Category Archives: Science

Arts, History, Science, Society

Quantum Leaps: Archimedes…

c. 287 – 212 BC

“Give me a place to stand on, and I will move the earth,” Archimedes is reputed to have declared to the people of Syracuse. The practicalities of an earth-bound life may have denied him that particular pedestal but arranging for his patron King Heiron to move a ship by pushing a small lever was considered only a slightly miraculous feat. With such audacious displays, along with his brilliance as an inventor, mechanical scientist and mathematician, it is no wonder Archimedes was so popular and highly regarded among his contemporaries.

The Mathematician

It was not only his peers, however, who benefited from Archimedes’ work. Many of his achievements are still with us today. First and foremost, Archimedes was an outstanding pure mathematician, “usually considered to be one of the greatest mathematicians of all time,’ according to the Oxford Dictionary of Scientists. He was, for example, the first to deduce that the volume of a sphere was 4πr³ x 3, where r is the radius. Other work in the same area, as outlined in his treatise On the Sphere and Cylinder, led him to deduce that a sphere’s surface area can be worked out by multiplying that of its greatest circle by four: or, similarly, a sphere’s volume is two-thirds that of its circumscribing cylinder. He calculated pi to be approximately 22/7, a figure that was widely used for the next 1500 years.

The Archimedes Principle

Archimedes also discovered the principle that an object immersed in a liquid is buoyed or thrust upwards by a force equal to the weight of the fluid it displaces. The volume of the displaced liquid is the same as the volume of the immersed object. Legend has it that he discovered this when set a challenge by King Heiron to find out whether one of his crowns was made of pure gold or was a fake. While contemplating the problem Archimedes took a bath and noticed that the more he immersed his body in the water, the more the water overflowed from the tub. He realised that if he immersed the crown in a container of water and measured the water that overflowed he would know the volume of the crown. By obtaining a volume of pure gold equivalent to the volume of water displaced by the crown and then weighing both the crown and the gold, he could answer the King’s question. On making this realisation, Archimedes is said to have leapt from his tub and run naked along the street shouting ‘Eureka!’, ‘I have found it!’

Levers and Pulleys

Indeed, it was the practical consequences of Archimedes’ work which mattered more to his contemporaries and for which he became famous.

One such practical demonstration allowed King Heiron to move a ship with a single small lever – which in turn was connected to a series of other levers. Mathematically, he understood the relationship between the lever length, fulcrum position, the weight to be lifted and the force required to move the weight. This meant he could successfully predict outcomes for any number of levers and objects to be lifted.

Likewise he came to understand and explain the principles behind the compound pulley, windless, wedge and screw, as well as finding ways to determine the centre of gravity in objects.

Archimedes goes to war

Perhaps the most important inventions to his peers, however, were the devices created during the Roman siege of Syracuse in the second Punic War. The Romans eventually seized Syracuse, due to neglect of the defences, and Archimedes was killed by a Roman soldier while hard at work on mathematical diagrams. His last words are reputed to have been, ‘Fellow, do not disturb my circles!’

Further achievements

Inventions

. Archimedes’ Screw: a device used to pump water out of ships, and also to irrigate fields.

. Archimedes’ Claw: a huge war machine designed to sink ships by grasping the prow and tipping them over, used in the defence of Syracuse.

. Compound pulley systems: enabled the lifting of enormous weights at a minimal expenditure of energy.

. The method of exhaustion: an integral-like limiting process used to compute the area and volume of two dimensional lamina and three-dimensional solids.

Discoveries

. Archimedes was responsible for the science of hydrostatics, the study of the displacement of bodies in water. He also discovered the principles of static mechanics and pycnometry (the measurement of the volume or density of an object).

. Known as the ‘father of integral calculus’, Archimedes’ reckonings were later used by, among others, Kepler, Fermat, Leibniz and Newton.

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Science, Society, Technology

Questions of Science: The Large Hadron Collider…

Why does the Large Hadron Collider (LHC) have to be so physically large, when it is only designed to detect particles that are extremely tiny?

The LHC is a synchrotron, a circular accelerator that uses carefully synchronised electromagnetic fields to accelerate particles to very high speeds. When this involves charged particles on a curved path they release synchrotron radiation, which wastes energy. This is not desirable because most of the particles that physicists are looking for, such as the recently discovered Higgs boson, have large masses and can only be created in high-energy collisions.

The large radius of the LHC’s track is big enough to limit the radial acceleration given to the particles, thus minimising the loss of energy the particles suffer as synchrotron radiation. The superconducting magnets used to control the flow and direction of the particles can accelerate them up to speeds in excess of 99.99 (but less than 100) per cent of the speed of light.

The magnets lie central in answering the question. The size of the LHC is actually a trade-off between three things: the magnets that are available; the energy (or velocity) it is necessary to give the particles; and, the feasible dimensions of the structure.

The faster the particles are moving, the more likely you are to see something interesting happen in a collision. So, it’s important to accelerate the particles, mainly protons, as much as possible.

The protons need to follow a circular path so they can be continuously accelerated by an electric field, and this is done using magnets positioned around the tunnel. The faster the protons travel, the stronger the magnetic fields need to be to keep them on track.

To increase energy there are two possible choices: make the magnets stronger or the accelerator ring larger, so that the particles’ path does not need to be bent so much. At some point there is either a technological or financial limit on the strength of the magnets, leaving the ring size as the only remaining variable.

However, to keep the costs of the project manageable, the LHC was built in an existing tunnel that housed a previous experiment, called the Large Electron-Positron Collider. So the energy to which protons can be accelerated was actually predetermined by limits of technology and funding.

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Science

Skimming pebbles on water. But how does it happen?

QUESTIONS OF SCIENCE

Intro: Neither medium seems especially elastic in nature, so how do the stones bounce on water? There are practical implications of the science, as this article reveals.

ONE

FOR the best results, a circular, flattish stone is best. It must be thrown so it is almost horizontal to the water’s surface, and also such that its trailing edge hits the water first. It is vital the action of the throwing imparts spin to the stone.

Any solid body moving through a liquid experiences forces that oppose its motion: these forces are proportional to the cross-sectional area of the body and the square of its speed. Although only a part of a skipping stone is actually moving through the water, with the rest travelling through air, these forces still have an effect on its forward motion.

There is a force exerted by the water at right angles to the spinning stone’s surface. It acts at the trailing edge of the spinning stone – because this is where the impact begins – and tends to turn the stone towards the horizontal. Because of its spin, though, the stone behaves like a gyroscope and refuses to change its orientation. Nevertheless, this force reduces the stone’s forward velocity somewhat.

There is also a force exerted by the water parallel to the stone’s surface, but this force is much smaller and so the stone’s velocity is barely changed by it on impact with the water. The net effect of these forces is that the stone flies from the water in a parabolic arc until it hits the water again and the whole process is repeated.

At each impact, the stone loses some of its kinetic energy, which is dissipated in the ripples that are created in the water. And as its velocity is gradually reduced, the impacts become closer together until the energy dissipated is greater than that lost in the impact and the stone sinks.

The minimum initial speed required by a stone varies with its inclination to the horizontal. Experiments show that skimming will not occur if the angle at which the flat surface of the stone hits the water is more than 45 degrees to the horizontal. The slowest speed for skimming is about 2.5 metres per second, when the inclination is about 20 degrees.

The fact that the water itself is not elastic is immaterial, but it is important that it gives way to the stone on impact.

Stones can also be skimmed on wet sand, and even on cloth-covered boards. In such cases, however, there is little or no give in the surface and the frictional grip at impact is sufficient to change the direction of the stone’s motion and also cause the stone to overcome the gyroscopic effect.

Skimming stones on water is an age-old pastime. The gunners of naval sailing ships worked out that it could be used to increase the range of cannonballs. These could be made to skip along the surface of the sea and hole enemy vessels near their waterline. However, to do this the cannon itself had to be near the sea surface, which meant the firing vessel needed perilously low ports. Indeed, some ships capsized after taking on water through those open ports. With cannonballs, the spin required by a skimming stone was unnecessary as their spherical symmetry precluded any gyroscopic effects.

Barnes Wallis’s famous bouncing bomb, created for the 1943 Dambusters raid on Germany in the Second World War, worked on the same principle. He had to use cylindrical bombs, though, so he figured out he could ensure their stability by giving them spin about a horizontal axis at right angles to their direction of motion on the water.

TWO

IN a paper published in the journal Nature, in 2004, the French physicist Lydéric Bocquet and his team of researchers revealed some of the secrets of successful stone skimming. They found that the optimum angle of attack is 20 degrees. So, even when the stone is thrown horizontally, the leading edge should be 20 degrees higher than the trailing edge. This maximises the number of jumps by limiting the contact time between the stone and the water, which is proportional to the energy dissipated.

The thrower also imparts spin to the pebble, providing a gyroscopic effect that stabilises its flight and preserves the original angle of attack when it bounces. In the absence of spin, the water would impart a torque (or turning force) on the stone and, because the trailing edge is the first to make contact with the water, this would tend to make it tumble.

The actual physics of stone skimming is not perfectly understood. However, the bounce could be understood as a result of the conservation of momentum and Newton’s third law: when the stone exerts a force on the water, the water exerts an equal and opposite force on the stone. This lifting force is proportional to the density of the water, the surface area that is wetted and the square of the forward speed of the stone. Also, the bow wave created ahead of the stone when it strikes the water might act like a water-ski jump – helping to launch the next hop. This minimises the contact time between the stone and the water, which in turn maximises the number of jumps.

Although ensuring the optimal angle of attack as the stone strikes the water, and imparting just enough spin to maintain stable flight are important, there are other factors. Selecting the correct size and shape of stone and having a fast throwing arm are examples.

Given that the urge to skim stones has been with us for thousands of years and the rules – getting the greatest distance or number of bounces – have remained unchanged since the ancient Greeks, perhaps this should become an Olympic sport. In the meantime, the current world record stands at 51 skips, set by Russell Byars in Pennsylvania on 19 July 2007.

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