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Encyclopedia > Torque
Relationship between force, torque, and momentum vectors in a rotating system

In physics, torque (or often called a moment) can informally be thought of as "rotational force" or "angular force" which causes a change in rotational motion. This force is defined by linear force multiplied by a radius. Look up torque in Wiktionary, the free dictionary. ... Image File history File links Torque_animation. ... This gyroscope remains upright while spinning due to its angular momentum. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... It has been suggested that this article or section be merged with torque. ...

The SI unit for torque is the newton metre (N m). In U.S. customary units, it is measured in foot pounds (ft·lbf) (also known as 'pounds feet'). The symbol for torque is τ, the Greek letter tau. The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ... Newton metre is the unit of moment (torque) in the SI system. ... U.S. customary units, also known in the United States as English units[1] (but see English unit) or standard units, are units of measurement that are currently used in the USA, in some cases alongside units from SI (the International System of Units â€” the modern metric system). ... In physics, a foot-pound (symbol ft·lbf or ft·lbf) is an Imperial and U.S. customary unit of mechanical work, or energy, although in scientific fields one commonly uses the equivalent metric unit of the joule (J). ... Look up Î¤, Ï„ in Wiktionary, the free dictionary. ... The Greek alphabet (Greek: ) is an alphabet consisting of 24 letters that has been used to write the Greek language since the late 8th or early 8th century BC. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel...

The concept of torque, also called moment or couple, originated with the work of Archimedes on levers. The rotational analogues of force, mass, and acceleration are torque, moment of inertia, and angular acceleration, respectively. It has been suggested that this article or section be merged with torque. ... For other meanings, see Couple A Couple is or are two equal and opposite forces whose lines of action do not coincide. ... For other uses, see Archimedes (disambiguation). ... For the Portuguese town and parish, see Lever, Portugal. ... For other uses, see Force (disambiguation). ... For other uses, see Mass (disambiguation). ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2), is the rotational analog of mass. ... Ð»Insert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non...

## Explanation

Mathematically, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product: For the cross product in algebraic topology, see KÃ¼nneth theorem. ...

$boldsymbol{tau} = mathbf{r} times mathbf{F}$

where

r is the particle's position vector relative to the fulcrum
F is the force acting on the particle,

or, more generally, torque can be defined as the rate of change of angular momentum, A position vector is a vector used to describe the spatial position of a point relative to a reference point called the origin (of the space). ... This gyroscope remains upright while spinning due to its angular momentum. ...

$boldsymbol{tau}=frac{mathrm{d}mathbf{L}}{mathrm{d}t}$

where

L is the angular momentum vector
t stands for time.

As a consequence of either of these definitions, torque is a vector, which points along the axis of the rotation it would tend to cause. This article is about vectors that have a particular relation to the spatial coordinates. ...

## Units

Torque has dimensions of force times distance and the SI units of torque are stated as "newton metres" (N m or N·m).[2] Even though the order of "newton" and "metres" are mathematically interchangeable, the BIPM (Bureau International des Poids et Mesures) specifies that the order should be N·m not m·N.[3] Distance is a numerical description of how far apart objects are at any given moment in time. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... Newton metre is the unit of moment (torque) in the SI system. ... The Bureau International des Poids et Mesures (International Bureau of Weights and Measures, or BIPM) is a standards organization, one of the three organizations established to maintain the SI system under the terms of the Metre Convention. ...

The joule, which is the SI unit for energy or work, is also defined as 1 N·m, but this unit is not used for torque. Since energy can be thought of as the result of "force times distance", energy is always a scalar whereas torque is "force cross distance" and so is a (pseudo) vector-valued quantity. Of course, the dimensional equivalence of these units is not simply a coincidence; a torque of 1 N·m applied through a full revolution will require an energy of exactly 2π joules. Mathematically, The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... In physics, mechanical work is the amount of energy transferred by a force. ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ...

$E= tau theta$

where

E is the energy
τ is torque
θ is the angle moved, in radians.

Other non-SI units of torque include "pound-force-feet" or "foot-pounds-force" or "ounce-force-inches" or "meter-kilograms-force". Some common angles, measured in radians. ... The pound-force is a non-SI unit of force or weight (properly abbreviated lbf or lbf). The pound-force is equal to a mass of one pound multiplied by the standard acceleration due to gravity on Earth (which is defined as exactly 9. ... A foot (plural: feet or foot;[1] symbol or abbreviation: ft or, sometimes, â€² â€“ a prime) is a unit of length, in a number of different systems, including English units, Imperial units, and United States customary units. ... An inch (plural: inches; symbol or abbreviation: in or, sometimes, â€³ - a double prime) is the name of a unit of length in a number of different systems, including English units, Imperial units, and United States customary units. ... The unit kilogram-force (kgf, often just kg) or kilopond (kp) is defined as the force exerted by one kilogram of mass in standard Earth gravity. ...

## Extended units in relation with rotation angles

As a consequence of the previous equation, if you introduce the radian (symbol "rad") as part of the dimensional units in the SI units system, the torque could be measured using "newton metres per radian", i.e. "N.m/rad" (or "joules per radian", symbol, "J/rad"), while the energy needed and spent to perform the rotation would be measured simply in "newton metres" or "joules". Some common angles, measured in radians. ...

In the strict SI system, angles are not given any dimensional unit, because they do not designate physical quantities, despite the fact that they are measurable indirectly simply by dividing two distances (the arc length and the radius): one way to conciliate the two systems would be to say that arc lengths are not measures of distances (given they are not measured over a straight line, and a full circle rotation returns to the same position, i.e. a null distance). So arc lengths should be measured in "radian meter" (rad.m), differently from straight segment lengths in "meters" (m). In such extended SI system, the perimeter of a circle whose radius is one meter, will be two pi rad.m, and not just two pi meters.

If you apply this measure to a rotating wheel in contact with a plane surface, the center of the wheel will move across a distance measured in meters with the same value, only if the contact is efficient and the wheel does not slide on it: this does not happen in practice, unless the surface of contact is constrained and is then not perfectly plane (and can resist to the horizontal linear forces applied to the irregularities of the pseudo-plane surface of movement and to the surface of the pseudo-circular rotating wheel); but then the system generates friction that looses some energy spent by the engine: this lost energy does not change the measurement of the torque or the total energy spent in the system but the effective distance that has been made by the center of the wheel.

The difference between the efficient energy spent by the engine and the energy produced in the linear movement is lost in friction and sliding, and this explains why, when applying the same non-null torque constantly to the wheel, so that the wheel moves at a constant speed according to the surface in contact, there may be no acceleration of the center of the wheel: in that case, the energy spent will be directly proportional to the distance made by the center of the wheel, and equal to the energy lost in the system by friction and sliding.

For this reason, when measuring the effective power produced by a rotating engine and the energy spent in the system to generate a movement, you will often need to take into account the angle of rotation, and then, adding the radian in the unit system is necessary as well as making a difference between the measurement of arcs (in radian meter) and the measurement of straight segment distances (in meters), as a way to effectively compute the efficiency of the mobile system and the capacity of a motor engine to convert between rotational power (in radian watt) and linear power (in watts): in a friction-free ideal system, the two measurements would have equal value, but this does not happen in practice, each conversion losing energy in friction (it's easier to limit all losses of energy caused by sliding, by introducing mechanical constraints of forms on the surfaces of contacts).

Depending on works, the extended units including radians as a fundamental dimension may or may not be used.

## Special cases and other facts

### Moment arm formula

Moment arm diagram

A very useful special case, often given as the definition of torque in fields other than physics, is as follows: Diagram showing moment arm. ... Diagram showing moment arm. ...

$boldsymbol{tau} = (textrm{moment arm}) cdot textrm{force}$

The construction of the "moment arm" is shown in the figure below, 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:

$boldsymbol{T} = (textrm{distance to center}) cdot textrm{force}$

For example, if a person places a force of 10 N on a spanner which is 0.5 m long, the torque will be 5 N·m, assuming that the person pulls the spanner by applying force perpendicular to the spanner.

### Force at an angle

If a force of magnitude F is at an angle θ from the displacement arm of length r (and within the plane perpendicular to the rotation axis), then from the definition of cross product, the magnitude of the torque arising is:

$boldsymbol tau=rF sintheta$

### 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: ΣH = 0 and ΣV = 0, and the torque a third equation: Στ = 0. That is, to solve statically determinate equilibrium problems in two-dimensions, we use three equations. In physics, static equilibrium, or neutral balance, exists when the forces (actions), and torques, on all components of a defined system are balanced such that no component is undergoing an acceleration relative to the designated frame of reference. ... In statics, a construction is statically indeterminate when the static equilibrium equations are not sufficient to calculate the reactions on that construction. ...

### Torque as a function of time

The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.

$boldsymbol{tau} ={mathrm{d}mathbf{L} over mathrm{d}t} ,!$

where

L is angular momentum.

Angular momentum on a rigid body can be written in terms of its moment of inertia $boldsymbol I ,!$ and its angular velocity $boldsymbol{omega}$: Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2), is the rotational analog of mass. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

$mathbf{L}=I,boldsymbol{omega} ,!$

so if $boldsymbol I ,!$ is constant,

$boldsymbol{tau}=I{mathrm{d}boldsymbol{omega} over mathrm{d}t}=Iboldsymbol{alpha} ,!$

where α is angular acceleration, a quantity usually measured in radians per second squared. Ð»Insert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non... Some common angles, measured in radians. ...

## Machine torque

Torque is part of the basic specification of an engine: the power output of an engine is expressed as its torque multiplied by its rotational speed. Internal-combustion engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000–6,000 rpm for a small car). The varying torque output over that range can be measured with a dynamometer, and shown as a torque curve. The peak of that torque curve usually occurs somewhat below the overall power peak. The torque peak cannot, by definition, appear at higher rpm than the power peak. For other uses, see Engine (disambiguation). ... In physics, power (symbol: P) is the rate at which work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time. ... An internal combustion engine is an engine that is powered by the expansion of hot combustion products of fuel directly acting within an engine. ... rpm or RPM may mean: revolutions per minute RPM Package Manager (originally called Red Hat Package Manager) RPM (movie) RPM (band), a Brazilian rock band RPM (magazine), a former Canadian music industry magazine In firearms, Rounds Per Minute: how many shots an automatic weapon can fire in one minute On... A dynamometer, or dyno for short, is a machine used to measure torque and rotational speed (rpm) from which power produced by an engine, motor or other rotating prime mover can be calculated. ...

Understanding the relationship between torque, power and engine speed is vital in automotive engineering, concerned as it is with transmitting power from the engine through the drive train to the wheels. Typically power is a function of torque and engine speed. The gearing of the drive train must be chosen appropriately to make the most of the motor's torque characteristics. Automotive engineering is a branch of Vehicle engineering, incorporating elements of mechanical, electrical, electronic, software and safety engineering as applied to the design, manufacture and operation of automobiles, buses and trucks and their respective engineering subsystems. ... â€œGearboxâ€ redirects here. ... In physics, power (symbol: P) is the rate at which work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time. ...

Steam engines and electric motors tend to produce maximum torque close to zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Therefore, these types of engines usually have quite different types of drivetrains from internal combustion engines. // The term steam engine may also refer to an entire railroad steam locomotive. ... For other kinds of motors, see motor. ...

## Relationship between torque, power and energy

If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through a rotational distance, it is doing work. Power is the work per unit time. However, time and rotational distance are related by the angular speed where each revolution results in the circumference of the circle being travelled by the force that is generating the torque. This means that torque that is causing the angular speed to increase is doing work and the generated power may be calculated as: For other uses, see Force (disambiguation). ... In physics, mechanical work is the amount of energy transferred by a force. ... In physics, power (symbol: P) is the rate at which work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time. ... Look up time in Wiktionary, the free dictionary. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency &#969; (also called angular speed) is a scalar measure of rotation rate. ... The circumference is the distance around a closed curve. ...

$mbox{Power}=mbox{torque} times mbox{angular speed} ,$

On the right hand side, this is a scalar product of two vectors, giving a scalar on the left hand side of the equation. Mathematically, the equation may be rearranged to compute torque for a given power output. However in practice there is no direct way to measure power whereas torque and angular speed can be measured directly. In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V &#8594; F, where V is a vector space and F its underlying field. ... This article is about vectors that have a particular relation to the spatial coordinates. ... A scalar may be: Look up scalar in Wiktionary, the free dictionary. ...

In practice, this relationship can be observed in power stations which are connected to a large electrical power grid. In such an arrangement, the generator's angular speed is fixed by the grid's frequency, and the power output of the plant is determined by the torque applied to the generator's axis of rotation. Look up grid in Wiktionary, the free dictionary. ... This article is about machines that produce electricity. ... For other uses, see Frequency (disambiguation). ...

Consistent units must be used. For metric SI units power is watts, torque is newton-metres and angular speed is radians per second (not rpm and not revolutions per second). For other uses, see Watt (disambiguation). ... Newton metre is the unit of moment (torque) in the SI system. ... Some common angles, measured in radians. ...

Also, the unit newton-metre is dimensionally equivalent to the joule, which is the unit of energy. However, in the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar. Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... This article is about vectors that have a particular relation to the spatial coordinates. ... A scalar may be: Look up scalar in Wiktionary, the free dictionary. ...

### Conversion to other units

For different units of power, torque, or angular speed, a conversion factor must be inserted into the equation. Also, if rotational speed (revolutions per time) is used in place of angular speed (radians per time), a conversion factor of must be added because there are radians in a revolution: Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency &#969; (also called angular speed) is a scalar measure of rotation rate. ... Rotational speed (sometimes called speed of revolution) indicates for example how fast the motor is running. ...

$mbox{Power} = mbox{torque} times 2 pi times mbox{rotational speed} ,$,

where rotational speed is in revolutions per unit time.

Useful formula in SI units:

$mbox{Power (kW)} = frac{ mbox{torque (Nm)} times 2 pi times mbox{rotational speed (rpm)}} {60000}$

where 60,000 comes from 60 seconds per minute times 1000 Watts per kilowatt.

Some people (e.g. American automotive engineers) use horsepower (imperial mechanical) for power, foot-pounds (lbf·ft) for torque and rpm (revolutions per minute) for angular speed. This results in the formula changing to: This article is about a unit of measurement. ...

$mbox{Power (hp)} approx frac{ mbox{torque(lbf}cdotmbox{ft)} times mbox{rotational speed (rpm)} }{5252}$

This conversion factor is approximate because the transcendental number π appears in it; a more precise value is 5252.113 122 032 55... It comes from 33,000 (ft·lbf./min) / 2π (radians/revolution). It also changes with the definition of the horsepower, of course; for example, using the metric horsepower, it becomes ~5180. When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

Use of other units (e.g. BTU/h for power) would require a different custom conversion factor. The British thermal unit (BTU) is a non-metric unit of energy, used in the United States and, to a certain extent, the UK. The SI unit is the joule (J), which is used by most other countries. ...

### Derivation

For a rotating object, the linear distance covered at the circumference in a radian of rotation is the product of the radius with the angular speed. That is: linear speed = radius x angular speed. By definition, linear distance=linear speed x time=radius x angular speed x time. The circumference is the distance around a closed curve. ... Some common angles, measured in radians. ...

By the definition of torque: torque=force x radius. We can rearrange this to determine force=torque/radius. These two values can be substituted into the definition of power: In physics, power (symbol: P) is the rate at which work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time. ...

$mbox{power} = frac{mbox{force} times mbox{linear distance}}{mbox{time}}=frac{left(frac{mbox{torque}}{r}right) times (r times mbox{angular speed} times t)} {t} = mbox{torque} times mbox{angular speed}$

The radius r and time t have dropped out of the equation. However angular speed must be in radians, 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 in the above derivation to give:

$mbox{power}=mbox{torque} times 2 pi times mbox{rotational speed} ,$

If torque is in lbf·ft and rotational speed in revolutions per minute, the above equation gives power in ft·lbf/min. The horsepower form of the equation is then derived by applying the conversion factor 33,000 ft·lbf/min per horsepower:

$mbox{power} = mbox{torque } times 2 pi times mbox{ rotational speed} cdot frac{mbox{ft}cdotmbox{lbf}}{mbox{min}} times frac{mbox{horsepower}}{33000 cdot frac{mbox{ft }cdotmbox{ lbf}}{mbox{min}} } approx frac {mbox{torque} times mbox{RPM}}{5252}$

because $5252.113555... = frac {33,000} {2 pi} ,$.

This gyroscope remains upright while spinning due to its angular momentum. ... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... It has been suggested that this article or section be merged with torque. ... A proof that torque is equal to the time-derivative of angular momentum can be stated as follows: The definition of angular momentum for a single particle is: where Ã— indicates the vector cross product. ... In physics, rigid body dynamics is the study of the motion of a rigid object. ... Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ... ZF torque converter A cut-away model of a torque converter A torque converter is a modified form of a hydrodynamic fluid coupling, and like the fluid coupling, is used to transfer rotating power from a prime mover, such as an internal combustion engine or electric motor, to a rotating... There are very few or no other articles that link to this one. ... A torque wrench is a wrench used to precisely set the torque of a fastening such as a nut or bolt. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...

## Notes

1. ^ Right Hand Rule for Torque. Retrieved on 2007-09-08.
2. ^ SI brochure Ed. 8, Section 5.1. Bureau International des Poids et Mesures (2006). Retrieved on 2007-04-01.
3. ^ SI brochure Ed. 8, Section 2.2.2. Bureau International des Poids et Mesures (2006). Retrieved on 2007-04-01.

Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 251st day of the year (252nd in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 91st day of the year (92nd in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 91st day of the year (92nd in leap years) in the Gregorian calendar. ...

## References

• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.

Results from FactBites:

 torque. The Columbia Encyclopedia, Sixth Edition. 2001-05 (244 words) The magnitude of the torque acting on a body is equal to the product of the force acting on the body and the distance from its point of application to the axis around which the body is free to rotate. The net torque acting on a body is always equal to the product of the body’s moment of inertia about its axis of rotation and its observed angular acceleration. Units of torque are units of force multiplied by units of distance, e.g., newton-meters, dyne-centimeters, and foot-pounds (or pound-feet).
 A.K.O. Inc. - What is torque? (680 words) Simple leverage is the same as torque and both are measured in terms of force and distance (distance is the length of the lever, force is the amount of pulling or pushing applied at the end of the lever). A torque wrench is used in conjunction with a threaded fastener for the single purpose of controlling the clamping ability of the fastener. Since the torque applied to the head of the fastener is directly proportional, or nearly so, to the load applied, it is possible to measure bolt stress by means of a torque wrench.
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