When dropped from a height of say 1 metre, the ball will bounce to a height of about 0.70 metre. 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. That's because the strings absorb some of the impact energy and then give almost all of that back to the ball. The amount of energy lost by the ball depends on its compression. When dropped from 100 inches on concrete, it compresses by about 6 mm. When dropped on the strings, it compresses by about 3 mm. The bigger the compression, the more energy is lost when the ball expands back to its original shape. That means that at high impact speeds, where the ball compresses more, the energy loss is even greater. Furthermore, the fraction of the ball's energy that is lost also increases as the the ball's energy is increased or as the compression is increased.