Simple_machine - Wikiparency

# Simple machine A **simple machine** is a [[Machine|mechanical device]] that changes the direction or the magnitude of a [[Force|force]]. The term usually refers to the six classical machines of Renaissance mechanics: the [[Lever|lever]], the [[Wheel_and_axle|wheel and axle]], the [[Pulley|pulley]], the [[Inclined_plane|inclined plane]], the [[Wedge|wedge]] and the [[Screw_mechanism|screw]]. Each is among the simplest [[Mechanism_(engineering)|mechanisms]] that provide [[Mechanical_advantage|mechanical advantage]], and every more complicated machine can be described as some combination of them. What the six have in common is more important than what separates them. None of them creates [[Energy|energy]]. Each takes in [[Work_(physics)|work]] at one point and delivers work at another, and if nothing is lost the two are equal, so a machine that multiplies force by a factor must move the load through a correspondingly shorter distance. The ideal factor is a pure ratio of lengths — two lever arms, two radii, the slope of a ramp, the pitch of a thread, the number of rope falls — and it can be read off the drawing before any force is measured. [[Friction|Friction]] does not change that geometry; it only raises the effort needed to realise it, and when it is severe enough it makes a machine refuse to run backwards at all. The same devices survive in modern kinematics, but not as a list of six. A lever, a pulley and a wheel and axle are all a body turning about a hinge; a wedge, a ramp and a screw are all a body sliding on a surface. Treating the joints rather than the parts as the primitives turns the classical list into the kinematic pairs from which [[Linkage_(mechanical)|linkages]], [[Cam_(mechanism)|cam and follower mechanisms]] and [[Gear_train|gear trains]] are built. The framework microsim *Simple machines: force multiplied, distance divided, work unchanged* puts one machine at a time on screen: the reader chooses the lever, pulley, inclined plane, wedge, screw or wheel and axle, sets its geometry, the load and the efficiency, and reads two groups of bars — the ideal effort, the real effort and the load on one side, the distance each travels on the other. Seven See-also variants open the same model on one machine each. ## History The idea of a simple machine is usually traced to [[Archimedes|Archimedes]] in the third century BC, who is credited with the analysis of the lever and with the remark, traditionally reported in the form "Give me a place to stand, and I will move the Earth," that expresses the unbounded force amplification a long enough arm would allow.[^history] Three of the classical machines — lever, pulley and screw — belong to that Archimedean tradition. [[Hero_of_Alexandria|Hero of Alexandria]], writing in the first century AD, treated five: the windlass, the lever, the pulley, the wedge and the screw.[^history] The inclined plane was added later, and the canonical set of six was settled by Renaissance writers; [[Galileo_Galilei|Galileo]] in *Le Meccaniche* argued that the machines were not six separate tricks but one principle applied six ways, the principle that force gained is distance lost.[^history] The list is conventional rather than fundamental, and its members overlap. A wedge is an inclined plane that moves instead of standing still, which is why some authors exclude it; a screw is an inclined plane wound around a cylinder, though it converts a [[Torque|torque]] rather than a straight push, which is why most authors keep it. A [[Windlass|windlass]] and a capstan are wheels and axles. The lasting value of the list is not taxonomic but practical: it is the shortest vocabulary in which the force-for-distance trade of any mechanism can be described, and it is still the vocabulary used when a mechanism is sized before it is drawn. ## Ideal simple machine A simple machine that dissipates nothing through [[Friction|friction]], [[Wear|wear]] or deformation is called ideal: energy is conserved through the device, the [[Power_(physics)|power]] in equals the power out, and its performance follows from its dimensions alone. With an effort F through a distance d_E and a load W through d_L, conservation of work gives `F d_E = W d_L`, and the **mechanical advantage** is the ratio `MA = W/F = d_E/d_L = a_E/a_L`, the same number read as a force ratio, a distance ratio, or a ratio of radii on a rotating machine. Because the ratio is geometric, each machine has a closed form: the [[Lever|lever]] gives `MA = a_E/a_L`, the effort arm over the load arm; the [[Wheel_and_axle|wheel and axle]] gives `R/r`, a lever taken all the way round; a [[Pulley|pulley]] system gives n, the number of rope falls supporting the load; an [[Inclined_plane|inclined plane]] gives `1/sin θ` for a load pushed along the slope; and a [[Screw_mechanism|screw]] turned by an effort at radius r gives `2πr/p` for pitch p.[^up077][^jensen215] The screw shows how large a geometric ratio can get. With a 2 mm pitch and an effort applied at a 20 mm radius, the ideal advantage is `2π × 20 / 2 = 62.8`: a newton at the handle balances 62.8 N at the nut. The same number follows from the lead-screw form `F = 2πT/l`, since a torque of 1 N × 0.02 m acting through a 2 mm lead gives 62.8 N.[^jensen165] Nothing is free: raising that load 0.1 m takes 6.28 m at the handle, twenty turns. A ramp is gentler in both senses — at 30° the advantage is `1/sin 30° = 2.00`, and the load travels only twice as far as it rises.[^up077] The wedge is the machine the framework's library does not name; the sim computes it in its own hooks as `MA = 1/tan(α/2)` for a symmetric wedge of included angle α, counting the total force parting both faces, which for a 15° wedge is 7.6.[^simspec] *Try: switch machine from the lever to the screw at a 2 mm pitch and watch the advantage jump from the lever's 3 to 62.8 while the effort-distance bar grows by the same factor and the two work totals stay level; then drop efficiency to 0.3 and see the real effort bar rise above the ideal one, and set the pulley to a single rope fall, where the effort is turned around but not multiplied.* ## Friction and efficiency Real machines rub. Part of the input [[Power_(physics)|power]] is dissipated as heat, so `P_in = P_out + P_fric`, and the [[Mechanical_efficiency|mechanical efficiency]] is the ratio of what comes out to what goes in, `η = P_out/P_in`, a number between 0 and 1. Because the geometry of the machine is unchanged by friction, the distance ratio stays what the drawing says it is and the whole loss appears in the force: the actual advantage is the ideal advantage scaled by the efficiency, `MA_real = η × MA_ideal`, the rule design texts state as the real advantage being the ideal one times the efficiency.[^jensen224] The consequences are easiest to see at the ends of the range. A lever with an arm ratio of 4 driven at 80 percent efficiency returns an actual advantage of 3.2 rather than 4 — a modest tax, and typical of machines whose joints roll or pivot.[^simspec] A sliding machine can do far worse. At 30 percent efficiency the 62.8 of an ideal screw falls to 18.8, and a single fixed pulley, whose ideal advantage is exactly 1 because it only redirects the rope, becomes a machine that costs more force than it saves: `η × MA = 0.3`, so the effort must exceed the load. That is not a defect of the model but the ordinary reason a rusty pulley is replaced rather than pulled harder. Efficiency is also what separates the two things the word "advantage" can mean. The ideal advantage is a property of the drawing; the actual advantage is a property of the built machine on the day it is measured, and it falls as surfaces wear, lubricant thins or a thread picks up dirt. The [[Mechanical_advantage|mechanical advantage]] variant of this section's sim shows both at once, the ideal ratio and the real one standing apart on the same bar panel, and the other variants open on one machine each — a class-1 [[Lever|lever]] at an arm ratio of 5, a [[Block_and_tackle|block and tackle]] with four supporting falls, a 20° ramp at `MA = 2.92`, a 15° [[Wedge|wedge]] at 7.6, a screw jack at 62.8 and a [[Wheel_and_axle|wheel and axle]] at `R/r = 5`.[^simspec] Nothing in those panels is an illustrative fit: every number on screen is a ratio returned by the framework's machine library, and only the drawing is scaled, with the force arrows capped so that a large advantage does not run off the frame.[^simspec] ## Compound machines A compound machine is a set of simple machines in series, the output force of one becoming the input force of the next. A bench [[Vise|vise]] is a lever — the handle — in series with a screw; a [[Gear_train|gear train]] is a series of wheels and axles; a hand-operated [[Crane_(machine)|crane]] is a windlass in series with a [[Block_and_tackle|block and tackle]]. Because each stage multiplies the force it receives, the ideal advantages multiply: `MA_compound = MA₁ × MA₂ × … × MA_N`. The efficiencies multiply as well, and that is the more demanding rule, because each stage keeps only its own fraction of what the stage before it delivered: `η_compound = η₁ × η₂ × … × η_N`. A vise whose handle lever has a ratio of 5 and whose screw has the 62.8 computed above has an ideal advantage of 314. If the lever is 90 percent efficient and the sliding thread only 30 percent — representative figures for a rolling joint and a dry square thread rather than measurements of any particular vise — the compound efficiency is 0.27 and the actual advantage is about 85. A 100 N pull on the handle still closes the jaws with something like 8.5 kN, which is why a vise holds, and the missing three quarters of the input is the heat in the thread. Two lessons follow for design. Adding stages is a cheap way to buy force and an expensive way to lose efficiency, so a drive is built from as few stages as the ratio allows; and the worst stage dominates, so effort spent improving a rolling stage that is already at 0.95 is wasted beside a sliding stage at 0.3. The same arithmetic governs [[Gear|gear]] reductions, where each mesh is typically 0.97 to 0.99 efficient and a three-stage train still returns more than 0.9, and worm drives, where one stage can cost more than half the input. ## Self-locking machines Most machines run in either direction. If the load on a lever is large enough it drives the lever backwards, doing work on whatever holds the handle; a raised weight on a rope will spin a windlass if the crank is let go. Machines in which this reversal cannot happen at any load, even with the input force removed, are called self-locking, non-reversible or non-overhauling. Self-locking appears mainly where the machine has large areas of sliding contact — the screw, the inclined plane and the wedge. A [[Screw_thread|threaded fastener]] is the everyday case: a [[Bolt_(fastener)|bolt]] can be turned in or out with a spanner, but no amount of tension along its axis will make it rotate. A [[Jackscrew|screw jack]] holds a car in the air with nothing but its own friction, and a [[Wedge|wedge]] driven into a log stays there when the hammer stops. The condition is stated in terms of efficiency alone: a machine is self-locking if and only if its efficiency is below 50 percent. Whether a given machine falls below that line depends on both the friction at its sliding surfaces and its ideal advantage: a screw locks when its lead angle is smaller than the friction angle `arctan μ`, and a finer thread — a larger ideal advantage — has a smaller lead angle, so the same pair of materials that overhauls on a coarse thread holds on a fine one. This is the one place in the subject where a low efficiency is the specification rather than a fault: a self-locking [[Leadscrew|leadscrew]] needs no brake, and a [[Worm_drive|worm drive]] is chosen for hoists precisely because the load cannot drive it backwards. ### Proof The criterion follows from the same bookkeeping that defines efficiency. Driving the machine forward through a small motion, the input does [[Work_(physics)|work]] `W_in = F d_E`, the load receives `W_out = η W_in`, and the difference `(1 − η) W_in` is lost to [[Friction|friction]]. Now let the load drive the machine backwards through the same geometry. The load now supplies `W_out` and the friction takes the same absolute amount as before, because the surfaces slide the same distances under the same normal forces, so the work returned at the input point is `W_back = W_out − (1 − η) W_in = W_out (1 − (1 − η)/η)`, and the efficiency of the reversed machine is `η_rev = (2η − 1)/η = 2 − 1/η`. The result is negative — meaning that no load, however large, can supply the friction losses of the return motion — exactly when `η < 0.5`. At η = 0.6 the reverse efficiency is 0.33 and the machine overhauls readily; at η = 0.5 it is zero, the marginal case; at η = 0.3 it is −1.33 and the machine is firmly locked. The asymmetry is worth stating plainly: a machine at η = 0.5 gives back half of what is put into it when driven forward, while the same friction absorbs everything the load can supply on the way back. ## Modern machine theory Modern [[Kinematics|kinematics]] does not treat machines as a catalogue of devices but as mechanical systems of [[Actuator|actuators]] and mechanisms that transmit forces and movement under sensors and controllers. The parts that matter are links — rigid bodies — and the joints that connect them, and a machine is analysed as the chain those joints form rather than as the objects a parts list would name. The shift is more than bookkeeping. A classical list answers the question "what kind of machine is this?", which a designer rarely needs to ask, while a joint-and-link description answers the questions that are actually asked of a mechanism: how many inputs does it need, what path does a given point trace, where does it lose the ability to transmit force, and which other mechanism does the same job with fewer parts. The three subsections below are that programme in order — the chains themselves, the classification Franz Reuleaux built from them, and the synthesis problem that runs the classification backwards from a required motion to the mechanism that produces it. ### Kinematic chains A [[Kinematic_chain|kinematic chain]] is an assembly of links connected by joints, and the simple machines are its elementary examples. The bearing at a lever's fulcrum, and the ones that let a pulley or a wheel and axle turn, are all the [[Kinematic_pair|kinematic pair]] called a [[Revolute_joint|revolute joint]]; the flat face of an inclined plane or a wedge is the [[Prismatic_joint|prismatic joint]]; the screw is the helical pair, a rotation and a translation locked together by the lead. Counting these joints gives a machine's mobility, the number of independent inputs it needs, by the standard planar formula `M = 3(L − 1) − 2J₁ − J₂` for L links with J₁ one-freedom and J₂ two-freedom joints. Chaining the elements produces the mechanisms that actually do work. Two levers joined by a link that carries the output of one to the input of the other form a [[Four-bar_linkage|four-bar linkage]], the simplest closed chain that moves: four links and four revolute joints give `M = 3(4 − 1) − 2 × 4 = 1`, so one crank angle fixes the whole assembly. Add links and the count grows to the [[Six-bar_linkage|six-bar linkage]], and, taken as an open chain with a motor at every joint, to the serial [[Robot|robot]] arm, whose [[Degrees_of_freedom_(mechanics)|degrees of freedom]] are counted by the same rule. ### Classification of machines The wish for a systematic way to invent machines, rather than merely to list them, is what produced this framing. [[Franz_Reuleaux|Franz Reuleaux]] assembled and studied several hundred elementary mechanisms in the nineteenth century — the surviving teaching models are held in Cornell University's KMODDL collection — and drew the conclusion that reorganised the field: a lever, a pulley and a wheel and axle are in essence one device, a body rotating about a hinge, and an inclined plane, a wedge and a screw are one device too, a body sliding on a surface.[^reuleaux][^kmoddl] What distinguishes machines, on this view, is not their parts but their connections. Starting from four joint types — the revolute, the sliding joint, the cam joint and the gear joint — and the links between them, the classical six become special cases of a much larger family that also contains [[Linkage_(mechanical)|linkages]], [[Cam_(mechanism)|cam and follower mechanisms]] and [[Gear_train|gear trains]]. The practical gain is that a designer can enumerate: given a required mobility and a set of available joints, the admissible chains can be listed and compared instead of recalled. ### Kinematic synthesis The design of mechanisms to produce a required movement and force transmission is [[Kinematic_synthesis|kinematic synthesis]], a body of geometric techniques for choosing link lengths, cam profiles and gear ratios to meet a specification. Its questions are the inverse of the ones above: not what a given four-bar does, but which four-bar carries a point along a wanted path, holds a wanted angle, or keeps its transmission angle inside a usable window through the whole cycle. The [[Four-bar_linkage|four-bar linkage]], the [[Cam_(mechanism)|cam]] and the [[Gear|gear]] train are the three standard answers, and each has its own article and its own microsim in this portal. ## See also - [[Mechanical_advantage]] - [[Lever]] - [[Pulley]] - [[Inclined_plane]] - [[Wedge]] - [[Screw_mechanism]] - [[Wheel_and_axle]] - [[Linkage_(mechanical)]] - [[Cam_(mechanism)]] - [[Gear_train]] - [[Mechanism_(engineering)]] - [[Rolamite]] ## References [^up077]: OpenStax. *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Portal Book 077 — the source the framework's `mech.machines` module cites for the ideal-advantage rule of each simple machine and for work in = work out. https://openstax.org/details/books/university-physics-volume-1 (chapter and page numbers not pinned for this section in the run's fact sheet). [^jensen215]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, pp. 215–218: mechanical advantage as a trade of force for distance, and the advantage of levers, pulleys, ramps and gears as ratios of geometry. [^jensen165]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, pp. 165, 182–183: the screw and the lead-screw relation `F = 2πT/l`. [^jensen224]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, p. 224: the real mechanical advantage is the ideal advantage times the efficiency. [^simspec]: Engineering portal sim spec `specs/sims/Simple_machine.json` with the framework module `mech.machines` (`leverAdvantage`, `pulleyAdvantage`, `inclineAdvantage`, `screwAdvantage`, `work`); real advantage = η × ideal, `F = W/(η MA)`, `d_E = MA × 0.1 m`. The wedge is a library gap: `mech.machines` has no wedge function, and `MA = 1/tan(α/2)` is computed in the sim's own hooks under a `// LIB GAP` comment. Build report, engineering run, September 18, 2026: no quantity on screen is ILLUSTRATIVE; the drawing conventions (arrow scale, the 1.5-unit cap on the force arrow, block spacing, the wedge halves hinged at the crack) are stated on the sim's sources sheet. [^history]: *Citation needed.* The attributions in this section — Archimedes on the lever in the third century BC and the "place to stand" remark, Hero of Alexandria's five machines, and Galileo's treatment of the six as one principle in *Le Meccaniche* — are standard in histories of mechanics but are not supported by any Portal Book carried in this run, whose sources are engineering texts rather than historical ones. The records that would settle them are Pappus of Alexandria's *Synagoge*, Book VIII, for the Archimedes remark; the surviving Arabic text of Hero's *Mechanica*; and a modern edition of Galileo's *Le Meccaniche* (composed c. 1600). [^reuleaux]: Reuleaux, Franz (1876). *The Kinematics of Machinery: Outlines of a Theory of Machines*. Translated and edited by Alexander B. W. Kennedy. London: Macmillan. (Translator, publisher and the size of Reuleaux's mechanism collection as recalled; the title, year and the joint-based classification are standard in histories of kinematics.) [^kmoddl]: Cornell University Library. *Kinematic Models for Design Digital Library (KMODDL)* — the Reuleaux collection of kinematic teaching models, with photographs and film of the working models. https://kmoddl.library.cornell.edu/  **Microsim — three.js (Wikitube framework):** *Simple machine* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Simple_machine.html" data-title="Simple machine"></div> *Built from `MICROSIM_GUIDE/specs/sims/Simple_machine.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Simple_machine) : [Wikitube](https://en.wikitube.io/wiki/Simple_machine) - skeleton pinned to revision 1369419169 (2026-09-18).