Inclined Plane - History

History

Stevin's proof
In 1586, Flemish engineer Simon Stevin (Stevinus) derived the mechanical advantage of the inclined plane by an argument that used a string of beads. He imagined two inclined planes of equal height but different slopes, placed back-to-back (above). A loop of string with beads at equal intervals is draped over the inclined planes, with part hanging down below. The beads resting on the planes act as loads on the planes, held up by the tension force in the string at point T. Stevin's argument goes like this:
  • The string must be stationary, in static equilibrium. If it was heavier on one side than the other, and began to slide right or left under its own weight, when each bead had moved to the position of the previous bead the string would be indistinguishable from its initial position and therefore would continue to be unbalanced and slide. This argument could be repeated indefinitely, resulting in a circular perpetual motion, which is absurd. Therefore it is stationary. with the forces on the two sides at point T (above) equal.
  • The portion of the chain hanging below the inclined planes is symmetrical, with an equal number of beads on each side. It exerts an equal force on each side of the string. Therefore, this portion of the string can be cut off at the edges of the planes (points S and V), leaving only the beads resting on the inclined planes, and this remaining portion will still be in static equilibrium.
  • Since the beads are at equal intervals on the string, the total number of beads supported by each plane, the total load, is proportional to the length of the plane. Since the input supporting force, the tension in the string, is the same for both, the mechanical advantage of each plane is proportional to its slant length.

Inclined planes have been used by people since prehistoric times to move heavy objects. The sloping roads and causeways built by ancient civilizations such as the Romans are examples of early inclined planes that have survived, and show that they understood the value of this device for moving things uphill. The heavy stones used in ancient stone structures such as Stonehenge are believed to have been moved and set in place using inclined planes made of earth, although it is hard to find evidence of such temporary building ramps. The Egyptian pyramids were constructed using inclined planes, Siege ramps enabled ancient armies to surmount fortress walls. The ancient Greeks constructed a paved ramp 6 km (3.7 miles) long, the Diolkos, to drag ships overland across the Isthmus of Corinth.

However the inclined plane was the last of the six classic simple machines to be recognised as a machine. This is probably because it is a passive, motionless device (the load is the moving part), and also because it is found in nature in the form of slopes and hills. Although they understood its use in lifting heavy objects, the ancient Greek philosophers who defined the other five simple machines did not include the inclined plane as a machine. This view persisted among a few later scientists; as late as 1826 Karl von Langsdorf wrote that an inclined plane "...is no more a machine than is the slope of a mountain. The problem of calculating the force required to push a weight up an inclined plane (its mechanical advantage) was attempted by Greek philosophers Heron of Alexandria (c. 10 - 60 AD) and Pappus of Alexandria (c. 290 - 350 AD), but they got it wrong.

It wasn't until the Renaissance that the inclined plane was classed with the other simple machines. The first correct analysis of the inclined plane appeared in the work of enigmatic 13th century author Jordanus de Nemore, however his solution was apparently not accepted by other philosophers of the time. Girolamo Cardano (1570) proposed the incorrect solution that the input force is proportional to the angle of the plane. Then at the end of the 16th century, three correct solutions were published within ten years, by Michael Varro (1584), Simon Stevin (1586), and Galileo Galilee (1592). Although it was not the first, the derivation of Flemish engineer Simon Stevin is the most well-known, because of its originality and use of a string of beads (see box). In 1600, Italian scientist Galileo Galilei included the inclined plane in his analysis of simple machines in Le Meccaniche ("On Mechanics"), showing its underlying similarity to the other machines as a force amplifier.

The first elementary rules of sliding friction on an inclined plane were discovered by Leonardo da Vinci (1452-1519), but remained unpublished in his notebooks. They were rediscovered by Guillaume Amontons (1699) and were further developed by Charles-Augustin de Coulomb (1785). Leonhard Euler (1750) showed that the tangent of the angle of repose on an inclined plane is equal to the coefficient of friction.

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