Editing Joule (section)

== Newton-metre and torque ==

{{Main|Newton-metre}}
In [[mechanics]], the concept of [[force]] (in some direction) has a close analogue in the concept of [[torque]] (about some angle):{{Citation needed|date=January 2024}}

{| class="wikitable"
|-
! Linear   !! Angular
|-
| Force    || Torque
|-
| [[Mass]] || [[Moment of inertia]]
|-
| Displacement
| Angle
|}

A result of this similarity is that the SI unit for torque is the [[newton-metre]], which works out [[algebra]]ically to have the same [[dimensional analysis|dimension]]s as the joule, but they are not interchangeable. The [[General Conference on Weights and Measures]] has given the unit of [[energy]] the name ''joule'', but has not given the unit of torque any special name, hence it is simply the newton-metre (N⋅m) – a compound name derived from its constituent parts.<ref name="BIPM2" /> The use of newton-metres for torque but joules for energy is helpful to avoid misunderstandings and miscommunication.<ref name="BIPM2">{{cite web |url = http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html |title=Units with special names and symbols; units that incorporate special names and symbols |publisher=[[International Bureau of Weights and Measures]] |access-date=18 March 2015 |archive-url = https://web.archive.org/web/20090628084157/http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html |archive-date=28 June 2009 |quote = A derived unit can often be expressed in different ways by combining base units with derived units having special names. Joule, for example, may formally be written newton metre, or kilogram metre squared per second squared. This, however, is an algebraic freedom to be governed by common sense physical considerations; in a given situation some forms may be more helpful than others. In practice, with certain quantities, preference is given to the use of certain special unit names, or combinations of unit names, to facilitate the distinction between different quantities having the same dimension.}}</ref>

The distinction may be seen also in the fact that energy is a [[scalar (physics)|scalar]] quantity – the [[dot product]] of a force [[Euclidean vector|vector]] and a displacement vector. By contrast, torque is a vector – the [[cross product]] of a force vector and a distance vector. Torque and energy are related to one another by the equation{{Citation needed|date=January 2024}}
<math display="block">E = \tau \theta\, ,</math>

where ''E'' is energy, ''τ'' is (the [[magnitude (mathematics)#Euclidean vector space|vector magnitude]] of) torque, and ''θ'' is the angle swept (in [[radian]]s). Since plane angles are dimensionless, it follows that torque and energy have the same dimensions.{{Citation needed|date=January 2024}}