torque

Historical terminology

The term torque (from Latin torquēre, 'to twist') is said to have been suggested by James Thomson and appeared in print in April, 1884. Usage is attested the same year by Silvanus P. Thompson in the first edition of Dynamo-Electric Machinery. Thompson describes his usage of the term as follows:

Just as the Newtonian definition of force is that which produces or tends to produce motion (along a line), so torque may be defined as that which produces or tends to produce torsion (around an axis). It is better to use a term which treats this action as a single definite entity than to use terms like "couple" and "moment", which suggest more complex ideas. The single notion of a twist applied to turn a shaft is better than the more complex notion of applying a linear force (or a pair of forces) with a certain leverage.

In mechanical engineering in the UK and the US, torque is generally referred to as moment of force, usually shortened to moment. This terminology can be traced back to at least 1811 in Siméon Denis Poisson's Traité de mécanique. An English translation of Poisson's work appeared in 1842.

Definition and relation to other physical quantities

Torque as a cross product between linear force and the radius about the rotational axis

The torque about an axis can be calculated by multiplying the linear force applied perpendicularly to a lever multiplied by its distance from the lever's fulcrum (the length of the lever arm).

Therefore, torque is defined as the product of the magnitude of the perpendicular component of the force and the distance of the line of action of a force from the point around which it is being determined.

In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement vector and the force vector. The direction of the torque can be determined by using the right-hand grip rule: if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque. It follows that the torque vector is perpendicular to both the position and force vectors, and defines the plane in which the two vectors lie. The resulting torque vector direction is determined by the right-hand rule. Therefore any force directed parallel to the particle's position vector does not produce a torque. The magnitude of torque applied to a rigid body depends on three quantities: the force applied, the lever arm vector connecting the point about which the torque is being measured to the point of force application, and the angle between the force and lever arm vectors. In symbols:

τ

=

r

×

F

⟹

τ

=

r

F

⊥

=

r

Relationship with the angular momentum

The net torque on a body determines the rate of change of the body's angular momentum,

τ

=

d

L

d

t

{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}

where

L

{\textstyle \mathbf {L} }

is the angular momentum vector and

t

{\textstyle t}

is time. For the motion of a point particle,

L

=

I

ω

,

{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}

where

I

=

m

r

2

{\textstyle I=mr^{2}}

is the moment of inertia and

Derivatives of torque

In physics, rotatum is the derivative of torque with respect to time

P

=

d

τ

d

t

,

{\displaystyle \mathbf {P} ={\frac {\mathrm {d} {\boldsymbol {\tau }}}{\mathrm {d} t}},}

where τ is torque.

This word is derived from the Latin word rotātus meaning 'to rotate'. The term rotatum is not universally recognized but is commonly used. There is not a universally accepted lexicon to indicate the successive derivatives of rotatum, even if sometimes various proposals have been made.

Using the cross product definition of torque, an alternative expression for rotatum is:

P

=

r

×

d

F

d

t

+

d

r

d

t

×

F

.

Relationship with power and energy

The law of conservation of energy can also be used to understand torque. If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass, the work W can be expressed as

W

=

∫

θ

1

θ

2

τ

d

θ

,

{\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \ \mathrm {d} \theta ,}

where τ is torque, and θ1 and θ2 represent (respectively) the initial and final angular positions of the body.

It follows from the work–energy principle that W also represents the change in the rotational kinetic energy Er of the body, given by

E

r

=

1

2

I

ω

2

,

{\displaystyle E_{\mathrm {r} }={\tfrac {1}{2}}I\omega ^{2},}

Principle of moments

The principle of moments, also known as Varignon's theorem (not to be confused with the geometrical theorem of the same name) states that the resultant torques due to several forces applied to about a point is equal to the sum of the contributing torques:

τ

=

r

1

×

F

1

+

r

2

×

F

2

+

…

+

r

N

×

F

N

.

{\displaystyle \tau =\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\ldots +\mathbf {r} _{N}\times \mathbf {F} _{N}.}

From this it follows that the torques resulting from N number of forces acting around a pivot on an object are balanced when

r

1

×

F

Units

Official SI literature indicates the newton-meter as the standard unit for torque, properly denoted using N⋅m; although this is dimensionally equivalent to the joule, which is not used for torque. More explicitly, torque can be interpreted as having the unit N⋅m/rad (equivalently J/rad), since it represents the rate of change of mechanical work with respect to the angular displacement. However, in the International System of Units (SI), the radian is defined as a dimensionless derived unit. Consequently, torque is conventionally expressed simply in newton-metres (N⋅m/1 = N⋅m), even though it is physically distinct from energy.

This interpretation of torque as energy per unit angular displacement (N⋅m/rad) can be verified from the work-torque relationship introduced in the previous section:

W

=

∫

θ

1

θ

2

τ

d

θ

{\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \ \mathrm {d} \theta }

where W is the mechanical work (J), τ is the torque (N⋅m/rad), θ1 and θ2 are the initial and final angular positions of the body (rad), respectively, and dθ is an infinitesimal angular displacement (rad)..

In the case of torque, the unit is assigned to a vector in three-dimensional space, whereas for energy, it is assigned to a scalar. Because of the implicit radian and the vector form of the torque, the dimensional equivalence of the newton-meter and the joule does not imply an equivalence of the physical quantities of torque and energy. The dimensional problem is addressed in orientational analysis, which treats the radian as a base unit rather than as a dimensionless unit.

Conversion to other units

A conversion factor may be necessary when using different units of power or torque. For example, if rotational speed (unit: revolution per minute or second) is used in place of angular speed (unit: radian per second), we must multiply by 2π radians per revolution. In the following formulas, P is power, τ is torque, and ν (Greek letter nu) is rotational speed.

P

=

τ

⋅

2

π

⋅

ν

{\displaystyle P=\tau \cdot 2\pi \cdot \nu }

Showing units:

P

W

=

τ

N

⋅

m

⋅

2

π

r

a

d

/

r

e

v

⋅

ν

r

e

Special cases and other facts

Moment arm formula

A very useful special case, often given as the definition of torque in fields other than physics, is as follows:

τ

=

(

moment arm

)

(

force

)

.

{\displaystyle \tau =({\text{moment arm}})({\text{force}}).}

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:

τ

=

(

distance to centre

)

(

force

)

.

{\displaystyle \tau =({\text{distance to centre}})({\text{force}}).}

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

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, 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

F

{\displaystyle \mathbf {F} }

is not zero, and

τ

1

{\displaystyle {\boldsymbol {\tau }}_{1}}

is the torque measured from

r

1

{\displaystyle \mathbf {r} _{1}}

, then the torque measured from

r

2

{\displaystyle \mathbf {r} _{2}}

is

τ

2

=

τ

1

+

(

r

2

−

r

1

)

×

Machine torque

Torque forms part of the basic specification of an engine: the power output of an engine is expressed as its torque multiplied by the angular speed of the drive shaft. 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). One can measure the varying torque output over that range with a dynamometer, and show it as a torque curve. 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). 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 bicycles: Bicycles are typically composed of two road wheels, front and rear gears (referred to as sprockets) meshing with a chain, and a derailleur mechanism if the bicycle's transmission system allows multiple gear ratios to be used (i.e. multi-speed bicycle), all of which attached to the frame. A cyclist, the person who rides the bicycle, provides the input power by turning pedals, thereby cranking the front sprocket (commonly referred to as 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 2π) and the torque at the spindle of the bicycle's crankset. The bicycle's 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)input pair is converted to a (torque, angular speed)output pair. By using a larger rear gear, or by switching to a lower gear in multi-speed bicycles, angular speed of the road wheels is decreased while the torque is increased, product of which (i.e. power) does not change.

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.