PHYSICS OF TENNIS
The behaviour of racquets and balls is governed mainly by Newton’s
three Laws of Tennis. These laws are fully explained in our book The Physics and
Technology of Tennis. The following is a summary of a few topics of interest to
give you a flavour of how physics relates to sport and to the real world of
everyday objects.
1. The sweet spots of a
racquet
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 player
will feel some jarring and vibration of the handle. 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 below.
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.
2. Vibration Nodes
The first two vibration modes of a freely suspended tennis racquet are
shown below. 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 180 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.
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 , 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.
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A metal tube will vibrate nicely when struck with a metal rod, but it does not vibrate when struck with a tennis ball. How come? The impact duration, about 0.005 sec, is too long to excite any vibration with a period shorter than about 0.002 sec, especially if the tube is struck at the fundamental vibration node (its sweet spot). |
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SEE a racquet vibrate
The vibration of a racquet is usually too small to see by
eye, but a cardboard racquet vibrates the same way and shows the vibration modes
clearly (see photo below). The top end was vibrated back and forth by about 1
mm at a frequency of about 30 Hz. The nodes in the frame and in the two strings
are easily seen. The photo on the left is with the vibration generator switched
off.
HEAR a
racquet vibrate
The vibration frequency of a racquet depends on the
stiffness of the frame. This is one way of determining whether you are using a
stiff or a flexible racquet. A stiff racquet vibrates at 180 Hz or more. A
flexible racquet vibrates at 140 Hz or less. QuickTime sound files for a
racquet held near the ear and tapped by finger or by a ball near the tip are
attached for frequencies of 120, 140, 160,
180, 200 and 220 Hz. These sounds are too low in
frequency to be heard properly using the internal speakers of a computer. You
will hear only some metallic clicks when the racquet is struck. Internal
speakers usually have almost no bass response. You will need to connect
external speakers, with fresh batteries if they are battery operated.
FEEL a racquet vibrate
The best way to feel a racquet vibrate is to hit a ball on
the tennis court. What is it that you “feel”? Does it feel good or bad? Why do different racquets feel different? Does a $400 racquet feel a lot better
than a $100 racquet, and if so in what way does it feel better? These are not
easy questions to answer. I have never heard a good explanation or description
in terms of the physics of feel. Players say that some racquets feel more
powerful than others, but how do they know unless they measure the ball speed,
which they don’t?
Can you feel the difference between a 140 Hz vibration and
a 160 Hz vibration? One way to test this is to put your hand on the speaker
used to play the sound files. It will help to use an external speaker rather
than the inbuilt speaker in your computer. You can feel a thump every time the
racquet is hit but that’s because you can easily feel the change in amplitude
(or size) of the vibration. I find that high frequencies tickle my hand more
than low frequencies, especially if the soft part of the palm of my hand rests
gently on the speaker.
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 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.
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.
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. 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. 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.
4. The dead spot of a racquet
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.
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. 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.
5. Tennis
balls
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.
6. Ball Spin
The modern game of tennis is dominated by the amount of spin that players can impart to the ball. The change from small 9 inch heads in wood racquets to large 10.5 inch heads in graphite racquets allowed players to hit with more topspin since the ball was less likely to clip the frame. This also allowed players to hit the ball harder since balls hit with topspin dive down more sharply onto the court after they clear the net. By hitting the ball harder, players generated even more topspin, which allowed them to hit the ball even harder. The modern game is played at a much faster pace than in the wood racquet era, not because modern racquets or players are more powerful but because racquet heads are now an inch or two wider, allowing players to hit the ball with much more topspin.
To extract as much topspin as possible from a stroke, players have learnt to swing up at the ball and to tilt the racquet head forward, That way, a ball coming off the court with topspin can be returned with topspin. The spin direction must be reversed to achieve this result. Two QuickTime movie files are attached showing how this is done, one by myself and one by Federer. The physics of each shot is the same, the only real difference being that Federer has a more elegant style. In my case, I filmed the shot at 25 frames/sec. The top half of the film is taken 10 ms before the bottom half so the magnitude and direction of spin could be measured more easily. The film of Federer was taken at about 1000 frames/sec judging by the fact that the ball sits on the strings for about 5ms and is seen for 4 or 5 frames on the strings.
7.
PowerPoint Presentations and Movies
I often get requests from students wanting ideas for experiments or
projects relating to the physics of tennis. I have prepared a few PowerPoint presentations on this subject, including some of
the basic physics behind each experiment. They can be viewed directly on the
web or downloaded as .ppt files. This page also contains some movies concerning
various tennis experiments.
7. Further reading
Brody, H. (1979) Physics of the tennis racket. American Journal of
Physics 47, 482-487.
Brody, H. (1981) Physics of the tennis racket II: The sweet spot. American
Journal of Physics, 49, 816-819.
Brody, H. (1987) Tennis Science for Tennis Players, University of Pennsylvania
Press.
Brody, H. (1995) How would a physicist design a tennis racket?. Physics Today,
48, 26-31.
Howard
Brody, Rod Cross and Crawford Lindsey, The Physics and Technology of Tennis
(available from www.racquettech.com)
Rod
Cross and Crawford Lindsey, Technical
Tennis (also available from www.racquettech.com)
Rod Cross,
Crawford Lindsey and Howard Brody at the TST2003 Tennis Conference organised by
the International Tennis Federation in London.