# 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.

The SI unit for torque is the newton meter (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.

## History of concept

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.

## Explanation

The force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque. For example, a force of three newtons applied two metres from the fulcrum exerts the same torque as one newton applied six metres from the fulcrum. This assumes the force is in a direction at right angles to the straight lever. 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. Your thumb points in the direction of the torque vector.[1]

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,

or, more generally, torque can be defined as the rate of change of angular momentum,

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.

## 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]

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,

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".

## 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".

In the strict SI system, angles are not given any dimensional unit, because they do not designate physical quantities, despite they are measurable indirectly simply by diving 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 liear 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 loosing 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:

### 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.

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

where
L is angular momentum.

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.

## 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.

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.

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.

Torque is also the easiest way to explain mechanical advantage in just about every simple machine.

## 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:

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 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.

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).

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.

### 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:

,

where rotational speed is in revolutions per unit time.

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 5252.113 122 032 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. 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. 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:

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 .

## 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.

## 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.

Torque can refer to:
• Torque, force at an angle that causes rotation
• Torc, an item of jewelry
• Torque (film), a 2004 film
• Apache Torque, an open-source object-relational mapper for Java licensed under the Apache license

Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
Principle of Moments redirects here. For the Robert Plant album, see The Principle of Moments. See also Moment (mathematics) for a more abstract concept of moments that evolved from this concept of physics.

International System of Units (abbreviated SI from the French Le Système international d'unités) is the modern form of the metric system.
Newton metre is the unit of moment (torque) in the SI system. The symbolic form is N m or N·m, [1] and sometimes hyphenated newton-metre.
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
The foot-pound force, or simply foot-pound (symbol: ft·lbf or ft·lb) is an English unit of work or energy. It is the amount of energy expended when a force of one pound acts through a distance of 1 foot along the direction of the force.
Tau (uppercase Τ, lowercase τ) is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300. This letter in English is pronounced taʊ
Greek alphabet
Child systems Gothic
Glagolitic
Cyrillic
Coptic
Old Italic alphabet
Latin alphabet

ISO 15924 Grek

Principle of Moments redirects here. For the Robert Plant album, see The Principle of Moments. See also Moment (mathematics) for a more abstract concept of moments that evolved from this concept of physics.

For other meanings, see Couple (disambiguation)

A Couple is a system of forces with a resultant moment but no resultant force. Another term for a couple is a pure moment. Its effect is to create rotation without translation.
Archimedes of Syracuse (Greek: Άρχιμήδης c. 287 BC – c. 212 BC) was an ancient Greek mathematician, physicist and engineer.
lever (from French lever, "to raise", c.f. a levant) is a rigid object that is used with an appropriate fulcrum or pivot point to multiply the mechanical force that can be applied to another object.
In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
acceleration is defined as the rate of change of velocity, or, equivalently, as the second derivative of position. It is thus a vector quantity with dimension length/time². In SI units, acceleration is measured in metres/second² (m·s-²).

Moment of inertia, also called mass moment of inertia and, sometimes, the
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha ().
Fulcrum may refer to one of the following.
• A fulcrum is the support or point of support on which a lever turns in raising or moving something.
• Fulcrum Wheels, a bicycle wheel manufacturer, based in Italy (owned by Campagnolo), which manufactures wheels for

The newton (symbol: N) is the SI derived unit of force, named after Sir Isaac Newton in recognition of his work on classical mechanics.

## Definition

A newton
1 metre =
SI units
1000 mm 0 cm
US customary / Imperial units
0 ft 0 in
The metre or meter[1](symbol: m) is the fundamental unit of length in the International System of Units (SI).
right angle is an angle of 90 degrees, corresponding to a quarter turn (that is, a quarter of a full circle). It can be defined as the angle such that twice that angle amounts to a half turn, or 180° [1].
cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result.
A position, location or radius vector is a vector which represents the position of an object in space in relation to an arbitrary inertial frame of reference, referred to as a reference or location "point" that exists in 2 or 3 dimensional space.
angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").
Si, si, or SI may refer to (all SI unless otherwise stated):

In language:
• One of two Italian words:
• (accented) for "yes"
• si