# Torque

Physical size
Surname Torque
Formula symbol ${\ displaystyle {\ vec {M}}}$
Size and
unit system
unit dimension
SI Nm = Nm ML 2T −2
cgs dyncm ML 2T −2
Torque vector In the case shown, the force acts perpendicular to the position vector${\ displaystyle {\ vec {M}}.}$${\ displaystyle {\ vec {F}}}$${\ displaystyle {\ vec {r}}.}$

The torque (also moment or moment of force, from the Latin momentum movement force) is a physical quantity in classical mechanics that describes the rotational effect of a force , a force couple or other force system on a body . It plays the same role in rotational movements as the force in linear movements . A torque can accelerate or brake the rotation of a body and bend the body ( bending moment ) or twist it ( torsion moment ). In drive shafts , the torque and the speed determine the power transmitted . Every torque can be described by a couple of forces. The torque of a pair of forces is independent of the reference point and can therefore be shifted as a free vector .

The internationally used unit of measurement for torque is the newton meter . As a formula symbol is common. If a force acts at right angles on the arm of a lever , the amount of torque is obtained by multiplying the amount of force with the length of the lever arm: ${\ displaystyle M}$ ${\ displaystyle M}$${\ displaystyle F}$${\ displaystyle h}$

${\ displaystyle M = h \ cdot F}$

${\ displaystyle M}$is the amount of the vector of the torque that results from the cross product of the position vector and the force vector: ${\ displaystyle {\ vec {M}}}$

${\ displaystyle {\ vec {M}} = {\ vec {r}} \ times {\ vec {F}}}$

It is the position vector from the reference point of the torque to the point of application of force. The direction of the torque vector indicates the direction of rotation of the torque. The reference point can be freely selected; it does not have to be the point around which the body rotates (in some cases there is no such point) and it does not have to be a point on the body on which the force acts. The torque of a single force, like the angular momentum, is thus only defined with respect to one point, which is sometimes also explicitly stated: ${\ displaystyle {\ vec {r}}}$

${\ displaystyle {\ vec {M}} ^ {(A)}}$with reference point .${\ displaystyle A}$

If several forces ( ) act on different points , the total torque is the vector sum of the individual torques: ${\ displaystyle {\ vec {F}} _ {i}}$${\ displaystyle i = 1,2, \ dotsc}$${\ displaystyle {\ vec {r}} _ {i}}$

${\ displaystyle {\ vec {M}} = \ sum _ {i} {\ vec {r}} _ {i} \ times {\ vec {F}} _ {i}}$

If two parallel forces act on a body, which have the same amount but opposite direction, and whose lines of action are a certain distance apart , they cause a torque with the amount . One then speaks of a couple of forces . ${\ displaystyle F}$${\ displaystyle h}$${\ displaystyle M = F \ cdot h}$

## Names and demarcation

### Torque as a first-order moment

The term “moment” is generally used for characteristics of distributions that relate to the shape

${\ displaystyle m_ {n} = \ int _ {\ mathbb {R}} x ^ {n} \ mathrm {d} \ mu (x)}$

bring it. In the case of a torque, the function that assigns a force to the location and the order is to be taken for the measure . The torque is therefore the first order moment ( dipole moment ) of a force distribution. ${\ displaystyle \ mu}$${\ displaystyle x}$${\ displaystyle n = 1}$

Instead of a force distribution, other physical quantities can also be considered and their distributions, as in a multipole expansion , generally developed according to moments. The resulting quantities that are not torques are also referred to with words that contain the ending -moment . Examples are the moment of area , the moment of inertia or the magnetic moment .

### Choice of words in science and technology

In the works of theoretical mechanics and physics, the physical quantity dealt with here is generally referred to as torque . In technical mechanics as well as in the DIN and VDI standards, the variable is usually generally referred to as the moment . It is also rarely referred to generally as torque, and sometimes the term torque is also rejected as "colloquial". Sometimes torque is used for the moment of a couple of forces. In most cases, torque is only used when the body in question rotates, for example when tightening screws or motor shafts, but not when there is a deformation (bending or torsional moment) or the effect is not yet known ( Moment).

In this article, the term torque is used in the general sense, synonymous with the moment in technical mechanics and is not limited to rotary movements or pairs of forces.

There are also a number of torques that are formed with the suffix -moment , such as the bending moment, the torsion moment or the drive moment. Designations such as bending torque or torsion torque are not used.

In English, the terms torque and moment are possible. Usage differs depending on British and US English and the technical and physical environment.

### Special torques in technology

Type of stress:

Type of movement:

Type of effect:

• Output torque: The torque that is measured on the shaft of a prime mover or on the output shaft of a gearbox . For the driven machine or the gearbox, it is the drive torque.
• Starting torque: The torque that a prime mover can provide from a standstill (more rarely also referred to as the breakaway torque), and / or that a machine or vehicle needs when starting up.
• Drive torque: The torque that acts on the input shaft of a machine or gearbox, on the wheel axle of a vehicle or on the axle of a propeller . For the driving engine or the driving gear it is the output torque.
• Torque or torque: The torque that when fixing (tightening) is applied a screw.
• Tipping moment: In mechanics, the moment that an upright object overturns. In electrical engineering, the maximum torque in the torque / speed curve of an asynchronous motor . See tipping point for details .
• Load torque : The torque that a work machine opposes the driving engine or the gearbox. For the engine or the gearbox, it is the output torque.
• Restraint moment : A moment that is generated at the restraint , i.e. the attachment of a body. It prevents the body from rotating.
• Offset moment: moment of a force with respect to the reference point for the force and moment equilibrium.

Other:

• Rated torque: The torque for a component in the construction dimensioned was.
• Nominal Moment : The moment a component was designed for.
• Specific torque: The torque per liter of displacement for piston engines. The maximum values ​​for four-stroke gasoline engines and for large four-stroke diesel engines are 200 Nm / dm³. Very large two-stroke marine diesel engines achieve 300 Nm / dm³.

## Types of torques

• the torque of a single force with respect to a point,
• the torque of a single force with respect to an axis and
• the torque of a force couple .

With the first two terms, the amount and direction of rotation of the torque depend on the reference piece (point or straight line). In the case of the force couple, however, the same total torque is always obtained regardless of the reference piece, if the torques of the individual forces of the force couple are considered and added.

For all three types, two different, equivalent approaches are possible:

• A mixed, geometrical and algebraic consideration in which the amount of torque is the product of the force and the lever arm. The plane of action and the direction of rotation result from geometric considerations.
• The second variant is purely analytical. The torque is regarded as a vector, which results as a vector product of the position vector and the force vector . The torque vector then indicates the amount, the plane of action and the direction of rotation.

Which consideration is more appropriate depends on the problem to be examined and on the mathematical knowledge of the user. If all acting forces are in the same plane, the geometrical-algebraic approach, which gets by with comparatively simple mathematics, is recommended. If the forces form a spatial system of forces , such an approach is possible, but difficult. The vector representation is then appropriate, but requires knowledge of more advanced concepts of mathematics such as the vector product. In addition, general mathematical relationships between torque and other physical quantities, such as those investigated in theoretical mechanics, can be represented more easily with vectors. In school books and introductory textbooks on technical mechanics, the geometrical-algebraic approach is initially preferred. In textbooks on theoretical mechanics and reference works on technical mechanics, however, the vector representation is widespread.

The following applies to the amount of torque for all three types: force times lever arm. A single torque acts in one plane and it is generally sufficient to consider this plane. The torque can then be specified by a single number, the sign of which indicates the direction of rotation. Torques that turn counterclockwise, i.e. in a mathematically positive sense, are usually counted positively. In the case of several torques that do not act in the same plane, it is more useful to describe them with their torque vector. This is perpendicular to the plane in which the torque acts.

Various ways are possible for the theoretical derivation of the torques. The torque of a single force can be defined based on the basic laws of mechanics. The torque of a pair of forces is then the sum of the torques of the two forces. Instead, considerations about the resultant of a force couple lead directly to its torque. The torque of an individual force is then obtained by shifting the force on a parallel line of action (offset torque, see shifting forces below ).

### Torque of a force with respect to a point

Force around a reference point

The torque or moment of a (single) force with respect to a point acts in the plane containing the force and the reference point. At this level, its amount is defined as the product of the lever arm and the amount of force : ${\ displaystyle A}$${\ displaystyle M}$${\ displaystyle a}$${\ displaystyle F}$

${\ displaystyle M = a \ cdot F}$

To avoid confusion with other torques, the reference point is also noted:

${\ displaystyle M ^ {(A)}}$or .${\ displaystyle M _ {(A)}}$

The lever arm is the vertical distance between the reference point and the line of action of the force. In general, this is not the direct connection line between the reference point and the point of application of the force. Since the lever arm does not change when the force is shifted along its line of action, its torque does not change either. The reference point itself can be freely selected. It does not have to be about the point around which the observed body rotates. This is partly not known and there is no such point in bodies that are firmly connected to their environment. The reference point does not have to be part of the body on which the force is acting. Both the amount and the direction of rotation of the torque depend on the choice of the reference point.

The vector definition is

${\ displaystyle {\ vec {M}} = {\ vec {M}} ^ {(A)} = {\ vec {r}} \ times {\ vec {F}}}$.

It is the vector product of the position vector , which points from the reference point to the point of application of the force, and the force vector . The magnitude of the position vector generally does not correspond to the lever arm. The amount of the torque vector can be calculated from the amounts of the position and force vectors and the angle between the two ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle {\ vec {F}}}$${\ displaystyle \ varphi}$

${\ displaystyle M = | {\ vec {M}} | = F \ cdot r \ cdot \ sin \ varphi = | {\ vec {F}} | \ cdot | {\ vec {r}} | \ cdot \ sin \ varphi}$

It is therefore true . ${\ displaystyle a = r \ cdot \ sin \ varphi}$

Often the torque is always related to the origin by convention : ${\ displaystyle O}$

${\ displaystyle {\ vec {M}} = {\ vec {M}} ^ {(O)} = {\ vec {r}} \ times {\ vec {F}}}$

The position vector then points from the origin to the point of application of the force. ${\ displaystyle {\ vec {r}}}$

The torque vector is perpendicular to the plane in which the torque acts, and thus also perpendicular to the plane that is spanned by the force and position vector. Its amount, i.e. its length, corresponds to the amount of torque and the area of ​​the parallelogram, which is formed by the position and force vector. The direction of rotation results from the right-hand rule : If you grasp the torque vector in your right hand in such a way that the thumb points in the direction of the arrowhead, then the other fingers indicate the direction of rotation.

### Torque of a force with respect to an axis

When torque is applied to a force with respect to an axis, the point of the axis that is closest to the point of application of the force is selected as the reference point. The distance between the point of application and the axis is then the lever arm. For the calculation, the force can be projected into a plane that is perpendicular to the axis, and then the projected force can be used to form the torque with respect to the point at which the axis penetrates the plane. Alternatively, the torque of the original force with respect to any point on the straight line can be formed. Then the torque vector is projected into a plane that is perpendicular to the straight line.

### Torque of a force couple

Power couple

A pair of forces consists of two forces that are on parallel lines of action, have the same amount and point in opposite directions. Unlike a single force, it cannot move a body, but it tries to turn it. Couples of forces are often present when bodies rotate; However, one of the two forces is often not immediately recognizable because it is mostly a constraining force . The amount of torque that is generated by a couple of forces can be calculated as the product of the amount of one of the two forces and the distance between their lines of action: ${\ displaystyle F}$${\ displaystyle a}$

${\ displaystyle M = F \ cdot a}$

The torque vector of the force couple can be calculated by:

${\ displaystyle {\ vec {M}} = {\ vec {r}} \ times {\ vec {F}}}$

The position vector points from any point on the line of action of one force to any point on the line of action of the other force. The vector that connects the points of application of the two forces is often used. ${\ displaystyle {\ vec {r}}}$

The effect of pairs of forces differs from individual forces in some important points, which is why the torques of pairs of forces also differ from other torques:

• The torque of a force couple is independent of reference points. This means that a couple of forces can be shifted to any location without changing their effect or torque.
• A couple of forces can be replaced by its torque without changing the effect on the body on which it acts. A single force, on the other hand, cannot be replaced by its torque.
• The torque vector of a pair of forces can be shifted to any location. It is a free vector . In contrast, the torque vector of a force is an axial vector . It can only be moved along the straight line that it defines.

### Derivations and relationships between the types of torque

There are various ways of deriving the torques based on the basic laws of mechanics.

In theoretical mechanics

In theoretical mechanics, Newton's second law is usually assumed in the form "force equals mass times acceleration":

${\ displaystyle {\ vec {F}} = m \, {\ ddot {\ vec {r}}} (t)}$

The vector points at every point in time from the origin to the location of the mass point, which is also the point of application of the force. The derivative of the position vector with respect to time gives the speed , which is indicated by a point, the second derivative gives the acceleration , which is characterized by two points. If the above equation is vectorially multiplied by the position vector from the left, the result is the torque of the force with respect to the origin on the left and the time derivative of the angular momentum on the right : ${\ displaystyle {\ vec {r}} (t)}$${\ displaystyle t}$${\ displaystyle {\ dot {\ vec {r}}} (t)}$${\ displaystyle {\ ddot {\ vec {r}}} (t)}$${\ displaystyle {\ vec {L}} (t)}$

${\ displaystyle {\ vec {r}} (t) \ times {\ vec {F}} = {\ vec {r}} (t) \ times m \, {\ ddot {\ vec {r}}} ( t)}$
${\ displaystyle {\ vec {M}} (t) = {\ dot {\ vec {L}}} (t)}$

The torque of a couple of forces results from the addition of the torques of the two forces: ${\ displaystyle {\ vec {M}} _ {K}}$

${\ displaystyle {\ vec {M}} _ {K} = {\ vec {M}} _ {1} + {\ vec {M}} _ {2} = {\ vec {r}} _ {1} \ times {\ vec {F}} _ {1} + {\ vec {r}} _ {2} \ times {\ vec {F}} _ {2}}$

Since it applies in the force couple , it also follows ${\ displaystyle {\ vec {F}} _ {2} = - {\ vec {F}} _ {1}}$

${\ displaystyle {\ vec {M}} _ {K} = ({\ vec {r}} _ {2} - {\ vec {r}} _ {1}) \ times {\ vec {F}} _ {1}}$,

in accordance with the above definition of the torque of a force couple, because . ${\ displaystyle {\ vec {r}} = {\ vec {r}} _ {2} - {\ vec {r}} _ {1}}$

In technical mechanics

In technical mechanics, considerations about the resultant of force systems lead directly to the torque of a force couple. The torque of a single force can be derived from this.

With the parallelogram of forces , two forces with a common point of application can be replaced by a resultant force. If the two forces act on a rigid body, they can also be combined if only the lines of action of the two forces intersect, as the forces can then be shifted to the point of intersection without changing the effect on the body. With parallel forces, however, there is no point of intersection. If the two forces are of unequal strength, however, an intersection point can be found and a resulting force can be formed by adding two more forces whose resulting force is zero. For the couple of forces, however, there is no point of intersection, but a different couple of forces, possibly at a different location and with rotated lines of action at a different distance from one another and with a different strength of the two oppositely equal forces. The product force times the distance between the lines of action , i.e. the torque, always remains constant. The pair of forces cannot be replaced by a single resulting force, but only by another pair of forces with the same torque. The couple of forces can therefore be replaced quite generally by its torque.

Fig. 8: A single force (a, black) is equivalent to an offset force (c, green) and a dislocation moment (c, red).

The torque of a single force with respect to a point results from the torque of a pair of forces using the dislocation torque (see shifting of forces below ). The line parallel to the line of action through the reference point is viewed as the line of action of two oppositely equal forces of the same magnitude as the individual force. The individual force is combined with the corresponding new force to form a force couple and this is then replaced by its torque. The result corresponds to the displacement of the original individual force and the addition of the torque of a force couple. The latter is the offset moment.

## Representations and notations

There are numerous notations for torques in equations and representations in drawings. If a plane is shown in drawings in which the torque acts, it is usually represented by a curved arrow that can range between a quarter circle and a three-quarter circle. The tip then indicates the direction of rotation. In three-dimensional representations, arrows are used as three-quarter circles, which rotate around certain axes, or straight arrows, which indicate the torque vectors. As is generally the case with vectors, these can be represented by a simple arrow. Since forces and torques occur simultaneously in many mechanical problems, the torque vectors are also marked with a double point to avoid confusion.

## Dependence on the reference point

In systems that are not in equilibrium, the value of the torque generally depends on the choice of the reference point. If the reference point is shifted by the distance , the torque with respect to the new reference point has the value ${\ displaystyle {\ vec {s}}}$${\ displaystyle {\ vec {M}} '}$

${\ displaystyle {\ vec {M}} '= {\ vec {M}} - {\ vec {s}} \ times {\ vec {F}}.}$

Here is the resulting force , ie the sum of all individual forces . ${\ displaystyle \ textstyle {\ vec {F}} = \ sum _ {i} {\ vec {F}} _ {i}}$${\ displaystyle {\ vec {F}} _ {i}}$

If the resulting force is zero, the body does not experience any acceleration and the center of gravity does not change its speed or direction of movement. The force only changes the angular momentum. In this case the torque is independent of its reference point and can be moved freely without changing the effect on the body. Since (at least) two forces are required for this situation, which have the same amount but an opposite direction and whose lines of action have a certain distance , one speaks of a force couple . The force couple causes a torque with the amount . ${\ displaystyle F}$${\ displaystyle a}$${\ displaystyle M = aF}$

## Unit of measurement

The unit of measurement of torque in SI is the newton meter (Nm). With the basic units of kilograms, meters and seconds:

${\ displaystyle 1 \ \ mathrm {Nm} = 1 \ {\ frac {\ mathrm {kg \, m ^ {2}}} {\ mathrm {s ^ {2}}}}}$

The unit of mechanical work is also the newton meter. Nevertheless, torque and work are different physical quantities that cannot be converted into one another, which is why the unit of work can be called a joule ( ), but that of torque cannot. Work is performed when a force (component) acts parallel to the movement when moving along a path. With torque, on the other hand, the force acts perpendicular to the path formed by the lever arm. The work is a scalar quantity . The torque, on the other hand, is a pseudo vector . ${\ displaystyle 1 \ \ mathrm {J} = 1 \ \ mathrm {Nm}}$

The phrase “work = force times distance” corresponds to “work = torque times angle”. In order to show this relationship, the unit can also be used for the torque as energy per angle

${\ displaystyle 1 \ {\ frac {\ mathrm {J}} {\ mathrm {rad}}}}$

can be used, the direction of the vector then pointing in the direction of the axis of rotation. Here, the unit Radiant for plane angle. ${\ displaystyle \ mathrm {rad}}$

In technical documents and on nameplates, the torque is specified in Nm. Other units used are e.g. B. or combinations of other (weight) force and length units. ${\ displaystyle {\ text {ft}} \ cdot {\ text {lb}}}$

Torques can be added to a resultant torque, much like forces can be added to a resultant force. If all torques are taken into account, one also speaks of the total torque. The set of moments makes relationships between the resulting force and the resulting torque .

### Total torque

The individual torques of two forces can be added if they refer to the same point : ${\ displaystyle A}$

${\ displaystyle {\ vec {M}} ^ {(A)} = {\ vec {r}} _ {A, F1} \ times {\ vec {F}} _ {1} + {\ vec {r} } _ {A, F2} \ times F_ {2}}$

If any number of forces are present, the total torque is the sum of all torques. When they are related to the origin, it results

${\ displaystyle {\ vec {M}} = {\ vec {M}} _ {\ mathrm {Ges}} = \ sum _ {i} {\ vec {r}} _ {i} \ times {\ vec { F}} _ {i}}$.

The vector points from the origin to the base of the force . If pairs of forces have been replaced by their torques , these must also be added: ${\ displaystyle {\ vec {r}} _ {i}}$${\ displaystyle {\ vec {F}} _ {i}}$${\ displaystyle {\ vec {M}} ^ {\ mathrm {KP}}}$

${\ displaystyle {\ vec {M}} = {\ vec {M}} _ {\ mathrm {Ges}} = \ sum _ {i} {\ vec {r}} _ {i} \ times {\ vec { F}} _ {i} + \ sum _ {j} {\ vec {M}} _ {j} ^ {\ mathrm {KP}}}$

### Moment set of statics

The moment theorem of statics says that the moment of the resulting force has the same effect on a body as the total moment, which results from the sum of the individual moments: ${\ displaystyle F _ {\ mathrm {R}}}$

${\ displaystyle a _ {\ mathrm {R}} \ cdot F _ {\ mathrm {R}} = a_ {1} \ cdot F_ {1} + a_ {2} \ cdot F_ {2} + \ dotsb + a_ {n } \ cdot F_ {n}}$

The resulting force, which is formed from all existing forces, must have the same effect on a body as the individual forces. The vector addition of the individual forces results in the amount and direction of the resulting force, but neither its point of application nor its line of action. These are determined using the moment set. The resulting force must lie on the line of action on which it generates the same moment as the individual forces.

The torque set is particularly important when checking the torque equilibrium or for calculating unknown forces using the torque equilibrium. Forces that are inclined to the coordinate axes in space can then be split into several forces that are perpendicular to the axes. Their moments can be calculated more easily. The moments caused by these force components correspond in sum to the moment caused by the original force.

## balance

When a body is in mechanical equilibrium , it does not change its state of motion. So it is neither accelerated nor decelerated.

A body in equilibrium, it is located both in the equilibrium of forces , as well as in torque balance or equilibrium of moments with respect to an arbitrary point : ${\ displaystyle A}$

${\ displaystyle \ sum _ {i} {{\ vec {M}} _ {i} ^ {(A)}} = {\ vec {0}}}$

This applies to any point A and thus even to points that are outside the body. There is a point at which the lines of action of as many forces as possible intersect. In these, the length of the lever arm is zero, resulting in zero torque. These torques do not appear in the equation, which simplifies the calculation. If there is only one unknown force among these forces, it can be calculated immediately. Sometimes it can be useful to determine several torque equilibria, if this allows a different unknown force to be calculated for each.

If a body is in torque equilibrium with respect to one point, one cannot conclude from this that it is also overall in equilibrium, and just as little that it is in torque equilibrium with respect to other points. For example, if only a single force acts, it is in torque equilibrium with respect to a point on the line of action of this force, but not in torque equilibrium with respect to points away from this line and not in total equilibrium, since a force acts for which there is no counterforce gives. However, a body is overall in equilibrium within a plane if it is in torque equilibrium with respect to three different points, provided that these three points are not on a straight line.

## Shifting forces

A force arrow can be moved along its line of action without restriction, without changing its effect on a rigid body. In the position where the distance vector is perpendicular to the line of action of the force arrow, it is called the lever arm . In terms of amount, the following applies: "Torque equals lever arm times force". With two acting forces (which are then referred to as force and load ), the torque equilibrium is equivalent to the lever law : ${\ displaystyle {\ vec {r}}}$

Low force times force = arm force times load.

(Note that, strictly speaking, only the amounts are the same, because the two torques are in opposite directions and therefore have different signs.)

If a force is shifted perpendicular to its line of action by the distance onto a parallel line of action, the torque it causes changes compared to the reference point. A force may therefore only be shifted in such a way if a torque is also introduced that compensates for this change. This is referred to as the displacement torque or the displacement torque and has the amount . ${\ displaystyle a}$${\ displaystyle F \ cdot a}$

## dynamics

The dynamics deals with states which are not in equilibrium. According to Newton's 2nd law , a resulting external force on a body leads to a change in speed ( acceleration ). Analogously to this, a resulting external torque means a change in the angular velocity ( angular acceleration ). Torques inside the body (bending or torsional moment) do not play a role in the change in movement. The inertia behavior with regard to the rotation depends not only on the mass of a body, but also on its spatial distribution. This is expressed in terms of the moment of inertia . For a rotation around a fixed axis, the following applies to the torque in the direction of this axis: ${\ displaystyle {\ vec {\ omega}}}$ ${\ displaystyle {\ vec {\ alpha}} = {\ dot {\ vec {\ omega}}}}$ ${\ displaystyle I}$

${\ displaystyle M = I \, \ alpha}$

It should be noted that the moment of inertia does not only depend on the position of the axis of rotation (see Steiner's theorem ), but also on its direction. If one wants to formulate the above equation in a more general way for any spatial direction, one has to use the inertia tensor instead: ${\ displaystyle \ mathbf {I}}$

${\ displaystyle {\ vec {M}} = \ mathbf {I} \, {\ vec {\ alpha}}}$

The relationship between torque and rate of change of angular momentum ( twist, momentum ) can be expressed as: ${\ displaystyle {\ vec {L}}}$

${\ displaystyle {\ vec {M}} = {\ frac {\ mathrm {d} {\ vec {L}}} {\ mathrm {d} t}}}$

This equation is in the engineering mechanics as angular momentum, angular momentum, torque rate or angular momentum rate , respectively. ( Angular momentum also stands for the angular momentum conservation rate , currently set is also available for the moment set from the static .)

In the two-dimensional special case , a torque merely accelerates or decelerates a rotational movement. In the general three-dimensional case, however, it can also change the direction of the axis of rotation (see e.g. precession ).

## Correspondences between linear motion and rotary motion

The torque increases in the classical mechanics for rotary motion a similar role as the force for linear motion: ${\ displaystyle {\ vec {M}}}$${\ displaystyle {\ vec {F}}}$

Straight-line movement Rotary motion
job Strength times away
${\ displaystyle W = F \ cdot \ Delta s}$
Torque times angle of rotation ( radians )
${\ displaystyle W = M \, \ Delta \ varphi}$
general:
${\ displaystyle W = \ int {\ vec {F}} ({\ vec {s}}) \ cdot \ mathrm {d} {\ vec {s}}}$
general:
${\ displaystyle W = \ int {\ vec {M}} ({\ vec {\ varphi}}) \ cdot \ mathrm {d} {\ vec {\ varphi}}}$
power Force times speed
${\ displaystyle P = {\ vec {F}} \ cdot {\ vec {v}}}$
Torque times angular velocity
${\ displaystyle P = {\ vec {M}} \ cdot {\ vec {\ omega}}}$
Static balance Balance of forces
${\ displaystyle \ sum {\ vec {F}} _ {i} = {\ vec {0}}}$
Torque balance
${\ displaystyle \ sum {\ vec {M}} _ {i} = {\ vec {0}}}$
Accelerated movement Mass times acceleration
${\ displaystyle {\ vec {F}} = m \, {\ vec {a}}}$
Inertia tensor times angular acceleration
${\ displaystyle {\ vec {M}} = \ mathbf {I} \, {\ vec {\ alpha}}}$
Rate of change of momentum
${\ displaystyle {\ vec {F}} = {\ frac {\ mathrm {d} {\ vec {p}}} {\ mathrm {d} t}}}$
Rate of change of angular momentum
${\ displaystyle {\ vec {M}} = {\ frac {\ mathrm {d} {\ vec {L}}} {\ mathrm {d} t}}}$
1. a b These simplified formulas apply to a constant force along a path in the direction of force or a constant torque around an axis in the direction of rotation. The general formulas in the line below are to be used for variable forces and torques or for oblique arrangements.

## Measurement of the torque

### Resting body

The rotating body is kept at rest by a static counter-torque . The torque to be measured and acting on the resting body is the same as the counter-torque that is generated, for example, with a lever, and its value is the product of the lever arm length and the counter-force at the end of the lever.

### Rotating body

The torque acting on a rotating shaft at a certain speed is measured with a brake dynamometer, for example a Prony bridle or a water vortex brake . This braking device connected to the shaft absorbs the entire transmitted power and measures the torque at the same time.

For example, a prime mover , on whose shaft the torque is to be measured, or the braking device are rotatably mounted about the axis of rotation of the shaft and the counteracting circumferential force is measured at the free end of a lever arm attached to the machine or the braking device.

The measurement is repeated several times and a torque / speed characteristic is generated.

The torque changing the rotational speed can be determined by measuring the angular acceleration  if the moment of inertia is known. The evaluation is carried out with the formula ${\ displaystyle \ alpha}$ ${\ displaystyle I}$

${\ displaystyle M = I \, \ alpha}$.

## Torques on selected machines

### Electric motors

Torque characteristics of an asynchronous motor .
Upper curve: delta connection
Middle curve: star connection

The asynchronous motor in the form of a squirrel cage rotor is a frequently used electric motor. The figure shows the torque typically generated when operating on the power grid (frequency and voltage constant) as a function of the speed. The motor can only be operated for a long time in the small speed range to the right of the tipping points K1 or K2 on the steeply sloping curve. To the left of the tipping points is the approach area, which must always be passed through as quickly as possible. When starting, the asynchronous motor has poor efficiency, high starting current and low torque. In order to avoid these disadvantages, various measures are used, for example the star-delta starting circuit or operation on a frequency converter . With the latter, the start succeeds with more than the nominal torque, so that the motor can also be used in vehicle drives.

A motor that is also frequently used is the series-wound direct current motor , which has a particularly high starting torque. It is therefore used for handheld devices, washing machines or rail drives.

### Internal combustion engines

Characteristic curves of two internal combustion engines

In automobile brochures, it is common for internal combustion engines to only specify their maximum value together with the corresponding speed instead of the torque / speed characteristic recorded in full load operation (see figure "Characteristic curves of two internal combustion engines").

Since the speed is again included as a linear factor in the equation for the power , the maximum power is at a higher speed than the maximum torque (see figure).

The following formula applies to the torque of two-stroke engines: ${\ displaystyle M}$

${\ displaystyle M = {\ frac {V_ {h} p_ {e}} {2 \ pi}}}$

Here is the stroke volume and the mean pressure of the burned fuel, i.e. the work performed in the cycle as “force times distance”. ${\ displaystyle V_ {h}}$${\ displaystyle p_ {e}}$${\ displaystyle V_ {h} p_ {e}}$

The following applies to the torque of four-stroke engines:

${\ displaystyle M = {\ frac {V_ {h} p_ {e}} {4 \ pi}}}$

Because with two revolutions per working cycle, the work per revolution is halved compared to the two-stroke engine.

Numerical example
Torque and power of a four-stroke engine

A series vehicle with 2000 cm³ (= 0.002 m³) displacement, whose four-stroke engine reaches a mean pressure of 9 bar (= 900,000 Pa ; 1 Pa = 1 N / m²) at a speed of 2000 / min,  calculated in SI units:

${\ displaystyle M = {\ frac {0 {,} 002 \, \ mathrm {m} ^ {3} \ cdot 900,000 \, {\ frac {\ mathrm {N}} {\ mathrm {m} ^ {2} }}} {4 \ pi}} = 143 \, \ mathrm {Nm}}$

The equation for the power during a rotary movement is (see above ; ... speed, number of revolutions per period) ${\ displaystyle n}$

${\ displaystyle P = 2 \ pi \ n \ M \}$

and as a function of speed

${\ displaystyle P (n) = 2 \ pi \ n \ M (n)}$.

${\ displaystyle M (n)}$is the speed-dependent torque characteristic for a specific motor. It is obtained by measurement.

An internal combustion engine that delivers a torque of 143 Nm at a speed of 2000 revolutions per minute has the  following power in this operating state:

${\ displaystyle P = 2 \ pi \ cdot {\ frac {2000} {60 \ \ mathrm {s}}} \ cdot 143 \ \ mathrm {Nm} \ \ approx 30 \ \ mathrm {kW}}$ .

### Hydraulic motors

The hydraulic power of a hydraulic motor is calculated from the pressures and at the motor inlet or outlet and the volume of oil swallowed ( is the volume per revolution): ${\ displaystyle P}$${\ displaystyle p_ {1}}$${\ displaystyle p_ {2}}$${\ displaystyle Q = q \ cdot n}$${\ displaystyle q}$

${\ displaystyle P = (p_ {1} -p_ {2}) \, qn}$

From the equation for the power during a rotary movement (see above )

${\ displaystyle P = 2 \ pi Mn}$

the torque follows:

${\ displaystyle M = {\ frac {(p_ {1} -p_ {2}) \, q} {2 \ pi}}}$

## literature

Commons : Torque  - collection of images, videos and audio files
Wiktionary: Torque  - explanations of meanings, word origins, synonyms, translations

## Individual evidence

1. The online dictionary. In: de.pons.com. PONS GmbH , accessed on April 23, 2017 .
2. Palle ET Jørgensen, Keri A. Kornelson, Karen L. Shuman: Iterated Function Systems, Moments, and Transformations of Infinite Matrices . In: Memoirs of the American Mathematical Society . American Mathematical Society, 2011, ISBN 0-8218-8248-1 , pp. 2 ( limited preview in Google Book search).
3. Electronic keyword search in:
• Bartelmann, Feuerbacher, Krüger, Lüst, Rebhan, Wipf (eds.): Theoretical Physics. Springer, 2015.
• Achim Feldmeier: Theoretical Mechanics - Analysis of Movement. 2013.
• Honerkamp, ​​Römer: Classical Theoretical Physics. Springer, 4th edition, 2012.
• Wolfgang Nolting: Basic Course Theoretical Mechanics 1 - Classical Mechanics. Springer, 10th edition, 2013.
• Norbert Straumann: Theoretical Mechanics. Springer, 2nd edition, 2015.
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5. Spura: Technical Mechanics 1 - Stereostatics. Springer, 2016, p. 43.
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7. Mahnken: Textbook of Technical Mechanics - Statics. Springer, 2012, p. 98.
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• Dankert, Dankert: Technical Mechanics. Springer, 7th edition, 2013.
• Wittenburg et al. (Hrsg.): The engineering knowledge - technical mechanics. Springer, 2014.
• Gross, Hauger, Schröder, Wall: Technical Mechanics 1 - Statics. Springer, 11th edition, 2011.
• Sayir, Dual, Kaufmann, Mazza: Engineering Mechanics 1 - Basics and Statics. Springer, 3rd edition, 2015.
• Spura: Technical Mechanics 1 - Stereostatics. Springer, 2016.
• Richard, Sander: Technical mechanics - statics. Springer, 5th edition, 2016.
• Dreyer: Technical Mechanics - Kinetics, Kinematics. Springer, 11th edition, 2012.
9. Jayendran: Mechanical Engineering - Fundamentals of mechanical engineering in English. Teubner, 2006, pp. 9, 67, 226, 233.
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Dankert, Dankert: Technische Mechanik. Springer, 7th edition, 2013, pp. 20, 23.
11. ^ Gross, Hauger, Schröder, Wall: Technical Mechanics 1 - Statics. Springer, 11th edition, 2011, p. 51, 54.
Mahnken: Textbook of technical mechanics - statics. Springer, 2012, p. 98, 103.
Spura: Technical Mechanics 1 - Stereostatics. Springer, 2016, pp. 43, 46.
12. Dieter Meschede (Ed.): Gerthsen Physik. Springer, 25th edition, 2015, p.
72.Bartelmann, Feuerbacher, Krüger, Lüst, Rebhan, Wipf (eds.): Theoretical Physics. Springer, 2015, p. 28.
Achim Feldmeier: Theoretical Mechanics - Analysis of Movement. 2013, p. 83.
Torsten Fließbach: Mechanics - Textbook on Theoretical Physics I. Springer, 7th edition, 2015, p. 18.
13. ^ Wittenburg et al. (Ed.): The engineering knowledge - technical mechanics. Springer, 2014, p. 13.
14. Mahnken: Textbook of Technical Mechanics - Statics. Springer, 2012, p. 145.
15. Sayir, Dual, Kaufmann, Mazza: Engineering Mechanics 1 - Fundamentals and Statics. Springer, 3rd edition, 2015.
16. Böge (Ed.): Manual of mechanical engineering. Springer, 21st edition, 2013, p. C2.
17. Dieter Meschede (Ed.): Gerthsen Physik. Springer, 25th edition, 2015, p. 73 f.
18. Achim Feldmeier: Theoretical Mechanics - Analysis of Movement. 2013, pp. 238-240.
19. ^ Gross, Hauger, Schröder, Wall: Technical Mechanics 1 - Statics. Springer, 11th edition, 2011, p. 73.
20. torque. In: Lexicon of Physics. Retrieved October 28, 2016 .
21. The International System of Units (SI) . German translation of the BIPM brochure "Le Système international d'unités / The International System of Units (8e édition, 2006)". In: PTB-Mitteilungen . tape 117 , no. 2 , 2007, p. 21 ( Online [PDF; 1.4 MB ]).
22. Böge: Technical Mechanics. Springer, 31st edition, p. 46.
23. Dankert, Dankert: Technical Mechanics. Springer, 7th edition, 2013, p. 24.
24. Mahnken, p. 24.
25. Böge (Ed.): Manual of mechanical engineering. Springer, 21st edition, 2013, p. C3.
26. Dankert, Dankert: Technical Mechanics. Springer, 7th edition, 2013, p. 571.
27. a b Gross et al: Technical Mechanics 3. Kinetics. Springer, 13th edition, 2014, p. 61.
28. ^ Conrad Eller: Holzmann / Meyer / Schumpich. Technical mechanics. Kinematics and kinetics. Springer, 12th edition, 2016, p. 127.
29. The measured values ​​are temporal mean values ​​over a full working cycle, i.e. over one revolution of the crankshaft in a two-stroke engine , over two revolutions in a four-stroke engine .