Inclined plane
An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical simple machines defined by Renaissance scientists. Inclined planes are used to move heavy loads over vertical obstacles. Examples vary from a ramp used to load goods into a truck, to a person walking up a pedestrian ramp, to an automobile or railroad train climbing a grade.
Moving an object up an inclined plane requires less force than lifting it straight up, at a cost of an increase in the distance moved. The mechanical advantage of an inclined plane, the factor by which the force is reduced, is equal to the ratio of the length of the sloped surface to the height it spans. Owing to conservation of energy, the same amount of mechanical energy is required to lift a given object by a given vertical distance, disregarding losses from friction, but the inclined plane allows the same work to be done with a smaller force exerted over a greater distance.
The angle of friction, also sometimes called the angle of repose, is the maximum angle at which a load can rest motionless on an inclined plane due to friction without sliding down. This angle is equal to the arctangent of the coefficient of static friction μs between the surfaces.
Two other simple machines are often considered to be derived from the inclined plane. The wedge can be considered a moving inclined plane or two inclined planes connected at the base. The screw consists of a narrow inclined plane wrapped around a cylinder.
The term may also refer to a specific implementation; a straight ramp cut into a steep hillside for transporting goods up and down the hill. This may include cars on rails or pulled up by a cable system; a funicular or cable railway, such as the Johnstown Inclined Plane.
Uses
Inclined planes are widely used in the form of loading ramps to load and unload goods on trucks, ships and planes. Wheelchair ramps are used to allow people in wheelchairs to get over vertical obstacles without exceeding their strength. Escalators and slanted conveyor belts are also forms of an inclined plane.In a funicular or cable railway a railroad car is pulled up a steep inclined plane using cables. Inclined planes also allow heavy fragile objects, including humans, to be safely lowered down a vertical distance by using the normal force of the plane to reduce the gravitational force. Aircraft evacuation slides allow people to rapidly and safely reach the ground from the height of a passenger airliner.Other inclined planes are built into permanent structures. Roads for vehicles and railroads have inclined planes in the form of gradual slopes, ramps, and causeways to allow vehicles to surmount vertical obstacles such as hills without losing traction on the road surface. Similarly, pedestrian paths and sidewalks have gentle ramps to limit their slope, to ensure that pedestrians can keep traction. Inclined planes are also used as entertainment for people to slide down in a controlled way, in playground slides, water slides, ski slopes and skateboard parks.
History
| Stevin's proof |
Image:StevinEquilibrium.svg|center|150px In 1586, Flemish engineer Simon Stevin 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 as in a prism. A loop of string with beads at equal intervals is draped over the inclined planes, with part of the string 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:
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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 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 and motionless device, 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 was attempted by Greek philosophers Heron of Alexandria and Pappus of Alexandria, but their solutions were incorrect.
It was not until the Renaissance that the inclined plane was solved mathematically and classed with the other simple machines. The first correct analysis of the inclined plane appeared in the work of 13th century author Jordanus de Nemore, however his solution was apparently not communicated to other philosophers of the time. Girolamo Cardano 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, Simon Stevin, and Galileo Galilei. 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. In 1600, Italian scientist Galileo Galilei included the inclined plane in his analysis of simple machines in Le Meccaniche, 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, but remained unpublished in his notebooks. They were rediscovered by Guillaume Amontons and were further developed by Charles-Augustin de Coulomb. Leonhard Euler showed that the tangent of the angle of repose on an inclined plane is equal to the coefficient of friction.
Terminology
Slope
The mechanical advantage of an inclined plane depends on its slope, meaning its gradient or steepness. The smaller the slope, the larger the mechanical advantage, and the smaller the force needed to raise a given weight. A plane's slope s is equal to the difference in height between its two ends, or "rise", divided by its horizontal length, or "run". It can also be expressed by the angle the plane makes with the horizontal,.Mechanical advantage
The mechanical advantage of a simple machine as defined as the ratio of the output force exerted on the load to the input force applied. The inclined plane the output load force is just the gravitational force of the load object on the plane, its weight . The input force is the force exerted on the object, parallel to the plane, to move it up the plane. The mechanical advantage isThe of an ideal inclined plane without friction is sometimes called ideal mechanical advantage while the MA when friction is included is called the actual mechanical advantage.
Frictionless inclined plane
If there is no friction between the object being moved and the plane, the device is called an ideal inclined plane. This condition might be approached if the object is rolling like a barrel, or supported on wheels or casters. Due to conservation of energy, for a frictionless inclined plane the work done on the load lifting it,, is equal to the work done by the input force,Work is defined as the force multiplied by the displacement an object moves. The work done on the load is equal to its weight multiplied by the vertical displacement it rises, which is the "rise" of the inclined plane
The input work is equal to the force on the object times the diagonal length of the inclined plane.
Substituting these values into the conservation of energy equation above and rearranging
To express the mechanical advantage by the angle of the plane, it can be seen from the diagram that
So
So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force from this equation is the force needed to hold the load motionless on the inclined plane, or push it up at a constant velocity. If the input force is greater than this, the load will accelerate up the plane. If the force is less, it will accelerate down the plane.