Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position (Forces and moments due to gravity not shown. Gravitation is a natural Phenomenon by which objects with Mass attract one another )

In physics, a torque (τ) (also called a moment) is a vector that measures the tendency of a force to rotate an object about some axis ^[1] (center). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector The magnitude of a torque is defined as the product of a force and the length of the lever arm ^[2] (radius). In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector Just as a force is a push or a pull, a torque can be thought of as a twist.

The SI unit for torque is newton meters (N m). Newton metre is the unit of moment ( Torque) in the SI system In U.S. customary units, it is measured in foot pounds (ft·lbf) (also known as 'pound feet'). US customary units, also known in the United States as English units or Imperial units (in reference to the British Empire) (but see English The foot-pound force, or simply foot-pound (symbol ft·lbf or ft·lb) is a unit of work or Energy (a scalar The symbol for torque is τ, the Greek letter tau. Tau (uppercase Τ, lowercase τ; Ταυ) is the 19th letter of the Greek alphabet. The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early

History

The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers. In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector A Couple is a system of Forces with a resultant moment but no Resultant force Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer The rotational analogues of force, mass, and acceleration are torque, moment of inertia, and angular acceleration, respectively. In Physics, a force is whatever can cause an object with Mass to Accelerate. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object This article is about the moment of inertia of a rotating object. Angular acceleration is the rate of change of Angular velocity over Time.

Explanation

The force applied to a lever, multiplied by its distance from the lever's fulcrum, the length of the lever arm, is its torque. For example, a force of three newtons applied two meters from the fulcrum exerts the same torque as one newton applied six meters from the fulcrum. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International This assumes the force is in a direction at right angles to the straight lever. In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle The direction of the torque can be determined by using the right hand rule: Using your right hand, curl your fingers in the direction of rotation, and stick your thumb out so it is aligned with the axis of rotation. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. Your thumb points in the direction of the torque vector. ^[3]

Mathematically, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:

where

r is the particle's position vector relative to the fulcrum

F is the force acting on the particle. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which A position, location or radius vector is a vector which represents the position of an object in space in relation to an arbitrary reference

The torque on a body determines the rate of change of its angular momentum,

where

L is the angular momentum vector

t stands for time. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

As can be seen from either of these relationships, torque is a vector, which points along the axis of the rotation it would tend to cause.

Units

Torque has dimensions of force times distance and the SI unit of torque is the "newton meter" (N m). Distance is a numerical description of how far apart objects are Newton metre is the unit of moment ( Torque) in the SI system ^[4] Even though the order of "newton" and "meter" are mathematically interchangeable, the BIPM (Bureau International des Poids et Mesures) specifies that the order should be N m not m N. The International Bureau of Weights and Measures ( Bureau international des poids et mesures, in French) is an international Standards organization, one N·m is also acceptable. ^[5]

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. The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, mechanical work is the amount of Energy transferred by a Force. 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. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an 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. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Mathematically,

where

E is the energy

τ is torque

θ is the angle moved, in radians. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

Other non-SI units of torque include "pound-force-feet" or "foot-pounds-force" or "ounce-force-inches" or "meter-kilograms-force". This article deals with the unit of force For the unit of mass see Pound (mass. A foot (plural feet or foot; symbol or abbreviation ft or sometimes &prime – the prime symbol) is a non-SI unit Inches redirects here To see the Les Savy Fav album see Inches. The unit kilogram-force ( kgf, often incorrectly just kg) or kilopond ( kp) is defined as the Force exerted by Earth's gravity

Extended units in relation with rotation angles

As a consequence of the previous equation, if you introduce the radian (rad) as part of the dimensional units in the SI units system, the torque could be measured using "newton meters per radian" (N m/rad), or "joules per radian" (J/rad), while the energy needed and spent to perform the rotation would be measured simply in "newton meters" or "joules". The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

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 loses 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:

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:

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:

τ = rFsinθ

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 Statics, a structure is statically indeterminate when the Static equilibrium equations are not sufficient for determining the internal forces and reactions on that

Torque as a function of time

The torque caused by the two opposing forces F_g and -F_g causes a change in the angular momentum L in the direction of that torque. This causes the top to precess. Precession refers to a change in the direction of the axis of a rotating object

Torque is the time-derivative of angular momentum, just as force is the time derivative of linear momentum:

where

L is angular momentum. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product

Angular momentum on a rigid body can be written in terms of its moment of inertia and its angular velocity :

so if is constant,

where α is angular acceleration, a quantity usually measured in radians per second squared. This article is about the moment of inertia of a rotating object. Do not confuse with Angular frequency The unit for angular velocity is rad/s Angular acceleration is the rate of change of Angular velocity over Time. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

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. An engine is a mechanical device that produces some form of output from a given input 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 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 internal combustion engine is an engine in which the Combustion of Fuel and an Oxidizer (typically air occurs in a confined space called a The varying torque output over that range can be measured with a dynamometer, and shown as a torque curve. For the dynamometer used in railroading see Dynamometer car. A dynamometer or "dyno" for short is a machine used to measure 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.

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. Modern automotive engineering is a branch of Vehicle engineering, incorporating elements of mechanical, electrical, electronic, software 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 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.

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). A steam engine is a Heat engine that performs Mechanical work using Steam as its Working fluid. An electric motor uses Electrical energy to produce Mechanical energy. Therefore, these types of engines usually have quite different types of drivetrains from internal combustion engines.

Torque is also the easiest way to explain mechanical advantage in just about every simple machine. In Physics and Engineering, mechanical advantage (MA is the factor by which a mechanism multiplies the force put into it In Physics, especially Mechanics, a simple machine is a mechanical device that changes the direction or magnitude of a Force.

Relationship between torque, power and energy

If a force is allowed to act through a distance, it is doing mechanical work. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics, mechanical work is the amount of Energy transferred by a Force. Similarly, if torque is allowed to act through a rotational distance, it is doing work. Power is the work per unit time. 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 For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of 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. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. The power injected by the applied torque may be calculated as:

On the right hand side, this is a scalar product of two vectors, giving a scalar on the left hand side of the equation. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R Mathematically, the equation may be rearranged to compute torque for a given 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 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. In Electricity generation, an electrical generator is a device that converts Mechanical energy to Electrical energy, generally using Electromagnetic Frequency is a measure of the number of occurrences of a repeating event per unit Time.

Consistent units must be used. For metric SI units power is watts, torque is newton meters and angular speed is radians per second (not rpm and not revolutions per second). The watt (symbol W) is the SI derived unit of power, equal to one Joule of energy per Second. Newton metre is the unit of moment ( Torque) in the SI system The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

Also, the unit newton meter is dimensionally equivalent to the joule, which is the unit of energy. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity However, in the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός

Conversion to other units

For different units of power, torque, or angular speed, a conversion factor must be inserted into the equation. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency Also, if rotational speed (revolutions per time) is used in place of angular speed (radians per time), a conversion factor of 2π must be added because there are 2π radians in a revolution:

,

where rotational speed is in revolutions per unit time. Rotational speed (sometimes called speed of revolution) indicates for example how fast a motor is running

Useful formula in SI units:

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 conversion factor is approximate because the transcendental number π appears in it; a more precise value is 5252. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 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.

Use of other units (e. g. BTU/h for power) would require a different custom conversion factor.

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. The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 That is: linear speed = radius x angular speed. By definition, linear distance=linear speed x time=radius x angular speed x time.

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:

The radius r and time t have dropped out of the equation. 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 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 2π in the above derivation to give:

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:

because .

See also

References

1. ^ Serway, R. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector Statics is the branch of Mechanics concerned with the analysis of loads ( Force, torque/moment) on Physical systems in Static equilibrium A torque converter is a modified form of Fluid coupling that is used to transfer rotating power from a prime mover, such as an Internal combustion engine A torque limiter is an automatic device that protects mechanical equipment or its work from damage by mechanical overload A torque wrench is a Tool used to precisely set the Torque of a fastener such as a nut or bolt. In Solid mechanics, torsion is the twisting of an object due to an applied Torque. A. and Jewett, Jr. J. W. (2003). Physics for Scientists and Engineers. 6th Ed. Brooks Cole. ISBN 0-53440-842-7.
2. ^ Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed. ). W. H. Freeman. ISBN 0-7167-0809-4.  
3. ^ Right Hand Rule for Torque. Retrieved on 2007-09-08. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 70 - Roman forces under Titus sack Jerusalem. 1264 - The Statute of Kalisz
4. ^ SI brochure Ed. 8, Section 5.1. Bureau International des Poids et Mesures (2006). Retrieved on 2007-04-01. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 527 - Byzantine Emperor Justin I names his nephew Justinian I as co-ruler and successor to the throne
5. ^ SI brochure Ed. 8, Section 2.2.2. Bureau International des Poids et Mesures (2006). Retrieved on 2007-04-01. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 527 - Byzantine Emperor Justin I names his nephew Justinian I as co-ruler and successor to the throne

External links

• Power and Torque Explained A clear explanation of the relationship between Power and Torque, and how they relate to engine performance.
• "Horsepower and Torque" An article showing how power, torque, and gearing affect a vehicle's performance.
• "Torque vs. Horsepower: Yet Another Argument" An automotive perspective
• a discussion of torque and angular momentum in an online textbook
• Torque and Angular Momentum in Circular Motion on Project PHYSNET.
• An interactive simulation of torque