Torque
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226084014
2008-07-16T19:27:18Z
Rracecarr
1284233
if they're good, does it matter that they're self-published? They are not being used as references.
{{otheruses}}
[[Image:Torque_animation.gif|frame|right|Relationship between force (F), torque (τ), and [[angular momentum|momentum]] vectors (p and L) in a rotating system. (Forces and moments due to [[gravity]] not shown.)]]
A ''' torque''' (τ) in [[physics]], also called a [[moment (physics)|moment]], is a [[spatial vector|vector]] that measures the tendency of a force to rotate an object about some axis <ref>Serway, R. A. and Jewett, Jr. J. W. (2003). ''Physics for Scientists and Engineers''. 6th Ed. Brooks Cole. ISBN 0-53440-842-7.</ref> (center). The magnitude of a torque is defined as the product of a force and the length of the [[lever arm]] <ref>{{cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.) | publisher=W. H. Freeman | year=2004 | id=ISBN 0-7167-0809-4}}</ref> (radius). Just as a force is a push or a pull, a torque can be thought of as a twist.
The [[SI units|SI unit]] for torque is [[newton meter]]s (Nm). In [[United States customary units|U.S. customary units]], it is measured in [[foot pound]]s (ft·lbf) (also known as 'pound feet'). The symbol for torque is ''[[tau|τ]]'', the [[Greek alphabet|Greek letter]] ''tau''.
== History ==
The concept of torque, also called [[moment (physics)|moment]] or [[couple (mechanics)|couple]], originated with the studies of [[Archimedes]] on [[lever]]s. 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]], the length of the lever arm, is its torque. A force of three [[newton]]s applied two [[meter]]s from the fulcrum, for example, exerts the same torque as one newton applied six meters from the fulcrum. This assumes the force is in a direction at [[right angle]]s to the straight lever. The direction of the torque can be determined by using the [[Right_hand_rule|right hand rule]]: curl the fingers of your right hand 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.<ref>{{cite web| url=http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html|title=Right Hand Rule for Torque |accessdate=2007-09-08}}</ref>
Mathematically, the torque on a particle (which has the position '''r''' in some reference frame) can be defined as the [[cross product]]:
:<math>\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}</math>
where
:'''r''' is the particle's [[position vector]] relative to the fulcrum
:'''F''' is the force acting on the particle.
The torque on a body determines the rate of change of its [[angular momentum]],
:<math>\boldsymbol{\tau}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}</math>
where
:'''L''' is the angular momentum vector
:'''t''' stands for time.
As can be seen from either of these relationships, torque is a [[Vector (spatial)|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]]" (Nm).<ref name="BIPM 5.1">{{cite web | title = SI brochure Ed. 8, Section 5.1 | publisher = Bureau International des Poids et Mesures | date= 2006 | url = http://www1.bipm.org/en/si/si_brochure/chapter5/5-1.html | accessdate = 2007-04-01}}</ref> 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 '''Nm''' not mN. N·m is also acceptable.<ref name="BIPM 2.2.2">{{cite web | title = SI brochure Ed. 8, Section 2.2.2 | publisher = Bureau International des Poids et Mesures | date= 2006 | url = http://www1.bipm.org/en/si/derived_units/2-2-2.html | accessdate = 2007-04-01}}</ref>
The [[joule]], which is the SI unit for [[energy]] or [[mechanical work|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 [[pseudovector|(pseudo) vector]]-valued quantity. The dimensional equivalence of these units, of course, 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,
:<math>E= \tau \theta\ </math>
where
:''E'' is the energy
:''τ'' is torque
:''θ'' is the angle moved, in [[radian]]s.
Other non-SI units of torque include "[[pound-force]]-[[foot (unit of length)|feet]]" or "foot-pounds-force" or "ounce-force-[[inch]]es" or "meter-[[kilogram-force|kilograms-force]]" or "kilogrammeter" (kgm).
== 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".
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===
[[Image:moment arm.png|thumb|right|250px|Moment arm diagram]]
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
<!--:''|τ|'' = moment arm × force-->
:<math>|\tau| = (\textrm{moment\ arm}) \cdot \textrm{force}</math>
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:
<!--:''|τ|'' = distance to centre × force-->
:<math>|\tau| = (\textrm{distance\ to\ center}) \cdot \textrm{force}</math>
For example, if a person places a force of 10 N on a spanner (wrench) 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:
:<math>\tau=rF \sin\theta</math>
===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: Σ''H'' = 0 and Σ''V'' = 0, and the torque a third equation: Σ''τ'' = 0. That is, to solve [[statically determinate]] equilibrium problems in two-dimensions, we use three equations.
===Torque as a function of time===
[[Image:PrecessionOfATop.svg|thumb|right|300px|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]].]]
Torque is the time-[[derivative]] of [[angular momentum]], just as force is the time derivative of [[momentum|linear momentum]]:
:<math>\boldsymbol{\tau} ={\mathrm{d}\mathbf{L} \over \mathrm{d}t} \,\!</math>
where
:'''L''' is angular momentum.
Angular momentum on a rigid body can be written in terms of its [[moment of inertia]] <math>\boldsymbol I \,\!</math> and its [[angular velocity]] <math>\boldsymbol{\omega}</math>:
:<math>\mathbf{L}=I\,\boldsymbol{\omega} \,\!</math>
so if <math>\boldsymbol I \,\!</math> is constant,
:<math>\boldsymbol{\tau}=I{\mathrm{d}\boldsymbol{\omega} \over \mathrm{d}t}=I\boldsymbol{\alpha} \,\!</math>
where '''α''' is [[angular acceleration]], a quantity usually measured in [[radian]]s per second squared.
==Machine torque==
Torque is part of the basic specification of an [[engine]]: the [[power (physics)|power]] output of an engine is expressed as its torque multiplied by its rotational speed of the axis. [[internal combustion|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 [[transmission (mechanics)|transmitting]] [[power (physics)|power]] from the engine through the drive train to the wheels. Power is typically 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 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). 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]].{{Fact|date=December 2007}}
==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 (physics)|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. The power injected by the applied torque may be calculated as:
:<math>\mbox{Power}=\mbox{torque} \cdot \mbox{angular speed} \,</math>
On the right hand side, this is a [[scalar product]] of two [[Vector (spatial)|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. 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 [[electrical generator|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 [[watt]]s, torque is [[newton meter]]s and angular speed is [[radian]]s per second (not rpm and not revolutions per second).
Also, the unit newton meter is [[dimensional analysis|dimensionally equivalent]] to the [[joule]], which is the unit of energy. However, in the case of torque, the unit is assigned to a [[Vector (spatial)|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 <math>2 \pi</math> must be added because there are <math>2 \pi</math> radians in a revolution:
:<math>\mbox{Power} = \mbox{torque} \times 2 \pi \times \mbox{rotational speed} \,</math>,
where rotational speed is in revolutions per unit time.
Useful formula in SI units:
:<math> \mbox{Power (kW)} = \frac{ \mbox{torque (N}\cdot\mbox{m)} \times 2 \pi \times \mbox{rotational speed (rpm)}} {60000} </math>
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:
:<math> \mbox{Power (hp)} = \frac{ \mbox{torque(lbf}\cdot\mbox{ft)} \times 2 \pi \times \mbox{rotational speed (rpm)} }{33000}. </math>
The constant below in, ft·lbf./min, changes with the definition of the horsepower; for example, using metric horsepower, it becomes ~32,550.
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 (physics)|power]]:
:<math>\mbox{power} = \frac{\mbox{force} \times \mbox{linear distance}}{\mbox{time}}=\frac{\left(\frac{\mbox{torque}}{r}\right) \times (r \times \mbox{angular speed} \times t)} {t} = \mbox{torque} \times \mbox{angular speed}</math>
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 <math>2 \pi</math> in the above derivation to give:
:<math>\mbox{power}=\mbox{torque} \times 2 \pi \times \mbox{rotational speed} \,</math>
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:
:<math>\mbox{power} = \mbox{torque } \times\ 2 \pi\ \times \mbox{ rotational speed} \cdot \frac{\mbox{ft}\cdot\mbox{lbf}}{\mbox{min}} \times \frac{\mbox{horsepower}}{33000 \cdot \frac{\mbox{ft }\cdot\mbox{ lbf}}{\mbox{min}} } \approx \frac {\mbox{torque} \times \mbox{RPM}}{5252} </math>
because <math>5252.113122... = \frac {33,000} {2 \pi} \,</math>.
==See also==
<div style="-moz-column-count:2; column-count:2;">
*[[Angular momentum]]
*[[Mechanical equilibrium]]
*[[Moment (physics)]]
*[[Proof of angular momentum]]
*[[Rigid body dynamics]]
*[[Statics]]
*[[Torque converter]]
*[[Torque limiter]]
*[[Torque wrench]]
*[[Torsion (mechanics)]]
</div>
==References==
<references/>
==External links==
*[http://www.epi-eng.com/ET-PwrTrq.htm Power and Torque Explained] A clear explanation of the relationship between Power and Torque, and how they relate to engine performance.
*[http://craig.backfire.ca/pages/autos/horsepower "Horsepower and Torque"] An article showing how power, torque, and gearing affect a vehicle's performance.
*[http://kevinthenerd.googlepages.com/torque_vs_hp.html "Torque vs. Horsepower: Yet Another Argument"] An automotive perspective
*[http://www.lightandmatter.com/html_books/2cl/ch05/ch05.html a discussion of torque and angular momentum in an online textbook]
*[http://www.physnet.org/modules/pdfmodules/m34.pdf ''Torque and Angular Momentum in Circular Motion ''] on [http://www.physnet.org Project PHYSNET].
*[http://www.phy.hk/wiki/englishhtm/Torque.htm An interactive simulation of torque]
[[Category:Fundamental physics concepts]]
[[Category:Engine technology]]
[[Category:Physical quantity]]
[[Category:Introductory physics]]
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