55-633: The Schiefe Ebene ( German pronunciation: [ˈʃiːfə ˈʔeːbənə] ; literally ' inclined plane ') is a steep incline on Bamberg–Hof section of the Ludwig South-North Railway in the region of Upper Franconia , in Bavaria , Germany . The Schiefe Ebene is located in the Landkreis (country district) of Kulmbach , beginning east of Neuenmarkt – Wirsberg station and ending at Marktschorgast . The route
110-532: 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
165-406: 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
220-476: A few hours (length 7–8 km, hilly terrain). On the road side of Marktschorgast station is an overview board of the trail; the first display board is on the road side of the former goods shed, about 20 m northeast (in the direction of Hof) next to the railway yard. On 27 December 1944 a serious accident occurred when a military train derailed in Neuenmarkt station, after its brakes had been unable to reduce
275-649: A horizontal table. Although the word horizontal is commonly used in daily life and language (see below), it is subject to many misconceptions. In general or in practice, something that is horizontal can be drawn from left to right (or right to left), such as the x-axis in the Cartesian coordinate system . The concept of a horizontal plane is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life. This dichotomy between
330-506: A larger scale the gravitational field of the Earth, which is at least approximately radial near the Earth, is not radial when it is affected by the Moon at higher altitudes. Neglecting the curvature of the earth, horizontal and vertical motions of a projectile moving under gravity are independent of each other. Vertical displacement of a projectile is not affected by the horizontal component of
385-556: 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,
440-440: 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 ,
495-437: A point P and designate a direction through P as vertical. A plane which contains P and is normal to the designated direction is the horizontal plane at P. Any plane going through P, normal to the horizontal plane is a vertical plane at P. Through any point P, there is one and only one horizontal plane but a multiplicity of vertical planes. This is a new feature that emerges in three dimensions. The symmetry that exists in
550-627: 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 . 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
605-409: 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
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#1732775584354660-399: 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,
715-424: 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 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;
770-552: 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
825-553: Is not electrified, but has been widened to two tracks. On the adjacent incline between Marktschorgast and Stammbach , the second track has been subsequently dismantled. On its way into the Franconian Forest mountains the ramp climbs 157.7 metres over a distance of 6.8 kilometres and therefore has an average incline of 23‰. It was built between 1844 and 1848 as part of the Ludwig South-North line and
880-431: Is small enough that sin θ≈tan θ. Vertical direction In astronomy , geography , and related sciences and contexts, a direction or plane passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be horizontal (or leveled ) if it is perpendicular to the vertical direction. In general, something that
935-488: 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
990-422: 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
1045-403: 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
1100-703: 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
1155-523: Is vertical can be drawn from up to down (or down to up), such as the y-axis in the Cartesian coordinate system . The word horizontal is derived from the Latin horizon , which derives from the Greek ὁρῐ́ζων , meaning 'separating' or 'marking a boundary'. The word vertical is derived from the late Latin verticalis , which is from the same root as vertex , meaning 'highest point' or more literally
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#17327755843541210-424: 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
1265-414: 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)
1320-503: 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
1375-512: The x -axis, in which case the y -axis is then automatically determined. There is no special reason to choose the horizontal over the vertical as the initial designation: the two directions are on par in this respect. The following hold in the two-dimensional case: Not all of these elementary geometric facts are true in the 3-D context. In the three-dimensional case, the situation is more complicated as now one has horizontal and vertical planes in addition to horizontal and vertical lines. Consider
1430-631: The 'turning point' such as in a whirlpool. Girard Desargues defined the vertical to be perpendicular to the horizon in his 1636 book Perspective . In physics, engineering and construction, the direction designated as vertical is usually that along which a plumb-bob hangs. Alternatively, a spirit level that exploits the buoyancy of an air bubble and its tendency to go vertically upwards may be used to test for horizontality. A water level device may also be used to establish horizontality. Modern rotary laser levels that can level themselves automatically are robust sophisticated instruments and work on
1485-608: The DRG Class 95 (ex- Prussian T 20 ) and DRG Class 96 (ex- Bavarian Gt 2x4/4 ), as well as DRG Classes 57 (ex- Prussian G 10 ) and 50 . In the early 1970s, the route was one of the last stamping grounds of the Deutsche Bundesbahn 's Class 01 express train steam locomotive. From 2001 to 2004 the Schiefe ;Ebene was also traversed by modern ICE-TD power cars ( DBAG Class 605 ) with tilting technology. Today
1540-533: The North Pole and as such has claim to be a horizontal plane. But it is. at the same time, a vertical plane for points on the equator. In this sense, a plane can, arguably, be both horizontal and vertical, horizontal at one place , and vertical at another . For a spinning earth, the plumb line deviates from the radial direction as a function of latitude. Only on the equator and at the North and South Poles does
1595-598: 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
1650-410: 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
1705-519: The apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from the fact that the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the world appears to be flat locally, and horizontal planes in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on
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1760-415: The applicable requirements, in particular in terms of accuracy. In graphical contexts, such as drawing and drafting and Co-ordinate geometry on rectangular paper, it is very common to associate one of the dimensions of the paper with a horizontal, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the horizontal direction is typically from the left side of
1815-465: 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
1870-422: The context of a 1-dimensional orthogonal Cartesian coordinate system on a Euclidean plane, to say that a line is horizontal or vertical, an initial designation has to be made. One can start off by designating the vertical direction, usually labelled the Y direction. The horizontal direction, usually labelled the X direction, is then automatically determined. Or, one can do it the other way around, i.e., nominate
1925-404: 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
1980-404: 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
2035-521: 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
2090-450: The launch velocity, and, conversely, the horizontal displacement is unaffected by the vertical component. The notion dates at least as far back as Galileo. When the curvature of the Earth is taken into account, the independence of the two motion does not hold. For example, even a projectile fired in a horizontal direction (i.e., with a zero vertical component) may leave the surface of the spherical Earth and indeed escape altogether. In
2145-620: The line is worked by the tilting DMUs of DB Class 612 , that serve the hourly RegionalExpress lines of Hof– Würzburg and Hof– Lichtenfels as well as the Interregio-Express line Nürnberg-Hof-Dresden. At the foot of the ramp in the former locomotive depot at Neuenmarkt-Wirsberg station – located in Neuenmarkt – is the German Steam Locomotive Museum ( Deutsches Dampflokomotiv-Museum ) or DDM . In
2200-413: 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
2255-444: 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 }
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2310-492: 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
2365-400: 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
2420-399: 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
2475-409: 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
2530-415: The plumb line align with the local radius. The situation is actually even more complicated because Earth is not a homogeneous smooth sphere. It is a non homogeneous, non spherical, knobby planet in motion, and the vertical not only need not lie along a radial, it may even be curved and be varying with time. On a smaller scale, a mountain to one side may deflect the plumb bob away from the true zenith . On
2585-465: 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 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
2640-487: The same fundamental principle. When the curvature of the Earth is taken into account, the concepts of vertical and horizontal take on yet another meaning. On the surface of a smoothly spherical, homogenous, non-rotating planet, the plumb bob picks out as vertical the radial direction. Strictly speaking, it is now no longer possible for vertical walls to be parallel: all verticals intersect. This fact has real practical applications in construction and civil engineering, e.g.,
2695-514: The speed of the train sufficiently coming down the Schiefe Ebene. The exact cause is unknown. The train was hauled by the locomotive 58 2813. The fireman, the engine driver and a non-commissioned officer died in the accident and an unknown number of people were injured. 50°04′31″N 11°36′53″E / 50.07528°N 11.61472°E / 50.07528; 11.61472 Inclined plane An inclined plane , also known as
2750-475: The tops of the towers of a suspension bridge are further apart than at the bottom. Also, horizontal planes can intersect when they are tangent planes to separated points on the surface of the Earth. In particular, a plane tangent to a point on the equator intersects the plane tangent to the North Pole at a right angle . (See diagram). Furthermore, the equatorial plane is parallel to the tangent plane at
2805-406: The two-dimensional case no longer holds. In the 2-dimension case, as mentioned already, the usual designation of the vertical coincides with the y-axis in co-ordinate geometry. This convention can cause confusion in the classroom. For the teacher, writing perhaps on a white board, the y -axis really is vertical in the sense of the plumbline verticality but for the student the axis may well lie on
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#17327755843542860-461: The upper station, in the former waiting room of Marktschorgast station, there is a small museum and information centre on the history of the Schiefe Ebene. Entrance is free. Marktschorgast station is also the start of a railway history trail that runs along the Schiefe Ebene and recounts the history of the line on display boards. The walk ends at Neuenmarkt-Wirsberg station. The trail should be walked with stout footwear and can be completed in
2915-443: 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
2970-486: 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
3025-478: Was opened on 1 November 1848. With its numerous supporting walls, cuttings and stone embankments it was one of the great engineering feats of its time. The cost of construction was 917,318 Gulden . In the era of steam, the ramp proved a real challenge for locomotives and crews. The majority of trains had to be reinforced by heavy pusher locomotives . These were stationed in the well-known locomotive depot at Neuenmarkt-Wirsberg. They included famous locomotives such as
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