1. PowerPoint Presentations
files can be viewed directly as PowerPoint presentations or downloaded as .ppt
files. They are designed mainly to assist students wanting ideas for tennis
More advanced projects for University
students can be found in the list of publications
and in the additional information provided below.
This is a photo
of the Stevens machine owned by Tennis Australia. It is still used in some
countries to measure ball stiffness as specified by the rules of tennis. It
includes a circular hole in each side to test whether the ball is too large,
too small or just right (according to the rules). The test procedure involves
(1) Squash the ball a few times in
a few directions to make sure it is round.
(2) Apply a force equivalent to a 8.165 kg or 18 lb
mass sitting on the ball.
(3) Wait 5 seconds then measure the deformation under
this load (called “forward deformation”)
(4) Continue to compress the ball until its initial
diameter has been reduced by one inch.
(5) Remove this compression until the load on the ball
is again 8.165 kg
(6) Wait 10 seconds then measure the deformation under this
load (called “return deformation”)
web site has
a movie showing the modern way to do it (using a robot). The
simplest way to do it at a University is to ask one of your Engineering
Departments to help you set up one of their small materials testing machines to
record the force and compression on whatever ball you want to test. They will
probably be happy to help you since they normally test less interesting objects
like blocks of concrete or metal rods.
BALL TEST RESULTS (July 2007)
Ball test results conducted in my
lab, plus wear tests on court, can
be found at
If you click on the authors names at the bottom of this article you
will discover lots of other interesting articles on the technical aspects of
tennis, all published in RSI magazine, but you won’t find any photos or stories
about famous players or their rankings. You will only find useful information
in RSI magazine.
hundreds of different tennis strings on the market that players can choose
from. Exactly why some players prefer one string to another is a bit of a mystery,
but a physical measurement of the properties of different strings provides some
of the answers. There are only four basic types of string, namely natural gut,
nylon, polyester and kevlar in order of increasing stiffness. Stiffness here
refers to how far a string will stretch longitudinally (lengthwise) when it is
pulled to a given tension. Tennis strings are a lot stiffer than rubber tubes
and a lot softer than steel guitar strings.
strings appeared on the market, players noticed an increase in ball spin. It
took a while for manufacturers to realise that slippery strings generate more
spin than rough textured strings. These days, most top players prefer to use
polyester strings since they a more slippery and snap back into place at the
end of each shot, giving the ball a bit more spin. That is essentially how spaghetti strings work.
When a string is installed in a tennis
racquet, it is pulled to a desired tension, typically about 60 lb or about 28
kg and then tied off at both ends. As soon as the string is tied off, the
tension starts dropping dramatically at first and then more slowly as the days
and weeks pass by. The tension also drops slightly every time the racquet is
used to hit a ball. Some of the
equipment I have used to pull and hit strings and to measure how far they
stretch lengthwise or sideways, is shown below. I use a sample of string about
40 cm long clamped at each end between metal jaws, with a load cell at one end
to measure the tension when the string is stretched either lengthwise or
sideways. A laser beam passing through a 1 mm grid is used to measure the
sideways stretch of the string when it is impacted with a hammer mounted as a
pendulum. The 1mm grid can be seen in the photo attached to the back end of the
hammer. The sideways force on the string is roughly equivalent to a 100 mph
Hammer pulled and held in position ready to
slam sideways into the string. The string is stretched lengthwise by means of a
screw thread attached to the blue clamp at the left hand end of the string.
Red clamp at right hand end of string,
attached to a load cell. As soon as the blue clamp is locked into a fixed
position, the tension starts dropping rapidly. When the hammer hits the string,
the tension increases rapidly and then drops back to its pre-impact value as
soon as the hammer bounces off the string. The hammer bounces off the string at
about 95% of its incident speed, regardless of string tension, string type or
string diameter. All tennis strings are essentially equal in this respect.
Measure String Tension ON-LINE!
Even though a
racquet stringer can string your racquet at any tension you like, the actual
string tension will be different. String tension drops rapidly as soon as the
string is tied off. Furthermore, a huge force is applied to the frame as each
new string is added, pulling the frame out of shape and affecting the tension
of the strings already installed. The tension drops over time until it is time
for a restring.
to hit their strings to hear the “ping” sound to determine whether the strings
are tight or loose. The ping
frequency of the whole stringbed is essentially the same as the vibration
frequency of a single string with a length equal to the square root of the area
of the stringbed. For example, if
area = 650 sq cm then length = 25.5 cm. QuickTime sound files for a 32 cm long
string at various tensions are attached. You can use these files to compare
with the ping frequency of your own racquet. Best results (the best “pings”)
are obtained by bouncing a golf ball off the strings, with string dampeners
removed. The tension in your racquet won’t be the same as the tension in the 32
cm string since the vibration frequency also depends on string diameter and
head area, but the result will be roughly correct. You can easily proportion the results since the ping
frequency is inversely proportional to string length. A 16 cm long string will
vibrate at twice the frequency of a 32 cm long string of the same diameter at
the same tension.
you can use the sound files to determine if the string tension has dropped
since the last time you tested them. In any given racquet, the ping frequency
of the stringbed is proportional to the square root of the string tension,
regardless of the string diameter and head size. But you can’t use this
technique to compare the string tension in your racquet with another racquet
unless you take into account the different head size, string diameter and
string type. For example, polyester is denser than nylon so a polyester string
will vibrate at a lower frequency than a nylon string of the same diameter and
length and tension.
The movies below are QuickTime movies. If
you don’t have QuickTime it is available for free for both Mac and Window
systems at http://www.apple.com/quicktime/download/mac.html
The 10Mb Balance and Swing Weight movies
might take a minute or two to download.
4. Measuring Balance & Swing
point of a racquet is the distance from the butt end of the handle to its
centre of mass or its centre of gravity (an old-fashioned term).
Swing Weight is
a term commonly used in sports to denote the moment of inertia of a bat or club
or racquet. It can be measured by mounting the implement as a pendulum and
timing the period for one oscillation. A more accurate result is obtained by
timing 10 oscillations and then divide the result by 10. The parallel axis
theorem can be used to work out the swing weight about any axis parallel to the
one shown in the movie (as described in the Physics
and Technology of Tennis book).
In tennis, swing weight refers to the moment of inertia about an axis 10
cm from the butt end, and is typically about 300 kg. cm2 for most racquets.
can be measured in terms of quantity called the ACOR which stands for Apparent
Coefficient of Restitution. The
ACOR itself is defined as the ratio of outgoing ball speed to incident ball
speed for a ball incident on a racquet initially at rest. A simple measurement
method is shown in the ACOR movie where I have filmed a ball incident at low
speed on a racquet hanging freely in a metal frame. There are 7 hooks at the
top of the ball pendulum, spaced 20 mm apart horizontally, so the ball can
impact different points on the strings from the centre to the edge of the
racquet. The hooks are mounted on a horizontal metal bar that can be moved to
16 different positions spaced 20 mm apart in the vertical direction. That way,
it is possible to map out the variation of ACOR over the whole string plane or
to map out vibrations in the handle (with a piezo disk attached to the handle).
The film was
taken from an angle to see the setup more clearly. To measure the incident and
rebound ball speed, the camera would be located so it views at right angles to
the path of the ball. The ACOR does NOT depend on whether the top end of the
handle is supported by a length of string (as in this movie) or whether it is
hinged or whether it is clamped. The ball rebounds before the reflected wave
off the handle gets back to the impact point, so the ball has no way of knowing
how the handle was supported.
Click photo or here
to load ACOR movie. Photo shows hooks to support the top end of the ball
pendulum. The “3-2-1-Go” countdown
helps me to find and download the portion of the film that I want to analyse.
Results of some of these
Court speed is measured officially
by the ITF using a Sestee machine. For research or club purposes, the speed of
a court or any other playing surface can be measured by filming the bounce of a
ball with a video camera. Some tips and various calculations required for such
a measurement are outlined in a pdf file that you can download.
See also the attached 0.5Mb movie or avi file showing a bounce on Rebound Ace, the
hardcourt surface that used to be used at the Australian Open. The top surface
is green acrylic paint mixed with fine sand to give a slightly rough surface.
Underneath is a layer of rubber to soften the impact under foot.
The movie is split into a top and a
bottom half. Each half is recorded at 25 frames/sec (40ms between each image)
but the top half is recorded 10 ms before the bottom half, and each half is
further split in half to give 50 fields/sec which makes it easier to determine
when the ball bounced and to measure the spin of the ball. In other words there
are 100 images/sec. The ball was incident at about 10 m/s with a small amount
of backspin. It is not necessary to measure ball spin to measure court speed
and bounce, but it helps to determine whether the ball was sliding throughout
the bounce (as required) or whether it gripped the court at some stage during
Hold a tennis racquet horizontally,
toss it in the air so it rotates once around a horizontal axis, and catch it by
the handle. The racquet will land upside-down, as shown in the movie. In the movie I taped a small red ping-pong
ball on one side of the racquet so it would be easier to see what was going on
and I filmed the event on the University grass courts with the physics building
in the background. The flipping-over effect is described by the “Tennis Racquet
Theorem”. A slow motion movie
version can be viewed here.
Any object has three perpendicular
axes in the x, y and z directions. In general, the moment of inertia (I) about
each axis is different, with Ix > Iy > Iz. If an object starts rotating
about the x or z axis it will continue to rotate about that axis without flipping
over. But if it starts to rotate about the y axis then it will end up spinning
about both the x and z axes as well.
It is the same with a book or a
packet of cornflakes. Only one of the three axes causes flipping, and it is the
one not with the smallest or the largest swing weight but the one with the
medium size swing weight. Flipping over can be good or bad depending on
circumstances. If the racquet happens to be a cat falling several floors out a
window, this flipping over effect is good for the cat. If the racquet happens
to be a fresh piece of buttered toast with jam on top, then the flipping over
effect is bad, especially for the carpet.
If you are an Olympic diver or gymnast, flipping over several times
while doing a few somersaults can earn you extra points, perhaps leading to
fame and fortune. If you are a tennis player, then watch out for the racquet
twisting in your hand as you swing it. That could be good or bad depending on
whether you want it to twist or not.
A double pendulum consists of one
pendulum mounted below another so that both can swing together but the motion
of one influences the other. If left to swing for a while the motion becomes
chaotic, but the first half cycle is quite reproducible. Common everyday examples include the
upper and lower arms and the upper and lower legs of a person, or a bat or
racquet or club connected to a human arm. The swing of a bat or club or racquet
can therefore be modelled by considering the motion of a double pendulum. The three
movies below show how this can be done experimentally for cases where
(a) a torque is applied to help
swing the upper pendulum (to simulate the effect of a muscle torque) See Movie (0.4Mb)
(b) the motion of each pendulum is
slowed to have a much longer period of oscillation by mounting each arm near
its centre of mass. See Movie (0.5Mb). This makes it
easier to analyse film taken at only 25 or 30 frames/sec.
(c) a near perfect bat (or club or
racquet) can be designed so that nearly all of the energy in the arm and the
bat ends up in the ball, without any follow-through of the bat or the arm. See Movie (0.5Mb). This requires a change to the rules of the
game, unfortunately, to allow for
heavier balls (around 1 lb).
9. Weird tennis action
Why do players stretch the fingers of their free hand? Don’t
ask me, I don’t know. Is it to increase the moment of inertia of the hand or
the arm to keep it steady? My
guess is that when players tense their muscles to hit a shot, they tense all
their muscles, not just the ones they need to tense (about 95% of them).