🎾 Tennis - Raquet - Physics - Pdf¶
Giới Thiệu¶
Tennis - Raquet - Physics - Pdf — tài liệu 7 trang từ thư viện sách tennis.
Chủ đề chính: Physics, Racquet
Tóm tắt nội dung (trích từ tài liệu gốc): Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html 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 a sweet spot. If a ball impacts at the sweet spot, the force transmitted to the hand is sufficiently s
Lưu ý: Nội dung dưới đây được trích xuất tự động từ PDF gốc tiếng Anh, giữ nguyên ngôn ngữ để bảo toàn độ chính xác kỹ thuật.
Nội Dung Gốc (Tiếng Anh)¶
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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 a 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. The sweet spot is a vibration node, located
near the centre of the strings. Another potential sweet spot is the centre of percussion (COP).
These and some other significant spots on a racquet are shown below.
Contrary to popular opinion, the sweet spot does not coincide with the point at which the ball
rebounds with maximum speed, nor does it locate the spot 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 COP shown in the diagram above is located close to the node point when the racquet is
freely suspended, but it shifts into the throat area of the racquet when the racquet is held in the
hand. Consequently, the COP shown in the diagram is NOT the sweet spot that players talk
about.
2. Vibration Nodes
The first two vibration modes of a freely suspended tennis racquet are shown below. A racquet
1 of 7 7/6/2016 9:04 AM
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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.
2 of 7 7/6/2016 9:04 AM
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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).
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 7/6/2016 9:04 AM
3 of 7
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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
4 of 7 7/6/2016 9:04 AM
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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 often regarded as a second sweet spot since the force on the hand
should be zero for an impact at the COP. However, the hand adds an additional mass of about
500 gram to the handle, and it shifts the location of the COP to a position near the throat area of
the racquet. The details of this effect are described in an Am J. Phys article that you can
download here.
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
5 of 7 7/6/2016 9:04 AM
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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. 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.
The amount of spin depends on a whole bunch of factors, including the speed, spin and angle of
the incident ball, the speed, approach angle and tilt of the racquet, the type of string etc. If the
ball is rising at the same speed as the racquet, then no topspin will be generated, as shown in this
movie clip, since the racquet does not brush upwards against the back of the ball. It is clear from
this result that more topspin will be generated if the player strikes the ball while the ball is
falling, rather than when the incoming ball is rising.
Detailed measurements and calculations of ball spin off different strings, as well as movie clips
showing the results, can be found at http://twu.tennis-warehouse.com/learning_center
/spinexperiment.php (NEW APRIL 2010)
6 of 7 7/6/2016 9:04 AM
Tennis Raquet Physics http://www.physics.usyd.edu.au/~cross/tennis.html
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.
7 of 7 7/6/2016 9:04 AM