Misplaced Pages

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36-641: The Mahanoy Plane was a railroad Incline plane located along northern edge of the borough of Frackville , Schuylkill County, Pennsylvania in the Coal Region of the United States. GPS coordinates of the abandoned site are, DMS: N 40° 47’ 14.817” W 76° 13’ 58.652” -or- DD: 40.7874493, -76.232959. The Mahanoy and Broad Mountain Railroad , predecessor of the Reading Company opened

72-532: 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

108-477: A funicular or cable railway , such as the Johnstown Inclined Plane . 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

144-556: 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 and 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,

180-409: Is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is which means that a 20 lb force will lift a 100 lb load. The Liverpool Minard inclined plane has the dimensions 1804 meters by 37.50 meters, which provides a mechanical advantage of so a 100 lb tension force on the cable will lift a 4810 lb load. The grade of this incline is 2%, which means the angle θ

216-399: Is dissipated as heat by friction, W fric {\displaystyle W_{\text{fric}}} , so less work is done on the load. Due to conservation of energy , the sum of the output work and the frictional energy losses is equal to the input work Therefore, more input force is required, and the mechanical advantage is lower, than if friction were not present. With friction,

252-552: Is not completely tight. The forces exerted by the hanging part of the chain need not be symmetrical because the hanging part need not retain its shape when let go. Even if the chain is released with a zero angular momentum, motion including oscillations is possible unless the chain is initially in its equilibrium configuration, a supposition which would make the argument circular. 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

288-488: Is sometimes called ideal mechanical advantage I M A {\displaystyle \mathrm {IMA} } while the MA when friction is included is called the actual mechanical advantage A M A {\displaystyle \mathrm {AMA} } . 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

324-422: Is the coefficient of static friction between the two surfaces, which varies with the material. When no input force is applied, if the inclination angle θ {\displaystyle \theta } of the plane is less than some maximum value ϕ {\displaystyle \phi } the component of gravitational force parallel to the plane will be too small to overcome friction, and

360-403: 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. Where there is friction between the plane and the load, as for example with a heavy box being slid up a ramp, some of the work applied by the input force

396-703: 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

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432-424: Is zero. Since the direction of the frictional force is opposite for the case of uphill and downhill motion, these two cases must be considered separately: The mechanical advantage of an inclined plane is the ratio of the weight of the load on the ramp to the force required to pull it up the ramp. If energy is not dissipated or stored in the movement of the load, then this mechanical advantage can be computed from

468-537: The Pennsylvania Historical & Museum Commission erected a historical marker noting the Mahanoy Plane's historic importance. Incline 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

504-414: 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)

540-503: The tangent of the angle of repose on an inclined plane is equal to the coefficient of friction . 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

576-482: The Mahanoy Plane on July 16, 1861. The Plane traversed Broad Mountain between the boroughs of Mahanoy Plane and Frackville . With a pitch of 28 degrees at its steepest point, the plane rose 524 feet (160 m) over a distance of 2,460 feet (750 m). A 2,500 horsepower (1,900 kW) engine could hoist a three car trip, equivalent to 200 tons in three minutes. After a fire in 1886, new 6,000 horsepower (4,500 kW) engines were installed. These were supposedly

612-598: 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

648-410: The angle the plane makes with the horizontal, θ {\displaystyle \theta } . The mechanical advantage M A {\displaystyle \mathrm {MA} } 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

684-465: The conservation of energy equation above and rearranging To express the mechanical advantage by the angle θ {\displaystyle \theta } of the plane, it can be seen from the diagram (above) 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 F i {\displaystyle F_{\rm {i}}} from this equation

720-404: The dimensions of the ramp. In order to show this, let the position r of a rail car on along the ramp with an angle, θ , above the horizontal be given by where R is the distance along the ramp. The velocity of the car up the ramp is now Because there are no losses, the power used by force F to move the load up the ramp equals the power out, which is the vertical lift of the weight W of

756-410: 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 ( work ) is required to lift a given object by a given vertical distance , disregarding losses from friction , but the inclined plane allows

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792-404: 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 F i {\displaystyle F_{\text{i}}} on the object times the diagonal length of the inclined plane. Substituting these values into

828-521: 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 . As pointed out by Dijksterhuis, Stevin's argument

864-405: 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;

900-413: The load is stationary, neither sliding up or down the plane, whereas with a frictionless inclined plane there is only one particular value of input force for which the load is stationary. A load resting on an inclined plane, when considered as a free body has three forces acting on it: Using Newton's second law of motion the load will be stationary or in steady motion if the sum of the forces on it

936-444: The load will only move if the net force parallel to the surface is greater than the frictional force F f {\displaystyle F_{\text{f}}} opposing it. The maximum friction force is given by where F n {\displaystyle F_{\text{n}}} is the normal force between the load and the plane, directed normal to the surface, and μ {\displaystyle \mu }

972-492: The load will remain motionless. This angle is called the angle of repose and depends on the composition of the surfaces, but is independent of the load weight. It is shown below that the tangent of the angle of repose ϕ {\displaystyle \phi } is equal to μ {\displaystyle \mu } With friction, there is always some range of input force F i {\displaystyle F_{\text{i}}} for which

1008-400: The load. The input power pulling the car up the ramp is given by and the power out is Equate the power in to the power out to obtain the mechanical advantage as The mechanical advantage of an inclined plane can also be calculated from the ratio of length of the ramp L to its height H, because the sine of the angle of the ramp is given by therefore, Example: If the height of a ramp

1044-597: The most powerful engines in the world, later surpassed only by the engines operating the locks on the Panama Canal . Approximately 800 to 900 railroad cars passed over the plane every twenty-four hours. The Mahanoy Plane ceased operation on February 27, 1932, due to the decline of Anthracite and the much easier route through the Mahanoy Tunnel . Foundations are all that remain and there are currently no plans of historical preservation or restoration. There

1080-399: 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, W out {\displaystyle W_{\text{out}}} , is equal to the work done by the input force, W in {\displaystyle W_{\text{in}}} Work is defined as

1116-409: The plane, its weight F w {\displaystyle F_{\text{w}}} . The input force is the force F i {\displaystyle F_{\text{i}}} exerted on the object, parallel to the plane, to move it up the plane. The mechanical advantage is The M A {\displaystyle \mathrm {MA} } of an ideal inclined plane without friction

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1152-452: 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 μ s between the surfaces. Two other simple machines are often considered to be derived from

1188-414: 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

1224-525: Was a previous “Mahanoy Plane” built in the same general area in 1834 by the Danville and Pottsville Railroad(D&P). This was incline plane Number 5 (of six) and was 345 feet high over 1,625 feet in length. Very quickly, however, the D&;P closed the eastern section of railroad. By 1844, the eastern section, according to Rupp's History of Schuylkill County, was "rotting in the sun." On September 8, 2007,

1260-443: Was apparently not communicated to 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 Galilei (1592). Although it was not the first, the derivation of Flemish engineer Simon Stevin

1296-486: Was attempted by Greek philosophers Heron of Alexandria (c. 10 - 60 CE) and Pappus of Alexandria (c. 290 - 350 CE), 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

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