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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, torque is the rotational equivalent of linear
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 ...
. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of the body. The concept originated with the studies by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
of the usage of
lever A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is div ...
s, which is reflected in his famous quote: "''Give me a lever and a place to stand and I will move the Earth''". Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Torque is defined as the product of the magnitude of the perpendicular component of the force and the distance of the
line of action In physics, the line of action (also called line of application) of a force ''(F)'' is a geometric representation of how the force is applied. It is the line through the point at which the force is applied in the same direction as the vector ...
of a force from the point around which it is being determined. The law of
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 ...
can also be used to understand torque. The symbol for torque is typically \boldsymbol\tau, the lowercase
Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...
''
tau Tau (uppercase Τ, lowercase τ, or \boldsymbol\tau; el, ταυ ) is the 19th letter of the Greek alphabet, representing the voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value of 300. The name in English ...
''. When being referred to as moment of force, it is commonly denoted by . In three dimensions, the torque is a
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
; for
point particles A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
, it is given by the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of the position vector ( distance vector) and the force vector. The magnitude of torque of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
depends on three quantities: the force applied, the ''lever arm vector'' connecting the point about which the torque is being measured to the point of force application, and the angle between the force and lever arm vectors. In symbols: :\boldsymbol \tau = \mathbf\times \mathbf\,\! :\tau = \, \mathbf\, \,\, \mathbf\, \sin \theta\,\! where *\boldsymbol\tau is the torque vector and \tau is the magnitude of the torque, * \mathbf is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied), * \mathbf is the force vector, * \times denotes the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
, which produces a vector that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to both and following the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
, * \theta is the angle between the force vector and the lever arm vector. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
for torque is the
newton-metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...
(N⋅m). For more on the units of torque, see '.


History

The term ''torque'' (from
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
'' torquēre'' "to twist") is said to have been suggested by James Thomson and appeared in print in April, 1884. Usage is attested the same year by
Silvanus P. Thompson Silvanus Phillips Thompson (19 June 1851 – 12 June 1916) was a professor of physics at the City and Guilds Technical College in Finsbury, England. He was elected to the Royal Society in 1891 and was known for his work as an electrical eng ...
in the first edition of ''Dynamo-Electric Machinery''. Thompson motivates the term as follows: Today, torque is referred to using different vocabulary depending on geographical location and field of study. This article follows the definition used in US physics in its usage of the word ''torque''.''Physics for Engineering'' by Hendricks, Subramony, and Van Blerk, Chinappi page 148
Web link
In the UK and in US
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
, torque is referred to as ''moment of force'', usually shortened to ''moment''.Kane, T.R. Kane and D.A. Levinson (1985). ''Dynamics, Theory and Applications'' pp. 90–99
Free download
.
That term has been attested in French since at least 1811 by
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
in ''Traité de mécanique''
An English translation
of that work appears in 1842.


Definition and relation to angular momentum

A force applied perpendicularly to a lever multiplied by its distance from the lever's fulcrum (the length of the
lever arm In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of ...
) is its torque. A force of three
newtons The newton (symbol: N) is the unit of force in the International System of Units (SI). It is defined as 1 kg⋅m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in r ...
applied two
metre The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
s from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. The direction of the torque can be determined by using the
right hand grip rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
: if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque. More generally, the torque on a point particle (which has the position r in some reference frame) can be defined as the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
: :\boldsymbol = \mathbf \times \mathbf, where F is the force acting on the particle. The magnitude ''τ'' of the torque is given by :\tau = rF\sin\theta, where ''F'' is the magnitude of the force applied, and ''θ'' is the angle between the position and force vectors. Alternatively, :\tau = rF_, where ''F'' is the amount of force directed perpendicularly to the position of the particle. Any force directed parallel to the particle's position vector does not produce a torque. It follows from the properties of the cross product that the ''torque vector'' is perpendicular to both the ''position'' and ''force'' vectors. Conversely, the ''torque vector'' defines the plane in which the ''position'' and ''force'' vectors lie. The resulting ''torque vector'' direction is determined by the right-hand rule. The net torque on a body determines the rate of change of the body's
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, :\boldsymbol = \frac where L is the angular momentum vector and ''t'' is time. For the motion of a point particle, :\mathbf = I\boldsymbol, where is the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
and ω is the orbital
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
pseudovector. It follows that :\boldsymbol_ = \frac = \frac = I\frac + \frac\boldsymbol = I\boldsymbol + \frac\boldsymbol = I\boldsymbol + 2rp_\boldsymbol, where ''α'' is the
angular acceleration In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceler ...
of the particle, and ''p'', , is the radial component of its
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
. This equation is the rotational analogue of
Newton's Second Law 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 ...
for point particles, and is valid for any type of trajectory. Note that although force and acceleration are always parallel and directly proportional, the torque ''τ'' need not be parallel or directly proportional to the angular acceleration ''α''. This arises from the fact that although mass is always conserved, the moment of inertia in general is not. In some simple cases like a rotating disc, the moment of inertia is a constant, the rotational Newton's Second Law can be \boldsymbol = I\boldsymbol where I = mr^2 and \boldsymbol\alpha = .


Proof of the equivalence of definitions

The definition of angular momentum for a single point particle is: \mathbf = \mathbf \times \mathbf where p is the particle's
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
and r is the position vector from the origin. The time-derivative of this is: \frac = \mathbf \times \frac + \frac \times \mathbf. This result can easily be proven by splitting the vectors into components and applying the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
. Now using the definition of force \mathbf = \frac (whether or not mass is constant) and the definition of velocity \frac = \mathbf \frac = \mathbf \times \mathbf + \mathbf \times \mathbf. The cross product of momentum \mathbf with its associated velocity \mathbf is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque ''τ'' = r × F. Therefore, torque on a particle is ''equal'' to the
first derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of its angular momentum with respect to time. If multiple forces are applied, Newton's second law instead reads , and it follows that \frac = \mathbf \times \mathbf_ = \boldsymbol_. This is a general proof for point particles. The proof can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass.


Units

Torque has the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of force times
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
, symbolically . Although those fundamental dimensions are the same as that for
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
or
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 tr ...
, official SI literature suggests using the unit ''
newton-metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...
'' (N⋅m) and never the
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
.From th
official SI website
, The International System of Units – 9th edition – Text in English Section 2.3.4: "...For example, the quantity torque is the cross product of a position vector and a force vector. The SI unit is newton metre. Even though torque has the same dimension as energy (SI unit joule), the joule is never used for expressing torque."
The unit ''newton metre'' is properly denoted N⋅m. The traditional imperial and U.S. customary units for torque are the pound foot (lbf-ft), or for small values the pound inch (lbf-in). In the US, torque is most commonly referred to as the foot-pound (denoted as either lb-ft or ft-lb) and the inch-pound (denoted as in-lb). Demonstration that, as in most US industrial settings, the torque ranges are given in ft-lb rather than lbf-ft. Practitioners depend on context and the hyphen in the abbreviation to know that these refer to torque and not to energy or moment of mass (as the symbolism ft-lb would properly imply).


Special cases and other facts


Moment arm formula

A very useful special case, often given as the definition of torque in fields other than physics, is as follows: :\tau = (\text) (\text). The construction of the "moment arm" is shown in the figure to the right, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three-dimensional cases. If the force is perpendicular to the displacement vector r, the moment arm will be equal to the distance to the centre, and torque will be a maximum for the given force. The equation for the magnitude of a torque, arising from a perpendicular force: :\tau = (\text) (\text). For example, if a person places a force of 10 N at the terminal end of a wrench that is 0.5 m long (or a force of 10 N acting 0.5 m from the twist point of a wrench of any length), the torque will be 5 N⋅m – assuming that the person moves the wrench by applying force in the plane of movement and perpendicular to the wrench.


Static equilibrium

For an object to be in static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments) about any point. For a two-dimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: and , and the torque a third equation: . That is, to solve
statically determinate In statics and structural mechanics, a structure is statically indeterminate when the static equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and Reaction (physics), reactions on tha ...
equilibrium problems in two-dimensions, three equations are used.


Net force versus torque

When the net force on the system is zero, the torque measured from any point in space is the same. For example, the torque on a current-carrying loop in a uniform magnetic field is the same regardless of the point of reference. If the net force \mathbf is not zero, and \boldsymbol_1 is the torque measured from \mathbf_1, then the torque measured from \mathbf_2 is \boldsymbol_2 = \boldsymbol_1 + (\mathbf_1 - \mathbf_2) \times \mathbf


Machine torque

Torque forms part of the basic specification of an
engine An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy. Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power gen ...
: the
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
output of an engine is expressed as its torque multiplied by the angular speed of the drive shaft.
Internal-combustion An internal combustion engine (ICE or IC engine) is a heat engine in which the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal combus ...
engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000–6,000 rpm for a small car). One can measure the varying torque output over that range with a
dynamometer A dynamometer or "dyno" for short, is a device for simultaneously measuring the torque and rotational speed (RPM) of an engine, motor or other rotating prime mover so that its instantaneous power may be calculated, and usually displayed by the ...
, and show it as a torque curve.
Steam engine A steam engine is a heat engine that performs mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a cylinder. This pushing force can be trans ...
s and
electric motor An electric motor is an Electric machine, electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a Electromagneti ...
s tend to produce maximum torque close to zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Reciprocating steam-engines and electric motors can start heavy loads from zero rpm without a
clutch A clutch is a mechanical device that engages and disengages power transmission, especially from a drive shaft to a driven shaft. In the simplest application, clutches connect and disconnect two rotating shafts (drive shafts or line shafts). ...
.


Relationship between torque, power, and energy

If a
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 ...
is allowed to act through a distance, it is doing
mechanical work In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stren ...
. Similarly, if torque is allowed to act through an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, the work ''W'' can be expressed as : W = \int_^ \tau\ \mathrm\theta, where ''τ'' is torque, and ''θ''1 and ''θ''2 represent (respectively) the initial and final angular positions of the body.


Proof

The work done by a variable force acting over a finite linear displacement s is given by integrating the force with respect to an elemental linear displacement \mathrm\mathbf :W = \int_^ \mathbf \cdot \mathrm\mathbf However, the infinitesimal linear displacement \mathrm\mathbf is related to a corresponding angular displacement \mathrm\boldsymbol and the radius vector \mathbf as :\mathrm\mathbf = \mathrm\boldsymbol\times\mathbf Substitution in the above expression for work gives :W = \int_^ \mathbf \cdot \mathrm\boldsymbol \times \mathbf The expression \mathbf\cdot\mathrm\boldsymbol\times\mathbf is a
scalar triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
given by \left mathbf\,\mathrm\boldsymbol\,\mathbf\right/math>. An alternate expression for the same scalar triple product is :\left mathbf \, \mathrm\boldsymbol\,\mathbf\right= \mathbf \times \mathbf \cdot \mathrm\boldsymbol But as per the definition of torque, :\boldsymbol = \mathbf \times \mathbf Corresponding substitution in the expression of work gives, :W = \int_^ \boldsymbol \cdot \mathrm\boldsymbol Since the parameter of integration has been changed from linear displacement to angular displacement, the limits of the integration also change correspondingly, giving :W = \int_^ \boldsymbol \cdot \mathrm\boldsymbol If the torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., \boldsymbol\cdot \mathrm\boldsymbol = \left, \boldsymbol\ \left, \mathrm\boldsymbol\\cos 0 = \tau \, \mathrm\theta giving :W = \int_^ \tau \, \mathrm\theta It follows from the
work–energy principle In physics, work is the energy transferred to or from an object via the application of force along a Displacement (vector), displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the pro ...
that ''W'' also represents the change in the
rotational kinetic energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the ...
''E''r of the body, given by :E_ = \tfracI\omega^2, where ''I'' is the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
of the body and ''ω'' is its
angular speed Angular may refer to: Anatomy * Angular artery, the terminal part of the facial artery * Angular bone, a large bone in the lower jaw of amphibians and reptiles * Angular incisure, a small anatomical notch on the stomach * Angular gyrus, a regio ...
.
Power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
is the work per unit
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
, given by :P = \boldsymbol \cdot \boldsymbol, where ''P'' is power, ''τ'' is torque, ''ω'' is the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
, and \cdot represents the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. Algebraically, the equation may be rearranged to compute torque for a given angular speed and power output. Note that the power injected by the torque depends only on the instantaneous angular speed – not on whether the angular speed increases, decreases, or remains constant while the torque is being applied (this is equivalent to the linear case where the power injected by a force depends only on the instantaneous speed – not on the resulting acceleration, if any). In practice, this relationship can be observed in
bicycle A bicycle, also called a pedal cycle, bike or cycle, is a human-powered or motor-powered assisted, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A is called a cyclist, or bicyclist. Bic ...
s: Bicycles are typically composed of two road wheels, front and rear gears (referred to as
sprockets A sprocket, sprocket-wheel or chainwheel is a profiled wheel with teeth that mesh with a chain, track or other perforated or indented material. The name 'sprocket' applies generally to any wheel upon which radial projections engage a chain passi ...
) meshing with a
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A c ...
, and a derailleur mechanism if the bicycle's transmission system allows multiple gear ratios to be used (i.e. multi-speed bicycle), all of which attached to the
frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...
. A
cyclist Cycling, also, when on a two-wheeled bicycle, called bicycling or biking, is the use of cycles for transport, recreation, exercise or sport. People engaged in cycling are referred to as "cyclists", "bicyclists", or "bikers". Apart from two ...
, the person who rides the bicycle, provides the input power by turning pedals, thereby cranking the front sprocket (commonly referred to as
chainring The crankset (in the US) or chainset (in the UK), is the component of a bicycle drivetrain that converts the reciprocating motion of the rider's legs into rotational motion used to drive the chain or belt, which in turn drives the rear whee ...
). The input power provided by the cyclist is equal to the product of angular speed (i.e. the number of pedal revolutions per minute times 2''π'') and the torque at the
spindle Spindle may refer to: Textiles and manufacturing * Spindle (textiles), a straight spike to spin fibers into yarn * Spindle (tool), a rotating axis of a machine tool Biology * Common spindle and other species of shrubs and trees in genus ''Euony ...
of the bicycle's
crankset The crankset (in the US) or chainset (in the UK), is the component of a bicycle drivetrain that converts the reciprocating motion of the rider's legs into rotational motion used to drive the chain or belt, which in turn drives the rear wheel. ...
. The bicycle's
drivetrain A drivetrain (also frequently spelled as drive train or sometimes drive-train) is the group of components that deliver mechanical power from the prime mover to the driven components. In automotive engineering, the drivetrain is the components o ...
transmits the input power to the road
wheel A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...
, which in turn conveys the received power to the road as the output power of the bicycle. Depending on the
gear ratio A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission ...
of the bicycle, a (torque, angular speed)input pair is converted to a (torque, angular speed)output pair. By using a larger rear gear, or by switching to a lower gear in multi-speed bicycles,
angular speed Angular may refer to: Anatomy * Angular artery, the terminal part of the facial artery * Angular bone, a large bone in the lower jaw of amphibians and reptiles * Angular incisure, a small anatomical notch on the stomach * Angular gyrus, a regio ...
of the road wheels is decreased while the torque is increased, product of which (i.e. power) does not change. For SI units, the unit of power is the
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
, the unit of torque is the
newton metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...
and the unit of angular speed is the
radian per second The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omega) ...
(not rpm and not revolutions per second). The unit newton-metre is dimensionally equivalent to the
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
, which is the unit of energy. In the case of torque, the unit is assigned to a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, whereas for
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
, it is assigned to a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
. This means that the dimensional equivalence of the newton-metre and the joule may be applied in the former, but not in the latter case. This problem is addressed in orientational analysis, which treats the radian as a base unit rather than as a dimensionless unit.


Conversion to other units

A conversion factor may be necessary when using different units of power or torque. For example, if
rotational speed Rotational frequency (also known as rotational speed or rate of rotation) of an object Rotation around a fixed axis, rotating around an axis is the frequency of rotation of the object. Its unit is revolution per minute (rpm), cycle per second ...
(revolutions per time) is used in place of angular speed (radians per time), we multiply by a factor of 2 radians per revolution. In the following formulas, ''P'' is power, ''τ'' is torque, and ''ν'' ( Greek letter nu) is rotational speed. :P = \tau \cdot 2 \pi \cdot \nu Showing units: : P _ = \tau _ \cdot 2 \pi _ \cdot \nu _ Dividing by 60 seconds per minute gives us the following. : P _ = \frac where rotational speed is in revolutions per minute (rpm). Some people (e.g., American automotive engineers) use
horsepower Horsepower (hp) is a unit of measurement of power, or the rate at which work is done, usually in reference to the output of engines or motors. There are many different standards and types of horsepower. Two common definitions used today are the ...
(mechanical) for power, foot-pounds (lbf⋅ft) for torque and rpm for rotational speed. This results in the formula changing to: : P _ = \frac . The constant below (in foot-pounds per minute) changes with the definition of the horsepower; for example, using metric horsepower, it becomes approximately 32,550. The use of other units (e.g.,
BTU The British thermal unit (BTU or Btu) is a unit of heat; it is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. It is also part of the United States customary units. The modern SI u ...
per hour for power) would require a different custom conversion factor.


Derivation

For a rotating object, the ''linear distance'' covered at the
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
of rotation is the product of the radius with the angle covered. That is: linear distance = radius × angular distance. And by definition, linear distance = linear speed × time = radius × angular speed × time. By the definition of torque: torque = radius × force. We can rearrange this to determine force = torque ÷ radius. These two values can be substituted into the definition of
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
: : \begin \text & = \frac \\ pt& = \frac t \\ pt& = \text \cdot \text. \end The radius ''r'' and time ''t'' have dropped out of the equation. However, angular speed must be in radians per unit of time, by the assumed direct relationship between linear speed and angular speed at the beginning of the derivation. If the rotational speed is measured in revolutions per unit of time, the linear speed and distance are increased proportionately by 2 in the above derivation to give: : \text = \text \cdot 2 \pi \cdot \text. \, If torque is in newton metres and rotational speed in revolutions per second, the above equation gives power in newton metres per second or watts. If Imperial units are used, and if torque is in pounds-force feet and rotational speed in revolutions per minute, the above equation gives power in foot pounds-force per minute. The horsepower form of the equation is then derived by applying the conversion factor 33,000 ft⋅lbf/min per horsepower: : \begin \text & = \text \cdot 2 \pi \cdot \text \cdot \frac \cdot \frac \\ pt& \approx \frac \end because 5252.113122 \approx \frac . \,


Principle of moments

The principle of moments, also known as
Varignon's theorem Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was ...
(not to be confused with the geometrical theorem of the same name) states that the resultant torques due to several forces applied to about a point is equal to the sum of the contributing torques: :\tau = \mathbf_1\times\mathbf_1 + \mathbf_2\times\mathbf_2 + \ldots + \mathbf_N\times\mathbf_N. From this it follows that the torques resulting from two forces acting around a pivot on an object are balanced when :\mathbf_1\times\mathbf_1 + \mathbf_2\times\mathbf_2 = \mathbf.


Torque multiplier

Torque can be multiplied via three methods: by locating the fulcrum such that the length of a lever is increased; by using a longer lever; or by the use of a speed reducing gearset or
gear box Propulsion transmission is the mode of transmitting and controlling propulsion power of a machine. The term ''transmission'' properly refers to the whole drivetrain, including clutch, gearbox, prop shaft (for rear-wheel drive vehicles), differe ...
. Such a mechanism multiplies torque, as rotation rate is reduced.


See also


References


External links


"Horsepower and Torque"
An article showing how power, torque, and gearing affect a vehicle's performance.

An automotive perspective
''Torque and Angular Momentum in Circular Motion ''
o
Project PHYSNET



Torque Unit Converter

A feel for torque
An order-of-magnitude interactive. {{Classical mechanics SI units Physical quantities Rotation Force Moment (physics)