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An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the
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 hor ...
, 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 machine A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. Usually the term refer ...
s 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 In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
than lifting it straight up, at a cost of an increase in the distance moved. The
mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for t ...
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 In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
, the same amount of
mechanical energy In Outline of physical science, physical sciences, mechanical energy is the sum of potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, t ...
(
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
) is required to lift a given object by a given
vertical distance Vertical position or vertical location, also known as vertical level or simply level, is a position along a vertical direction above or below a given vertical datum (reference level). Vertical distance or vertical separation is the distance betwe ...
, disregarding losses from
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
, but the inclined plane allows the same work to be done with a smaller force exerted over a greater distance. The
angle of friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
, 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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
without sliding down. This angle is equal to the
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
of the
coefficient of static friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
''μs'' between the surfaces. Two other simple machines are often considered to be derived from the inclined plane. The
wedge A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
can be considered a moving inclined plane or two inclined planes connected at the base. The
screw A screw and a bolt (see '' Differentiation between bolt and screw'' below) are similar types of fastener typically made of metal and characterized by a helical ridge, called a ''male thread'' (external thread). Screws and bolts are used to fa ...
consists of a narrow inclined plane wrapped around a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
. 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 A funicular (, , ) is a type of cable railway system that connects points along a railway track laid on a steep slope. The system is characterized by two counterbalanced carriages (also called cars or trains) permanently attached to opposite en ...
or
cable railway Cable may refer to: Mechanical * Nautical cable, an assembly of three or more ropes woven against the weave of the ropes, rendering it virtually waterproof * Wire rope, a type of rope that consists of several strands of metal wire laid into a hel ...
, such as the
Johnstown Inclined Plane The Johnstown Inclined Plane is a funicular in Johnstown, Cambria County, in the U.S. state of Pennsylvania. The incline and its two stations connect the city of Johnstown, situated in a valley at the confluence of the Stonycreek and the Li ...
.


Uses

Inclined planes are widely used in the form of ''loading ramps'' to load and unload goods on trucks, ships and planes.
Wheelchair ramp A wheelchair ramp is an inclined plane installed in addition to or instead of stairs. Ramps permit wheelchair users, as well as people pushing strollers, carts, or other wheeled objects, to more easily access a building, or navigate between ar ...
s are used to allow people in
wheelchair A wheelchair is a chair with wheels, used when walking is difficult or impossible due to illness, injury, problems related to old age, or disability. These can include spinal cord injuries ( paraplegia, hemiplegia, and quadriplegia), cerebr ...
s to get over vertical obstacles without exceeding their strength.
Escalator An escalator is a moving staircase which carries people between floors of a building or structure. It consists of a motor-driven chain of individually linked steps on a track which cycle on a pair of tracks which keep the step tread horizo ...
s and slanted
conveyor belt A conveyor belt is the carrying medium of a belt conveyor system (often shortened to belt conveyor). A belt conveyor system is one of many types of conveyor systems. A belt conveyor system consists of two or more pulleys (sometimes referred to ...
s are also forms of an inclined plane.In a
funicular A funicular (, , ) is a type of cable railway system that connects points along a railway track laid on a steep slope. The system is characterized by two counterbalanced carriages (also called cars or trains) permanently attached to opposite en ...
or
cable railway Cable may refer to: Mechanical * Nautical cable, an assembly of three or more ropes woven against the weave of the ropes, rendering it virtually waterproof * Wire rope, a type of rope that consists of several strands of metal wire laid into a hel ...
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 In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance ''normal'' is used in the geometric sense and means perpendicular, as oppose ...
of the plane to reduce the
gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
. Aircraft
evacuation slides An evacuation slide is an inflatable slide used to evacuate an aircraft quickly. An escape slide is required on all commercial (passenger carrying) aircraft where the door sill height is such that, in the event of an evacuation, passengers wo ...
allow people to rapidly and safely reach the ground from the height of a passenger
airliner An airliner is a type of aircraft for transporting passengers and air cargo. Such aircraft are most often operated by airlines. Although the definition of an airliner can vary from country to country, an airliner is typically defined as an ...
. Other inclined planes are built into permanent structures. Roads for vehicles and railroads have inclined planes in the form of gradual slopes, ramps, and
causeway A causeway is a track, road or railway on the upper point of an embankment across "a low, or wet place, or piece of water". It can be constructed of earth, masonry, wood, or concrete. One of the earliest known wooden causeways is the Sweet Tra ...
s to allow vehicles to surmount vertical obstacles such as hills without losing traction on the road surface. Similarly, pedestrian paths and
sidewalk A sidewalk (North American English), pavement (British English), footpath in Australia, India, New Zealand and Ireland, or footway, is a path along the side of a street, street, highway, terminals. Usually constructed of concrete, pavers, brick ...
s 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 slide Playground slides are found in parks, schools, playgrounds and backyards. The slide is an example of the simple machine known as the inclined plane, which makes moving objects up and down easier, or in this case more fun. The slide may be flat, ...
s,
water slide A water slide (also referred to as a flume, or water chute) is a type of slide designed for warm-weather or indoor recreational use at water parks. Water slides differ in their riding method and therefore size. Some slides require riders to si ...
s,
ski slope A ski is a narrow strip of semi-rigid material worn underfoot to glide over snow. Substantially longer than wide and characteristically employed in pairs, skis are attached to ski boots with ski bindings, with either a free, lockable, or parti ...
s and
skateboard park A skatepark, or skate park, is a purpose-built recreational environment made for skateboarding, BMX, Freestyle scootering, scootering, wheelchairs, and aggressive inline skating. A skatepark may contain half-pipes, handrails, funboxes, vert r ...
s.


History

Inclined planes have been used by people since prehistoric times to move heavy objects.Therese McGuire, ''Light on Sacred Stones'', in The sloping roads and
causeway A causeway is a track, road or railway on the upper point of an embankment across "a low, or wet place, or piece of water". It can be constructed of earth, masonry, wood, or concrete. One of the earliest known wooden causeways is the Sweet Tra ...
s 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 Stonehenge is a prehistoric monument on Salisbury Plain in Wiltshire, England, west of Amesbury. It consists of an outer ring of vertical sarsen standing stones, each around high, wide, and weighing around 25 tons, topped by connectin ...
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 The Egyptian pyramids are ancient masonry structures located in Egypt. Sources cite at least 118 identified "Egyptian" pyramids. Approximately 80 pyramids were built within the Kingdom of Kush, now located in the modern country of Sudan. Of ...
were constructed using inclined planes,
Siege A siege is a military blockade of a city, or fortress, with the intent of conquering by attrition warfare, attrition, or a well-prepared assault. This derives from la, sedere, lit=to sit. Siege warfare is a form of constant, low-intensity con ...
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 The Isthmus of Corinth (Greek: Ισθμός της Κορίνθου) is the narrow land bridge which connects the Peloponnese peninsula with the rest of the mainland of Greece, near the city of Corinth. The word "isthmus" comes from the Ancien ...
. However the inclined plane was the last of the six classic
simple machine A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. Usually the term refer ...
s 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, the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
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''".Karl von Langsdorf (1826) ''Machinenkunde'', quoted in 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 Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He i ...
(c. 10 - 60 CE) and
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
(c. 290 - 350 CE), but their solutions were incorrect.Egidio Festa and Sophie Roux, ''The enigma of the inclined plane'' in It wasn't until the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
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 Jordanus de Nemore (fl. 13th century), also known as Jordanus Nemorarius and Giordano of Nemi, was a thirteenth-century European mathematician and scientist. The literal translation of Jordanus de Nemore (Giordano of Nemi) would indicate that he w ...
, however his solution was apparently not communicated to other philosophers of the time.
Girolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
(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 Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
(1586), and
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
(1592). Although it was not the first, the derivation of Flemish engineer
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
on an inclined plane were discovered by
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
(1452-1519), but remained unpublished in his notebooks. They were rediscovered by
Guillaume Amontons Guillaume Amontons (31 August 1663 – 11 October 1705) was a French scientific instrument inventor and physicist. He was one of the pioneers in studying the problem of friction, which is the resistance to motion when bodies make contact. He is ...
(1699) and were further developed by
Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attrac ...
(1785).
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(1750) showed that the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
of the angle of repose on an inclined plane is equal to the
coefficient of friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
.


Terminology


Slope

The
mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for t ...
of an inclined plane depends on its ''
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
'', meaning its
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
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, ''θ''. :\theta = \tan^ \bigg( \frac \bigg) \,


Mechanical advantage

The
mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for t ...
''MA'' of a simple machine is defined as the ratio of the output force exerted on the load to the input force applied. For the inclined plane the output load force is just the gravitational force of the load object on the plane, its weight ''Fw''. The input force is the force ''Fi'' exerted on the object, parallel to the plane, to move it up the plane. The mechanical advantage is :\mathrm = \frac \, The MA of an ideal inclined plane without friction is sometimes called ''ideal mechanical advantage'' (IMA) while the MA when friction is included is called the ''actual mechanical advantage'' (AMA).


Frictionless inclined plane

If there is no
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
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 A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
, or supported on wheels or
caster A caster (or castor) is an undriven wheel that is designed to be attached to the bottom of a larger object (the "vehicle") to enable that object to be moved. Casters are used in numerous applications, including shopping carts, office chairs, t ...
s. Due to
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
, for a frictionless inclined plane the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
done on the load lifting it, ''W''out, is equal to the work done by the input force, ''W''in :W_ = W_ \, 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 :W_ = F_w \cdot \text \, The input work is equal to the force ''Fi'' on the object times the diagonal length of the inclined plane. :W_ = F_i \cdot \text \, Substituting these values into the conservation of energy equation above and rearranging :\text = \frac = \frac \, To express the mechanical advantage by the angle ''θ'' of the plane, it can be seen from the diagram ''(above)'' that :\sin \theta = \frac \, So :\text = \frac = \frac \, So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force ''Fi'' 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.


Inclined plane with friction

Where there is
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
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 is dissipated as heat by friction, ''W''fric, so less work is done on the load. Due to
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
, the sum of the output work and the frictional energy losses is equal to the input work :W_\text = W_\text + W_\text \, Therefore, more input force is required, and the mechanical advantage is lower, than if friction were not present. With friction, the load will only move if the net force parallel to the surface is greater than the frictional force ''Ff'' opposing it.This derives slightly more general equations which cover force applied at any angle: The maximum friction force is given by :F_f = \mu F_n \, where ''Fn'' is the
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance ''normal'' is used in the geometric sense and means perpendicular, as oppose ...
between the load and the plane, directed normal to the surface, and ''μ'' is the
coefficient of static friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
between the two surfaces, which varies with the material. When no input force is applied, if the inclination angle ''θ'' of the plane is less than some maximum value ''φ'' the component of gravitational force parallel to the plane will be too small to overcome friction, and 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
of the angle of repose ''φ'' is equal to ''μ'' :\phi = \tan^ \mu \, With friction, there is always some range of input force ''Fi'' for which 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.


Analysis

A load resting on an inclined plane, when considered as a
free body The term free body is usually associated with the motion of a free body diagram, a pictorial device used by physicists and engineers. In that context, a body is said to be "free" when it is singled out from other bodies for the purposes of dynamic o ...
has three forces acting on it: *The applied force, ''Fi'' exerted on the load to move it, which acts parallel to the inclined plane. *The weight of the load, ''Fw'', which acts vertically downwards *The force of the plane on the load. This can be resolved into two components: **The normal force ''Fn'' of the inclined plane on the load, supporting it. This is directed perpendicular (
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
) to the surface. **The frictional force, ''Ff'' of the plane on the load acts parallel to the surface, and is always in a direction opposite to the motion of the object. It is equal to the normal force multiplied by the
coefficient of static friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
μ between the two surfaces. Using
Newton's second law of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
the load will be stationary or in steady motion if the sum of the forces on it 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: *Uphill motion: The total force on the load is toward the uphill side, so the frictional force is directed down the plane, opposing the input force. :The mechanical advantage is :\mathrm = \frac = \frac \, :where \phi = \tan^ \mu \,. This is the condition for ''impending motion'' up the inclined plane. If the applied force ''Fi'' is greater than given by this equation, the load will move up the plane. *Downhill motion: The total force on the load is toward the downhill side, so the frictional force is directed up the plane. :The mechanical advantage is :\mathrm = \frac = \frac \, :This is the condition for impending motion down the plane; if the applied force ''Fi'' is less than given in this equation, the load will slide down the plane. There are three cases: :#\theta < \phi\,: The mechanical advantage is negative. In the absence of applied force the load will remain motionless, and requires some negative (downhill) applied force to slide down. :#\theta = \phi\,: The ' angle of repose'. The mechanical advantage is infinite. With no applied force, load will not slide, but the slightest negative (downhill) force will cause it to slide. :#\theta > \phi\,: The mechanical advantage is positive. In the absence of applied force the load will slide down the plane, and requires some positive (uphill) force to hold it motionless


Mechanical advantage using power

The
mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for t ...
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 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 :\mathbf = R (\cos\theta, \sin\theta), where ''R'' is the distance along the ramp. The velocity of the car up the ramp is now :\mathbf = V (\cos\theta, \sin\theta). 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 the load. The input power pulling the car up the ramp is given by :P_ = F V,\! and the power out is :P_ = \mathbf\cdot\mathbf = (0, W)\cdot V (\cos\theta, \sin\theta) = WV\sin\theta. Equate the power in to the power out to obtain the mechanical advantage as : \mathrm = \frac = \frac. 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 : \sin\theta = \frac, therefore, : \mathrm = \frac = \frac. Example: If the height of a ramp is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is : \mathrm = \frac = 5, 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 : \mathrm = \frac = 1804/37.50 = 48.1, 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 is small enough that sin θ=tan θ.


See also

*
Canal inclined plane An inclined plane is a type of cable railway used on some canals for raising boats between different water levels. Boats may be conveyed afloat, in caissons, or may be carried in cradles or slings. History Inclined planes have evolved over the ...
*
Grade (slope) The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero in ...
*
Inclined plane railroad Incline, inclined, inclining, or inclination may refer to: *Grade (slope), the tilt, steepness, or angle from horizontal of a topographic feature (hillside, meadow, etc.) or constructed element (road, railway, field, etc.) *Slope, the tilt, steepn ...
*
Ramp function The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for o ...
*
Schiefe Ebene The Schiefe Ebene ( 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. Location and construction The Schiefe Ebene is locat ...
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Stairs Stairs are a structure designed to bridge a large vertical distance between lower and higher levels by dividing it into smaller vertical distances. This is achieved as a diagonal series of horizontal platforms called steps which enable passage ...


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An interactive simulation of Physics inclined plane
{{Authority control Simple machines