What do Steve Backley, Johnny Wilkinson, Nigel Kennedy and the captain of the QE2 have in common? They have all experienced the ‘sweet spot’ – a term that I use to encompass those unique instances in sport, music and natural phenomena where everything just seems to come together to produce the perfect javelin throw, the perfectly struck kick, the perfect musical note or even an enormous ocean wave, higher than the QE2 itself. Sweet spots are rare because they only occur under very precise conditions. Physicist Len Fisher examines how science is now helping to unravel their secrets.
Launch it Like Backley
Some commentators at the Olympics spoke of javelin throwing as a simple matter of rapid arm speed and accuracy, but one perceptive commentator spoke of Backley as being ‘hot on rhythm and timing’. He was dead right. If speed and accuracy were all there was to it, Backley’s 1990 record throw of 90.98 metres would not have been a unique moment that he was never able to repeat. The secret of his great throw was not just arm speed, but the fact that he used perfect timing to make his body act like a whip during the throw. A well-designed whip narrows progressively towards the tip – a feature that allows the energy imparted to the handle to be transmitted undiminished to the tip, which moves at supersonic speeds (the ‘crack’ that you hear is actually a supersonic bang). Backley was similarly able to transmit his energy of motion along his body to his forearm and hence to the javelin at the moment of release in a perfectly timed ‘sweet spot’ moment. Such timing, which fast bowlers like Steve Harmison also use, requires an extraordinarily precise ‘feel’ that is difficult to gain and, as Backley can attest, very easy to lose.
Kick it Like Wilkinson; Bend it Like Beckham
Expert kickers like Johnny Wilkinson and David Beckham use a similar technique to javelin throwers and fast bowlers, turning their bodies into whips to deliver maximum energy to the ball. Wilkinson’s kicking coach, Dave Aldred, talks of ‘feel’, ‘focus’ and ‘bringing everything together’ in the kick. What he is really talking about is timing; the ability to coordinate all of the body’s movements so as to deliver maximum energy to the foot at the moment of impact. The size of the foot may also help. My physicist colleague Jeff Odell pointed out an unexpected similarity with snooker, where a correctly struck cue ball stops dead and transfers all of its energy to a centrally-struck object ball. This only happens when the two balls have the same weight. Is it a coincidence that a rugby ball weighs around as much as a human foot, while a javelin weighs about as much as a human forearm?
As an Australian, I was stunned by Johnny Wilkinson’s winning kick in the 2003 World Cup Final . I was even more stunned when Dave Aldred told me that it was a mis-kick, and not a ‘sweet spot’ moment at all. If Wilkinson could do that with a mis-kick, what hope is there for the rest of us?
Hit it like Henman
Today amateur tennis players can hit tennis balls as sweetly as did the champions of the past. How? Because modern racquets have enlarged sweet spots, produced by changing the size, shape and mass distribution in the heads and by varying the thickness and tension of the strings across the racquet. All of this means that the ball need not be struck so precisely.
Scientifically speaking, the ‘sweet spot’ (technically known as the centre of percussion) in sports like tennis or cricket is that point where all the power of the bat or racquet is transferred to a ball without jarring the hand and arm of the striker. Scientists use complicated mathematics to calculate its position, but there is a simpler way. Just hold the bat or racquet loosely between the fingers, and have a friend strike it where they think the sweet spot is. If they hit it too high, it will fly backwards out of your hand. If they hit it too low, it will try to rotate so that that bottom moves backwards but the handle moves forwardout of your hand. If they hit it on the sweet spot, though, it will swing like a pendulum, and won’t come out of your hand no matter how loosely you are holding it.
A Life on the Ocean Wave
Nature has her own sweet spots. None is more impressive than a rogue wave formed when several mid-ocean waves get together. This process usually produces a series of waves which are no more than ten metres high and which spread out and lose height as they travel. Very occasionally, circumstances conspire to produce a quite different sort of wave; a rogue wave (known technically as a non-linear dispersive wave), where the wave’s tendency to spread out is balanced by other factors, so that it maintains its shape and travels along unchanged in size or shape. Such waves can be up to thirty metres high, and are very different in shape to an ordinary wave, being almost flat at the front so that it is impossible for a normal small boat to ride up and over them. Few people have survived intimate contact with such waves. The British rowers who recently met one while attempting to cross the Atlantic were one of the fortunate few. Another survivor was Commodore Ron Warwick, who was captain of the QE2 when it was struck in mid-Atlantic by a rogue wave, with the wave slamming down on it with a force that bent the aft deck. He later told me that he was ‘glad that he hadn’t been in a smaller ship’.
A rogue wave is an example of a soliton – a single wave that can travel for large distances without losing its shape. Not all solitons are as frightening as a rogue wave. In fact, I have even surfed on one.
Riding the Severn Bore
My private soliton was the Severn bore, which travels up the River Severn several times each month at times dictated mainly by the phase of the moon. The water at the front of the wave piles up as the river narrows and becomes shallower, but this is balanced by loss of water from the back of the wave, so that the bore travels unchanged for many miles. The bores vary in height, but even the biggest is only a metre or so high, and easy to surf – or so I was assured by my mentor Phil Williams, who has stayed on one for a full five miles. He was disappearing in the distance as we passed under the bridge on which my wife was standing, only for my surfboard to reappear on the other side without me on it.
Bouncing Bridges and Bras
Many objects have a natural rate of vibration – a built-in sweet spot. Bridges, for example, often vibrate spontaneously in a vertical direction at one or two vibrations per second. If a group of people were to march across the bridge with their feet striking the road surface at this rate, the natural vibrations of the bridge would be reinforced, with potentially dangerous consequences. This is why it is usually illegal for groups of people to march across bridges. Sometimes it doesn’t need people, though. The Tacoma Narrows Bridge in Washington State, U.S.A. became a famous tourist attraction in the later 1930s because even the gentlest wind would reinforce the natural vibrations of its thin box-girder construction so that it visibly bucked and swayed. Then one day in 1940 the wind became a little stronger, and the bridge bucked itself to destruction – an event caught on camera, and still used as an awful warning to embryonic civil engineers.
Tall buildings also have a natural frequency at which they sway. Unfortunately, this frequency is often close to that of the lateral shearing vibrations generated by earthquakes, which can reinforce the building’s vibration until it collapses. Recognising this problem, engineers have come up with some ingenious solutions. One is to change the natural vibration frequency by putting a heavy swimming pool on top of the building. Another is to put the building on rollers, so that the earth can shake backwards and forwards underneath without affecting the building.
Even female athletes can have problems with unwanted reinforcement of vibration. A weight hung on a spring vibrates at a natural frequency determined by the size of the weight and the strength of the spring. Measurements performed by my colleague Jeff Odell have shown that, when the spring is a bra strap, the natural frequency tends to be close to that with which the feet strike the ground when running. The solution adopted by sports bra manufacturers is to make the spring stronger, so that the natural frequency is higher. My own solution, not yet adopted, is to make the bra strap hollow and fill it with oil, so that it acts like a shock absorber in a car, damping out the natural vibrations by dissipating the energy as heat when the oil is forced back and forth.
In the Balance
A pencil balanced on its point has a unique sweet spot where it will remain in balance, but the pencil must be dead vertical. The slightest deviation from this condition means that the pencil will fall over; in scientific terms, the equilibrium is unstable. Circus performers get around the problem (for example, when balancing a pencil on the chin) by moving the point of contact imperceptibly sideways so that it stays directly under the centre of gravity.
It would seem even harder to balance a spinning ball on a fingertip, but not so. The spinning ball acts as a gyroscope, with built-in stability due to its spin. The problem comes when there are multiple points of contact with the fingertip, creating ‘couples’ that twists the ball off its spinning axis. The answer? Balance the ball on the fingernail, so that it is contact with a single point instead of a broad area of the finger.
The hardest trick of all for the amateur would appear to be balancing on a high wire, but the scientific answer is simple. Just hang heavy weights on the ends of the pole so that they are below the level of the wire. Then your overall centre of gravity will be below the wire, and your position after any involuntary tilt will be restored by the weights.
Striking a Sour Note
A well-tuned piano should be the epitome of the sweet spot in music. Not so, as I found when talking to professional piano tuner David Widdicombe. Every piano in the world, he told me, is out of tune! A relatively simple mathematical argument (known to Bach and others) shows that pianos can never be perfectly tuned so that all of the notes are in a perfect mathematical relationship to each other. If all of the C’s are in tune, for example, some of the other notes in the scale are bound to be out of tune with them. The art of the piano tuner is to detune allof the notes along the keyboard equally so that nobody notices. It is a musical solution to an intractable mathematical problem.
Science may not be able to tune a piano, but it can certainly help to tune the human soprano voice to achieve that ideal ‘sweet spot’. A device invented by my Australian colleagues Professor Joe Wolfe and Dr. John Smith enables sopranos more accurately to ‘tune’ their vocal tracts to different notes by placing a tiny microphone in the mouth and watching how the harmonics displayed on a screen change as they change the shape of their mouths. It turns out that higher notes are best produced by lowering the jaw and widening the mouth – in other words, smiling. Next time you see an opera, watch the soprano. You will see that the higher she sings, the more she smiles!
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