In

Web link

/ref> In the UK and in US

Free download

For example, a rotational force applied to a shaft causing acceleration, such as a drill bit accelerating from rest, results in a moment called a ''torque''. By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the angular momentum of the beam is not changing, this bending moment is not called a ''torque''. Similarly with any force couple on an object that has no change to its angular momentum, such moment is also not called a ''torque''.

_{⊥} 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 _{, , } is the radial component of its

official SI website

"...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).

_{1} and ''θ''_{2} represent (respectively) the initial and final

_{r} of the body, given by
:$E\_\; =\; \backslash tfracI\backslash omega^2,$
where ''I'' is the _{input} pair is converted to a (torque, rpm)_{output} pair. By using a larger rear gear, or by switching to a lower gear in multi-speed bicycles,

"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

physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ...

, torque is the rotational equivalent of linear force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study. The concept originated with the studies by Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

of the usage of lever
A lever ( or ) is a simple machine
A simple machine is a mechanical device that changes the direction or magnitude of a force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge o ...

s. 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. Another definition of torque is the product of the magnitude of the force and the perpendicular 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 (mathematics), line through the point at which the force is applied in the same direction ...

of a force from the axis of rotation
Rotation around a fixed axis is a special case of rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotati ...

. The symbol for torque is typically $\backslash boldsymbol\backslash tau$ or , the lowercase Greek letter
The Greek alphabet has been used to write the Greek language
Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, Indo-European family of languages, nat ...

''tau
Tau (uppercase Τ, lowercase τ; el, ταυ ) is the 19th letter of the Greek alphabet
The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...

''. When being referred to as moment of force, it is commonly denoted by .
In three dimensions, the torque is a pseudovector
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

; for point particles
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...

, it is given by the cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of the position vector ( distance vector) and the force vector. The magnitude of torque of a rigid body
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and 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:
:$\backslash boldsymbol\; \backslash tau\; =\; \backslash mathbf\backslash times\; \backslash mathbf\backslash ,\backslash !$
:$\backslash tau\; =\; \backslash ,\; \backslash mathbf\backslash ,\; \backslash ,\backslash ,\; \backslash mathbf\backslash ,\; \backslash sin\; \backslash theta\backslash ,\backslash !$
where
*$\backslash boldsymbol\backslash tau$ is the torque vector and $\backslash tau$ is the magnitude of the torque,
*$\backslash mathbf$ is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied),
*$\backslash mathbf$ is the force vector,
*$\backslash times$ denotes the cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

, which produces a vector that is perpendicular
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

to both and following the right-hand rule
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

,
*$\backslash theta$ 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
The metric system is a that succeeded the decimal ...

for torque is the newton-metre
The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of acti ...

(N⋅m). For more on the units of torque, see .
Defining terminology

James Thomson, the brother ofLord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of ...

, introduced the term ''torque'' into English scientific literature in 1884. However, 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 148Web link

/ref> In the UK and in US

mechanical engineering
Mechanical engineering is an engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineerin ...

, torque is referred to as ''moment of force'', usually shortened to ''moment''. These terms are interchangeable in US physics and UK physics terminology, unlike in US mechanical engineering, where the term ''torque'' is used for the closely related "resultant moment of a couple".
Torque and moment in the US mechanical engineering terminology

In US mechanical engineering, ''torque'' is defined mathematically as the rate of change ofangular momentum
In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant.
In three , the ...

of an object (in physics it is called "net torque"). The definition of torque states that one or both of the angular velocity
In physics, angular velocity (\boldsymbol or \boldsymbol), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angu ...

or 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
In physics
Physics is the natural science that studies matter, its ...

of an object are changing. ''Moment'' is the general term used for the tendency of one or more applied force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s to rotate an object about an axis, but not necessarily to change the angular momentum of the object (the concept which is called ''torque'' in physics).Kane, T.R. Kane and D.A. Levinson (1985). ''Dynamics, Theory and Applications'' pp. 90–99Free download

For example, a rotational force applied to a shaft causing acceleration, such as a drill bit accelerating from rest, results in a moment called a ''torque''. By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the angular momentum of the beam is not changing, this bending moment is not called a ''torque''. Similarly with any force couple on an object that has no change to its angular momentum, such moment is also not called a ''torque''.

Definition and relation to angular momentum

A force applied perpendicularly to a lever multiplied by its distance from the (the length of thelever arm
In physics and mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Disp ...

) is its torque. A force of three newtons
The newton (symbol: N) is the International System of Units
International is an adjective (also used as a noun) meaning "between nations".
International may also refer to:
Music Albums
* International (Kevin Michael album), ''International'' ( ...

applied two meter
The metre ( Commonwealth spelling) or meter (American spelling
Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...

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 (vector space), orientation of axes in three-dimensional space.
Most of the various left-hand and right-hand rules arise from the fact that the th ...

: 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 , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

:
:$\backslash boldsymbol\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf,$
where r is the particle's position vector
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

relative to the fulcrum, and F is the force acting on the particle. The magnitude ''τ'' of the torque is given by
:$\backslash tau\; =\; rF\backslash sin\backslash theta,$
where ''r'' is the distance from the axis of rotation to the particle, ''F'' is the magnitude of the force applied, and ''θ'' is the angle between the position and force vectors. Alternatively,
:$\backslash tau\; =\; rF\_,$
where ''F''angular momentum
In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant.
In three , the ...

,
:$\backslash boldsymbol\; =\; \backslash frac$
where L is the angular momentum vector and ''t'' is time.
For the motion of a point particle,
:$\backslash mathbf\; =\; I\backslash boldsymbol,$
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
In physics
Physics is the natural science that studies matter, its ...

and ω is the orbital angular velocity
In physics, angular velocity (\boldsymbol or \boldsymbol), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angu ...

pseudovector. It follows that
:$\backslash boldsymbol\_\; =\; \backslash frac\; =\; \backslash frac\; =\; I\backslash frac\; +\; \backslash frac\backslash boldsymbol\; =\; I\backslash boldsymbol\; +\; \backslash frac\backslash boldsymbol\; =\; I\backslash boldsymbol\; +\; 2rp\_\backslash boldsymbol,$
where α is the angular acceleration
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...

of the particle, and ''p''linear momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum ( pl. momenta) is the product of the mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what ...

. This equation is the rotational analogue of Newton's Second Law
In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon in which a ...

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.
Proof of the equivalence of definitions

The definition of angular momentum for a single point particle is: :$\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash boldsymbol$ where p is the particle'slinear momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum ( pl. momenta) is the product of the mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what ...

and r is the position vector from the origin. The time-derivative of this is:
:$\backslash frac\; =\; \backslash mathbf\; \backslash times\; \backslash frac\; +\; \backslash frac\; \backslash times\; \backslash boldsymbol.$
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 (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...

. Now using the definition of force $\backslash mathbf=\backslash frac$ (whether or not mass is constant) and the definition of velocity $\backslash frac\; =\; \backslash mathbf$
:$\backslash frac\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash boldsymbol.$
The cross product of momentum $\backslash boldsymbol$ with its associated velocity $\backslash mathbf$ 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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of its angular momentum with respect to time.
If multiple forces are applied, Newton's second law instead reads , and it follows that
:$\backslash frac\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf\_\; =\; \backslash boldsymbol\_.$
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 thedimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of force times distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

, symbolically . Although those fundamental dimensions are the same as that for energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

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
* Work (physics), the product of ...

, official literature suggests using the unit ''newton metre
The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of ac ...

'' (N⋅m) and never the joule
The joule ( ; symbol: J) is a derived unit of energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

.From thofficial SI website

"...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: :$\backslash tau\; =\; (\backslash text)\; (\backslash text).$ 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: :$\backslash tau\; =\; (\backslash text)\; (\backslash text).$ 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 exactly 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 instatic equilibrium
In classical mechanics, a particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, phys ...

, 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: Σ''H'' = 0 and Σ''V'' = 0, and the torque a third equation: Σ''τ'' = 0. That is, to solve statically determinateIn statics
Statics is the branch of mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to obj ...

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 your point of reference. If the net force $\backslash mathbf$ is not zero, and $\backslash boldsymbol\_1$ is the torque measured from $\backslash mathbf\_1$, then the torque measured from $\backslash mathbf\_2$ is $$\backslash boldsymbol\_2\; =\; \backslash boldsymbol\_1\; +\; (\backslash mathbf\_1\; -\; \backslash mathbf\_2)\; \backslash times\; \backslash mathbf$$Machine torque

Torque forms part of the basic specification of anengine
An engine or motor is a machine
A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine that uses Power (physics), power to apply For ...

: the power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...

output of an engine is expressed as its torque multiplied by its rotational speed of the axis. Internal-combustion
An internal combustion engine (ICE) 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 combustion engine, t ...

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
In physics and mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specificall ...

, and show it as a torque curve.
Steam engine
from Stott Park Bobbin Mill, Cumbria, England
A steam engine is a heat engine
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energ ...

s and 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
Transmission may refer to:
Science and technology
* Power transmissionPower transmission is the movement of energy from its place of generation to a location wh ...

.
Relationship between torque, power, and energy

If aforce
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

is allowed to act through a distance, it is doing mechanical work
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

. Similarly, if torque is allowed to act through a rotational distance, 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
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...

, the work ''W'' can be expressed as
:$W\; =\; \backslash int\_^\; \backslash tau\backslash \; \backslash mathrm\backslash theta,$
where ''τ'' is torque, and ''θ''angular position
Changing orientation of a rotating
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imagina ...

s 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 $\backslash mathrm\backslash mathbf$ :$W\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf$ However, the infinitesimal linear displacement $\backslash mathrm\backslash mathbf$ is related to a corresponding angular displacement $\backslash mathrm\backslash boldsymbol$ and the radius vector $\backslash mathbf$ as :$\backslash mathrm\backslash mathbf\; =\; \backslash mathrm\backslash boldsymbol\backslash times\backslash mathbf$ Substitution in the above expression for work gives :$W\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash boldsymbol\; \backslash times\; \backslash mathbf$ The expression $\backslash mathbf\backslash cdot\backslash mathrm\backslash boldsymbol\backslash times\backslash mathbf$ is ascalar triple product
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

given by $\backslash left;\; href="/html/ALL/s/mathbf\backslash ,\backslash mathrm\backslash boldsymbol\backslash ,\backslash mathbf\backslash right.html"\; ;"title="mathbf\backslash ,\backslash mathrm\backslash boldsymbol\backslash ,\backslash mathbf\backslash right">mathbf\backslash ,\backslash mathrm\backslash boldsymbol\backslash ,\backslash mathbf\backslash right$work-energy theorem
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

that ''W'' also represents the change in the rotational kinetic energy ''E''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
In physics
Physics is the natural science that studies matter, its ...

of the body and ''ω'' is its angular speed Angular frequency ''ω'' (in radians per second), is larger than frequency ''ν'' (in cycles per second, also called Hertz, Hz), by a factor of . This figure uses the symbol ''ν'', rather than ''f'' to denote frequency.
In .
Power (physics)">Power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...

is the work per unit time, given by
:$P\; =\; \backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol,$
where ''P'' is power, ''τ'' is torque, ''ω'' is the angular velocity
In physics, angular velocity (\boldsymbol or \boldsymbol), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angu ...

, and $\backslash cdot$ represents the scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

.
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 bike or cycle, is a human-powered transport, human-powered or motorized bicycle, motor-powered, bicycle pedal, pedal-driven, single-track vehicle, having two bicycle wheel, wheels attached to a bicycle frame, frame, ...

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, or cogs, 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 ...

) meshing with a circular chain
A chain is a wikt:series#Noun, 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 (physics), compression but line (g ...

, 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 cyclist
Cycling, also called bicycling or biking, is the use of bicycle
A bicycle, also called a bike or cycle, is a human-powered transport, human-powered or motorized bicycle, motor-powered, bicycle pedal, pedal-driven, single-track vehic ...

, the person who rides the bicycle, provides the input power by turning pedals, thereby cranking the front sprocket (commonly referred to as chainring
Deore right crankset, showing crank arm, spider, three chainrings and chainring guard
File:Belt-drive crankset.JPG, Belt-driven bicycle, Belt-drive crankset on a Trek Bicycle Corporation, Trek District
The crankset (in the US) or chainset (in the ...

). The input power provided by the cyclist is equal to the product of cadence
In Western musical theory, a cadence (Latin ''cadentia'', "a falling") is "a melodic or harmonic configuration that creates a sense of resolution inality or pause.Don Michael Randel (1999). ''The Harvard Concise Dictionary of Music and Musician ...

(i.e. the number of pedal revolutions per minute) and the torque on 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
Double-acting stationary steam engine demonstrating conversion of reciprocating motion to rotary motion. The p ...

. The bicycle's drivetrain
The drivetrain, also frequently spelled as drive train, or sometimes drive-train, is the group of components of a motor vehicle that deliver power to the driving wheels. This excludes the engine or motor that generates the power. In contrast, t ...

transmits the input power to the road wheel
File:Roue primitive.png, An early wheel made of a solid piece of wood
A wheel is a circular component that is intended to rotate on an axle
An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the ...

, 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
A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine
A molecular machine, nanite, or nanomachin ...

of the bicycle, a (torque, rpm)angular speed Angular frequency ''ω'' (in radians per second), is larger than frequency ''ν'' (in cycles per second, also called Hertz, Hz), by a factor of . This figure uses the symbol ''ν'', rather than ''f'' to denote frequency.
In [ hysics, angular freque ...

of the road wheels is decreased while the torque is increased, product of which (i.e. power) does not change.
Consistent units must be used. For metric SI units, power is watts, torque is newton metre
The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of ac ...

s and angular speed is radian
The radian, denoted by the symbol \text, is 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 sy ...

s per second (not rpm and not revolutions per second).
Also, the unit newton metre is dimensionally equivalent to the joule
The joule ( ; symbol: J) is a derived unit of energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

, which is the unit of energy. However, in the case of torque, the unit is assigned to a vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

, whereas for energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

, 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 as ...

. 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 radians as a base unit rather than a dimensionless unit.
Conversion to other units

A conversion factor may be necessary when using different units of power or torque. For example, ifrotational speed
Rotational speed (also known as speed of revolution or rate of rotation), of an object rotating around an axis is the number of turns of the object divided by time, specified as revolutions per minute
Revolutions per minute (abbreviated r ...

(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.
:$P\; =\; \backslash tau\; \backslash cdot\; 2\; \backslash pi\; \backslash cdot\; \backslash nu$
Showing units:
:$P\; ()\; =\; \backslash tau\; \backslash cdot\; 2\; \backslash pi\; \backslash cdot\; \backslash nu$
Dividing by 60 seconds per minute gives us the following.
:$P\; ()\; =\; \backslash frac$
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
A unit of measurement is a definite magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the ...

(mechanical) for power, foot-pounds (lbf⋅ft) for torque and rpm for rotational speed. This results in the formula changing to:
:$P\; ()\; =\; \backslash frac\; .$
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
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physica ...

per hour for power) would require a different custom conversion factor.
Derivation

For a rotating object, the ''linear distance'' covered at thecircumference
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...

:
: $\backslash begin\; \backslash text\; \&\; =\; \backslash frac\; \backslash \backslash $& = \frac t \\& = \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:
: $\backslash text\; =\; \backslash text\; \backslash cdot\; 2\; \backslash pi\; \backslash cdot\; \backslash text.\; \backslash ,$
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:
: $\backslash begin\; \backslash text\; \&\; =\; \backslash text\; \backslash cdot\; 2\; \backslash pi\; \backslash cdot\; \backslash text\; \backslash cdot\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash \backslash $& \approx \frac
\end
because $5252.113122\; \backslash approx\; \backslash frac\; .\; \backslash ,$
Principle of moments

The Principle of Moments, also known as Varignon's theorem (not to be confused with the geometrical theorem of the same name) states that the sum of torques due to several forces applied to ''a single'' point is equal to the torque due to the sum (resultant) of the forces. Mathematically, this follows from: :$(\backslash mathbf\backslash times\backslash mathbf\_1)\; +\; (\backslash mathbf\backslash times\backslash mathbf\_2)\; +\; \backslash cdots\; =\; \backslash mathbf\backslash times(\backslash mathbf\_1+\backslash mathbf\_2\; +\; \backslash cdots).$ From this it follows that if a pivoted beam of zero mass is balanced with two opposed forces then: :$(\backslash mathbf\backslash times\backslash mathbf\_1)\; =\; (\backslash mathbf\backslash times\backslash mathbf\_2)\; .$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 orgear box
A transmission is a machine
A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine that uses Power (physics), power to apply Force, forc ...

. 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