The
physics of pole-vaulting
Pole-vaulting was not one
of the original Olympic sports in ancient Greece. Some
think that the sport was derived from the Dutch habit
of dyke-jumping, although one of the earliest pole-vaulting
stands was built in Germany in 1791. The objective is,
obviously, to get the athlete's centre of mass over
the highest bar possible. However, today's pole-vaulters
use a quite different technique to that used 100 years
ago, when athletes went over the bar with their feet
pointing downwards. Athletes now do a complex gymnastic
manoeuvre, turning upside down as the jump takes place
(figure 2a). We shall see that this is a direct result
of the technology used.
(figure 2)
The rules for pole-vaulting
that are set by the International Amateur Athletic Federation
(IAAF) the sport's governing body are extremely
liberal. There is no restriction on the length of the
pole, the materials from which it is constructed or
its energy-storage capacity. The only stipulation is
that poles should be generally smooth and not be covered
with too much adhesive tape.
Poles were originally made
out of solid wood, probably hickory. Slightly more flexible
bamboo poles were introduced in the early 1900s, mostly
by American vaulters, who dominated the sport at the
time. Basic mechanics tells us that the highest stresses
occur on the outside of a bent beam, and that a symmetrically
bent object, such as a pole, actually has a zone down
the middle (known as the neutral axis) where the stresses
are very low and even zero. There is therefore no need
for the pole to have any mass down the centre. Bamboo,
which is a naturally hollow material, is much lighter
per unit length than a solid pole yet provides
the same maximum stress. This enables an athlete carrying
a bamboo pole to either take a faster run-up or to use
a slightly longer pole.
The use of bamboo poles led
to a steady increase in the winning height of the Olympic
pole-vaulting competition (figure 2b). However, the
improvements were starting to level off by the mid-1950s,
and in the early 1960s bamboo began to be replaced by
glass-fibre poles. This led to a dramatic increase in
the winning heights. Glass-fibre poles consist of long
filaments of glass fibre ranging from 320
µm in diameter embedded in a matrix of less stiff
polymer resin. The material can readily be fabricated
into different shapes and has a high stiffness-to-weight
ratio.
Essentially pole-vaulting
involves the conversion of the kinetic energy of the
running athlete to the potential energy of the jump
using strain energy stored in the pole (the energy stored
in elastic deformation). Consider an athlete of mass
m = 80 kg running at a speed v = 10 m s1, who has
kinetic energy of 1/2mv2 = 4000 J. If this energy is
converted with 100% efficiency into potential energy
mgh, where g is the acceleration due to gravity and
h is the height jumped, then the athlete can climb a
height of 4000/mg, or just over 5 m. However, in reality,
most pole-vaulters can jump heights of nearly 6 m. So
where does the extra energy required to propel the athlete
to these greater heights come from?
It turns out that the extra
energy comes from the athleticism of the vaulter bending
the pole. Energy is stored in the pole as it is bent
or strained by the athlete's muscles, and returned to
the vaulter as the pole recoils. The strain energy comes
from the work done by the muscles of the athlete as
he or she takes off, carrying out work on the pole as
it is bent. The maximum strain energy of the pole is
ms2/2rE, where s is the maximum or "failure" stress
on the outside of the pole, r is its density and E is
its Young's modulus (i.e. its "stiffness").
Bamboo has a relatively low
Young's modulus and density, and a moderate failure
stress. Glass fibre also has a low Young's modulus and
density but a much higher failure stress than
bamboo. In fact, the maximum strain energy that can
be stored in a glass-fibre pole before it breaks is
about 2500 J, compared with just 100 J for bamboo. One
consequence of this is that glass-fibre poles can be
bent through much larger angles before breaking, which
is why athletes can use them to jump gymnastically over
the bar.
If we assume that the efficiency
of a glass-fibre pole is 50%, an extra 2500 J of stored
energy would be enough to get the athlete's centre of
mass, feet first, over the bar. In other words, our
athlete would have a kinetic energy of 4000 J plus a
strain energy from the pole of 1250 J, giving a total
energy of 5250 J. If all this is converted into potential
energy, the athlete would climb a height of 5250/80g
~ 6.5 metres.
The advances in pole-vaulting
performance are certainly not as good now as they were
in the 1960s. That has not, however, stopped researchers
from searching for further improvements, which have
included introducing carbon fibres to make the pole
still stronger and lighter (figure 2c), and allowing
the pole to vary in thickness along its length. The
latter innovation stemmed from work done in 1996 by
Stuart Burgess, then at the Department of Engineering
at Cambridge University. He demonstrated theoretically
and experimentally that, during a jump, the greatest
bending moments and hence the greatest stresses
are at the middle of a pole, while the lowest
stresses are at the ends. In other words, the ends of
a pole can be made narrower and hence lighter
without compromising the pole's performance. Burgess
therefore optimized the thickness of a pole so that
it tapered towards its end, giving a mass saving. Also,
the bending moment and hence the failure stress
increases with the stiffness of the pole for a
given section of the pole and radius of bend. The ideal
pole therefore has a low stiffness, a low mass and a
high failure stress.
Clearly, pole-vaulting is
an example of a sport in which technology has been used
to improve athletic performance. As the Olympic winning
heights in the discipline level off, it will be interesting
to see if our ingenuity can provide another technological
leap to allow pole-vaulters to jump even higher.
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