Inclined plane
Instrumented inclined plane for experimentation.
Classification	Simple machine
Industry	Construction

Layout of the cable drive system for the Liverpool Minard inclined plane.

The Johnstown Inclined Plane in Johnstown, Pennsylvania.

An inclined plane is a flat surface that lies at an angle relative to level ground used to form a ramp to raise and lower a load, such as a wheelchair ramp. Renaissance scientists added the inclined plane to the five simple machines described by Hero of Alexandria.^[1] The force required to lift the load is reduced by pulling it up the ramp. Inclined plane can also refer to a particular type of implementation, a straight ramp cut into a hillside for moving loads up and down the hill. These may include cars riding on rails pulled up and lowered down with a cable system, such as the Johnstown Inclined Plane.

Assuming the movement of the load does not dissipate or store energy, the mechanical advantage of an inclined plane is the ratio of the length of the ramp to the height lifted.^[2][3]

Ramps and screws[link]

An inclined plane is one of the six simple machines identified by Renaissance scientists. The inclined plane is a ramp that assists moving heavy objects up and down heights.

Portable ramps allow easy loading and unloading of high-decked trucks. Siege ramps gave ancient armies the ability to bring war machines to the top of fortress walls. Slides allow fragile objects, including humans, to be rapidly and safely lowered down a vertical rise by resisting gravitational force with the normal force provided by the stiff surface of the inclined plane. Airplane rescue slides allow people to rapidly and safely reach the ground from the height of a passenger airplane.

The screw is an inclined plane wrapped around a cylinder. The Archimedes screw has been used since ancient times to move a variety of materials such as water, minerals and grain.^[4]

Mechanical advantage[link]

Key: N = Normal force that is perpendicular to the plane, W=mg, where m = mass, g = gravity, and θ (theta) = Angle of inclination of the plane

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 the dimensions of the ramp.

In order to show this, let the position of a rail car on along a ramp at an angle,  θ, above the horizontal be given by the Cartesian coordinates

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r} = R (\cos\theta, \sin\theta),

where R is the distance along the ramp. The velocity of the car up the ramp is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} = 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

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{\mathrm{in}} = FV,\!

and the power out is

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{\mathrm{out}} = \mathbf{W}\cdot\mathbf{v} = (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

Failed to parse (Missing texvc executable; please see math/README to configure.): MA = \frac{W}{F} = \frac{1}{\sin\theta}.

The mechanical advantage of an inclined 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\theta = \frac{H}{L},

therefore,

Failed to parse (Missing texvc executable; please see math/README to configure.): MA = \frac{W}{F} = \frac{L}{H}.

Example: If the height of a ramp is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is

Failed to parse (Missing texvc executable; please see math/README to configure.): MA = \frac{W}{F} = 5.

See also[link]

• Simple machine
• Mechanical advantage
• Frictionless plane
• Grade (slope)
• Canal inclined plane
• Inclined plane railroad
• Schiefe Ebene

External links[link]

Wikimedia Commons has media related to: Inclined planes

• An interactive simulation of Physics inclined plane

References[link]

Table of simple mechanisms, from Chambers' Cyclopedia, 1728.^[1] Simple machines provide a "vocabulary" for understanding more complex machines.

A simple machine is a mechanical device that changes the direction or magnitude of a force.^[2] In general, they can be defined as the simplest mechanisms that provide mechanical advantage (also called leverage).^[3]

Usually the term refers to the six classical simple machines which were defined by Renaissance scientists:^[4]

• Lever
• Wheel and axle
• Pulley
• Inclined plane
• Wedge
• Screw

A simple machine is an elementary device that has a specific movement (often called a mechanism), which can be combined with other devices and movements to form a machine. Thus simple machines are considered to be the "building blocks" of more complicated machines. This analytical view of machines as decomposable into simple machines first arose in the Renaissance as a neoclassical amplification of ancient Greek texts on technology,^[5] and is still a central part of engineering in today's age of applied science. For example, wheels, levers, and pulleys are all used in the mechanism of a bicycle.^[6][7] Between the simple machines and complex assemblies, several intermediate classes can be defined, which may be termed "compound machines"^[8][3][9] or "machine elements".^[10] The mechanical advantage of a compound machine is simply the product of the mechanical advantages of the simple machines of which it is composed.

Various authors have compiled lists of simple machines and machine elements, sometimes lumping them together under a single term such as "simple machines",^[1] "basic machines",^[6] "compound machines",^[8] or "machine elements"; the use of the term "simple machines" in this broader sense is a departure from the neoclassical sense of the six essential simple machines, which is why many authors prefer to avoid its use, preferring the other terms (such as "machine element"). In all cases, the theme of an analytical and synthetic connection from simple to compound to complex is at work. A page from a 1728 text by Ephraim Chambers^[1] (in the figure to the right) shows more machine elements. By the late 1800s, Franz Reuleaux^[11] identified hundreds of machine elements (calling them "simple machines"). Models of these devices can be found at Cornell University's KMODDL website.^[12]

History[link]

The idea of a "simple machine" originated with the Greek philosopher Archimedes around the 3rd century BC, who studied the "Archimedean" simple machines: lever, pulley, and screw.^[3][13] He discovered the principle of mechanical advantage in the lever.^[14] Later Greek philosophers defined the classic five simple machines (excluding the inclined plane) and were able to roughly calculate their mechanical advantage.^[5] Heron of Alexandria (ca. 10–75 AD) in his work Mechanics lists five mechanisms that can "set a load in motion"; lever, windlass, pulley, wedge, and screw,^[13] and describes their fabrication and uses.^[15] However the Greeks' understanding was limited to the statics of simple machines; the balance of forces, and did not include dynamics; the tradeoff between force and distance, or the concept of work.

During the Renaissance the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how much useful work they could perform, leading eventually to the new concept of mechanical work. In 1586 Flemish engineer Simon Stevin derived the mechanical advantage of the inclined plane, and it was included with the other simple machines. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche ("On Mechanics").^[16][17] He was the first to understand that simple machines do not create energy, only transform it.^[16]

The classic rules of sliding friction in machines 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).^[18]

Mechanical advantage[link]

A simple machine has an applied force that works against a load force. If there are no friction losses, the work done on the load is equal to the work done by the applied force. This allows an increase in the output force at the cost of a proportional decrease in the distance moved by the load. The ratio of the output force to the input force is the mechanical advantage of the machine.

If the simple machine does not dissipate or absorb energy, then its mechanical advantage can be calculated from the machine's geometry. For example, the mechanical advantage of a lever is equal to the ratio of its lever arms. A simple machine with no friction or elasticity is often called an ideal machine.^[19][20]

The mechanical advantage of an ideal simple machine is easily obtained from the requirement the rate of energy in, or power in, equals the rate of energy out, or power out, that is

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{in} = P_{out}.\!

Because power is the product of a force and the velocity of its point of application, the applied force times the velocity the input point moves, v_in, must be equal to the load force times the velocity the load moves, v_out, given by

Failed to parse (Missing texvc executable; please see math/README to configure.): F_{in}v_{in} = F_{out}v_{out}.\,

So the ratio of output to input force, the mechanical advantage, of a frictionless machine is equal to the "velocity ratio"; the ratio of input velocity to output velocity:

Failed to parse (Missing texvc executable; please see math/README to configure.): MA = \frac{F_{out}}{F_{in}} = \frac{v_{in}}{v_{out}} \,

      (Ideal Mechanical Advantage)

In the screw, which uses rotational motion, the input force should be replaced by the torque, and the velocity by the angular velocity the shaft is turned.

Compound machine[link]

A compound machine is a machine formed from a set of simple machines connected in series with the output force of one providing the input force to the next. For example a bench vise consists of a lever (the vise's handle) in series with a screw, and a simple gear train consists of a number of gears (wheels and axles) connected in series.

The mechanical advantage of a compound machine is the ratio of the output force exerted by the last machine in the series divided by the input force applied to the first machine, that is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{MA}_{compound} = \frac {F_{\mathrm{out} n}} {F_{\mathrm{in} 1}}. \,

Because the output force of each machine is the input of the next, Failed to parse (Missing texvc executable; please see math/README to configure.): F_{out 1}=F_{in 2} \,

and Failed to parse (Missing texvc executable; please see math/README to configure.):  F_{out k} =F_{in k+1} \,

, this mechanical advantage is also given by,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{MA}_{compound} = \frac {F_{out 1}} {F_{in 1}} \frac {F_{out 2}} {F_{in 2}} \ldots\frac {F_{out n}} {F_{in n}}. \,

Thus, the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of the series of simple machines that form it,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{MA}_{compound} = \mathrm{MA}_1 \mathrm{MA}_2 \ldots\mathrm{MA}_n \,

Energy losses and efficiency[link]

Machines lose energy through friction, deformation and wear, which is dissipated as heat. This means the power out of the machine is less than power in. The ratio of power out to power in is the efficiency η of the machine, and is a measure of the energy losses,

Failed to parse (Missing texvc executable; please see math/README to configure.): \eta = \frac{P_{out}}{P_{in}}. \,

The velocity ratio of a machine is fixed by its dimensions, so it is the mechanical advantage that is reduced by the losses, that is

Failed to parse (Missing texvc executable; please see math/README to configure.): MA = \frac{F_{out}}{F_{in}} = \eta \frac{v_{in}}{v_{out}}.

So in non-ideal machines, the mechanical advantage is always less than the velocity ratio by the product with the efficiency η. So a machine that includes losses such as friction, deformation and wear, will not be able to move as large a load as a corresponding ideal machine using the same input force.

The efficiency of a compound machine is the product of the efficiencies of the series of simple machines that form it,

Failed to parse (Missing texvc executable; please see math/README to configure.): \eta_{compound} = \eta_1 \eta_2 \ldots \eta_n.\,

Kinematic chains[link]

Illustration of a four-bar linkage from Kinematics of Machinery, 1876

Simple machines are elementary examples of kinematic chains that are used to model mechanical systems ranging from the steam engine to robot manipulators. The bearings that form the fulcrum of a lever and that allow the wheel and axle and pulleys to rotate are examples of a kinematic pair called a hinged joint. Similarly, the flat surface of an inclined plane and wedge are examples of the kinematic pair called a sliding joint. The screw is usually identified as its own kinematic pair called a helical joint.

Two levers, or cranks, are combined into a planar four-bar linkage by attaching a link that connects the output of one crank to the input of another. Additional links can be attached to form a six-bar linkage or in series to form a robot.^[19]

Classification of machines[link]

The identification of simple machines arises from a desire for a systematic method to invent new machines. Therefore, an important concern is how simple machines are combined to make more complex machines. One approach is to attach simple machines in series to obtain compound machines.

However, a more successful strategy was identified by Franz Reuleaux, who collected and studied over 800 elementary machines. He realized that a lever, pulley and wheel and axle are in essence the same device, a body rotating about a hinge, and that an inclined plane, wedge and screw are similarly a block sliding on a flat surface.^[21]

This realization shows that it is the joints, or the connections that provide movement, that are the primary elements of a machine. Starting with four types of joints, the rotary joint, sliding joint, cam joint and gear joint, and related connections such as cables and belts, it is possible to understand a machine as an assembly of solid parts that connect these joints.^[19]

See also[link]

• Machine (mechanical)
• Mechanism (engineering)
• Linkage (mechanical)
• Four-bar linkage
• Six-bar linkage
• Gear train
• Mechanical advantage
• Power (physics)

References[link]