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 .M ...
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
, 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
Moment or Moments may refer to:
* Present time
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* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
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:
:
:
where
*
is the torque vector and
is the magnitude of the torque,
*
is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied),
*
is the force vector,
*
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 ...
,
*
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 translationof that work appears in 1842.
Definition and relation to angular momentum
![Torque, position, and force](https://upload.wikimedia.org/wikipedia/commons/2/2e/Torque%2C_position%2C_and_force.svg)
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 the ...
) 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 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 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 ...
:
:
where F is the force acting on the particle. The magnitude ''τ'' of the torque is given by
:
where ''F'' is the magnitude of the force applied, and ''θ'' is the angle between the position and force vectors. Alternatively,
:
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 ...
,
:
where L is the angular momentum vector and ''t'' is time.
For the motion of a point particle,
:
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
:
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
where
and
.
Proof of the equivalence of definitions
The definition of angular momentum for a single point particle is:
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:
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
(whether or not mass is constant) and the definition of velocity
The cross product of momentum
with its associated velocity
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. F ...
of its angular momentum with respect to time.
If multiple forces are applied, Newton's second law instead reads , and it follows that
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
![moment arm](https://upload.wikimedia.org/wikipedia/commons/a/ae/Moment_arm.svg)
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
:
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:
:
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
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero ...
, 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
is not zero, and
is the torque measured from
, then the torque measured from
is
Machine torque
![Torque Curve](https://upload.wikimedia.org/wikipedia/commons/2/21/Torque_Curve.svg)
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
:
where ''τ'' is torque, and ''θ''
1 and ''θ''
2 represent (respectively) the initial and final
angular position
In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies.
More specifically, it refers to the imagin ...
s of the body.
Proof
The work done by a variable force acting over a finite linear displacement
is given by integrating the force with respect to an elemental linear displacement
:
However, the infinitesimal linear displacement
is related to a corresponding angular displacement
and the radius vector
as
:
Substitution in the above expression for work gives
:
The expression
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