Editing Torque (section)

== Special cases and other facts ==
=== Moment arm formula ===
[[File:moment arm.svg|thumb|right|Moment arm diagram]]
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:

<math display="block">\tau = (\text{moment arm}) (\text{force}).</math>

The construction of the "moment arm" is shown in the figure to the right, along with the vectors '''r''' and '''F''' mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three-dimensional cases. If the force is perpendicular to the displacement vector '''r''', the moment arm will be equal to the distance to the centre, and torque will be a maximum for the given force. The equation for the magnitude of a torque, arising from a perpendicular force:

<math display="block">\tau = (\text{distance to centre}) (\text{force}).</math>

For example, if a person places a force of 10&nbsp;N at the terminal end of a wrench that is 0.5&nbsp;m long (or a force of 10&nbsp;N acting 0.5&nbsp;m from the twist point of a wrench of any length), the torque will be 5&nbsp;N&sdot;m – assuming that the person moves the wrench by applying force in the plane of movement and perpendicular to the wrench.

[[File:PrecessionOfATop.svg|thumb|right|The torque caused by the two opposing forces '''F'''<sub>g</sub> and −'''F'''<sub>g</sub> causes a change in the angular momentum '''L''' in the direction of that torque. This causes the top to [[precess]].]]

=== 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: {{math|1=Σ''H'' = 0}} and {{math|1=Σ''V'' = 0}}, and the torque a third equation: {{math|1=Σ''τ'' = 0}}. That is, to solve [[statically determinate]] equilibrium problems in two-dimensions, three equations are used.

=== Net force versus torque ===
When the net force on the system is zero, the torque measured from any point in space is the same. For example, the torque on a current-carrying loop in a uniform magnetic field is the same regardless of the point of reference. If the net force <math>\mathbf{F}</math> is not zero, and <math>\boldsymbol{\tau}_1</math> is the torque measured from <math>\mathbf{r}_1</math>, then the torque measured from <math>\mathbf{r}_2</math> is
<math display="block">\boldsymbol{\tau}_2 = \boldsymbol{\tau}_1 + (\mathbf{r}_2 - \mathbf{r}_1) \times \mathbf{F}</math>

=== Machine torque ===
[[File:Torque Curve.svg|thumb|Torque curve of a motorcycle ("BMW K 1200 R 2005"). The horizontal axis shows the [[rotational speed]] (in [[Revolutions per minute|rpm]]) that the [[crankshaft]] is turning, and the vertical axis is the torque (in [[newton-metre]]s) that the engine is capable of providing at that speed.]]

Torque forms part of the basic specification of an [[engine]]: the [[power (physics)|power]] output of an engine is expressed as its torque multiplied by the angular speed of the drive shaft. [[Internal combustion|Internal-combustion]] engines produce useful torque only over a limited range of [[rotational speed]]s (typically from around 1,000–6,000&nbsp;[[rpm]] for a small car). One can measure the varying torque output over that range with a [[dynamometer]], and show it as a torque curve. [[Steam engine]]s and [[electric motor]]s tend to produce maximum torque close to zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Reciprocating steam-engines and electric motors can start heavy loads from zero rpm without a [[clutch]].

In practice, the relationship between power and torque can be observed in [[bicycle]]s: Bicycles are typically composed of two road wheels, front and rear gears (referred to as [[sprockets]]) meshing with a [[bicycle chain|chain]], and a [[derailleur gears|derailleur mechanism]] if the bicycle's transmission system allows multiple gear ratios to be used (i.e. [[Single-speed bicycle#Advantages and disadvantages versus multi-speed bicycles|multi-speed bicycle]]), all of which attached to the [[bicycle frame|frame]]. A [[cyclist]], the person who rides the bicycle, provides the input power by turning pedals, thereby [[Crank (mechanism)|cranking]] the front sprocket (commonly referred to as [[Crankset#Chainring|chainring]]). The input power provided by the cyclist is equal to the product of angular speed (i.e. the number of pedal revolutions per minute times 2''π'') and the torque at the [[Axle|spindle]] of the bicycle's [[crankset]]. The bicycle's [[bicycle drivetrain systems|drivetrain]] transmits the input power to the road [[wheel]], which in turn conveys the received power to the road as the output power of the bicycle. Depending on the [[gear ratio]] of the bicycle, a (torque, angular speed)<sub>input</sub> pair is converted to a (torque, angular speed)<sub>output</sub> pair. By using a larger rear gear, or by switching to a lower gear in multi-speed bicycles, [[angular frequency|angular speed]] of the road wheels is decreased while the torque is increased, product of which (i.e. power) does not change.

=== Torque multiplier ===
{{Main|Torque multiplier}}
Torque can be multiplied via three methods: by locating the fulcrum such that the length of a lever is increased; by using a longer lever; or by the use of a speed-reducing gearset or [[gear box]].  Such a mechanism multiplies torque, as rotation rate is reduced.