[ and UK physics terminology, unlike in US mechanical engineering, where the term torque is used for the closely related "resultant moment of a couple".][
]
Torque and moment in the US mechanical engineering terminology
In US mechanical engineering, torque is defined mathematically as the rate of change of angular momentum of an object (in physics it is called "net torque"). The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the general term used for the tendency of one or more applied to rotate an object about an axis, but not necessarily to change the angular momentum of the object (the concept which is called torque in physics).[
Kane, T.R. Kane and D.A. Levinson (1985). Dynamics, Theory and Applications pp. 90–99: Free download.
]
For example, a rotational force applied to a shaft causing acceleration, such as a drill bit accelerating from rest, results in a moment called a torque. By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the angular momentum of the beam is not changing, this bending moment is not called a torque. Similarly with any force couple on an object that has no change to its angular momentum, such moment is also not called a torque.
Definition and relation to angular momentum
A force applied perpendicularly to a lever multiplied by its distance from the Lever (the length of the lever arm) is its torque. A force of three newtons applied two from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. 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.
More generally, the torque on a point particle (which has the position r in some reference frame) can be defined as the cross product:
- $\backslash boldsymbol\{\backslash tau\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{F\},$
where r is the particle's position vector relative to the fulcrum, and F is the force acting on the particle. The magnitude τ of the torque is given by
- $\backslash tau\; =\; rF\backslash sin\backslash theta,\backslash !$
where r is the distance from the axis of rotation to the particle, F is the magnitude of the force applied, and θ is the angle between the position and force vectors. Alternatively,
- $\backslash tau\; =\; rF\_\{\backslash perp\},$
where F_{⊥} is the amount of force directed perpendicularly to the position of the particle. Any force directed parallel to the particle's position vector does not produce a torque.
It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors. Conversely, the torque vector defines the plane in which the position and force vectors lie. The resulting torque vector direction is determined by the right-hand rule.[
]
The net torque on a body determines the rate of change of the body's angular momentum,
- $\backslash boldsymbol\{\backslash tau\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{L\}\}\{\backslash mathrm\{d\}t\}$
where L is the angular momentum vector and t is time.
For the motion of a point particle,
- $\backslash mathbf\{L\}\; =\; I\backslash boldsymbol\{\backslash omega\},$
where is the moment of inertia and ω is the orbital angular velocity pseudovector. It follows that
- $\backslash boldsymbol\{\backslash tau\}\_\{\backslash mathrm\{net\}\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{L\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash mathrm\{d\}(I\backslash boldsymbol\{\backslash omega\})\}\{\backslash mathrm\{d\}t\}\; =\; I\backslash frac\{\backslash mathrm\{d\}\backslash boldsymbol\{\backslash omega\}\}\{\backslash mathrm\{d\}t\}\; +\; \backslash frac\{\backslash mathrm\{d\}I\}\{\backslash mathrm\{d\}t\}\backslash boldsymbol\{\backslash omega\}\; =\; I\backslash boldsymbol\{\backslash alpha\}\; +\; \backslash frac\{\backslash mathrm\{d\}(mr^2)\}\{\backslash mathrm\{d\}t\}\backslash boldsymbol\{\backslash omega\}\; =\; I\backslash boldsymbol\{\backslash alpha\}\; +\; 2rp\_\backslash boldsymbol\{\backslash omega\},$
where α is the angular acceleration of the particle, and p_{||} is the radial component of its linear momentum. This equation is the rotational analogue of Newton's Second Law for point particles, and is valid for any type of trajectory. Note that although force and acceleration are always parallel and directly proportional, the torque τ need not be parallel or directly proportional to the angular acceleration α. This arises from the fact that although mass is always conserved, the moment of inertia in general is not.
Proof of the equivalence of definitions
The definition of angular momentum for a single point particle is:
- $\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash boldsymbol\{p\}$
where p is the particle's linear momentum and r is the position vector from the origin. The time-derivative of this is:
- $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{L\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash frac\{\backslash mathrm\{d\}\backslash boldsymbol\{p\}\}\{\backslash mathrm\{d\}t\}\; +\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{r\}\}\{\backslash mathrm\{d\}t\}\; \backslash times\; \backslash boldsymbol\{p\}.$
This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force $\backslash mathbf\{F\}=\backslash frac\{\backslash mathrm\{d\}\backslash boldsymbol\{p\}\}\{\backslash mathrm\{d\}t\}$ (whether or not mass is constant) and the definition of velocity $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{r\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash mathbf\{v\}$
- $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{L\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{F\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash boldsymbol\{p\}.$
The cross product of momentum $\backslash boldsymbol\{p\}$ with its associated velocity $\backslash mathbf\{v\}$ is zero because velocity and momentum are parallel, so the second term vanishes.
By definition, torque τ = r × F. Therefore, torque on a particle is equal to the
first derivative of its angular momentum with respect to time.
If multiple forces are applied, Newton's second law instead reads , and it follows that
- $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{L\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{F\}\_\{\backslash mathrm\{net\}\}\; =\; \backslash boldsymbol\{\backslash tau\}\_\{\backslash mathrm\{net\}\}.$
This is a general proof for point particles.
The proof can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass.
Units
Torque has the dimension of force times distance, symbolically . Although those fundamental dimensions are the same as that for energy or mechanical work, official SI literature suggests using the unit newton metre (N⋅m) and never the joule.[From the official SI website: "...For
example, the quantity torque is the cross product of a position vector and a force vector.
The SI unit is newton metre. Even though torque has the same dimension as energy (SI unit
joule), the joule is never used for expressing torque."] The unit newton metre is properly denoted N⋅m.
The traditional Imperial and U.S. customary units for torque are the pound foot (lbf-ft), or for small values the pound inch (lbf-in). Confusingly, in US practice torque is most commonly referred to as the foot-pound (denoted as either lb-ft or ft-lb) and the inch-pound (denoted as in-lb)[ Demonstration that, as in most US industrial settings, the torque ranges are given in ft-lb rather than lbf-ft.]
. Practitioners depend on context and the hyphen in the abbreviation to know that these refer to torque and not to energy or moment of mass (as the symbolism ft-lb would properly imply).
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:
- $\backslash tau\; =\; (\backslash text\{moment\; arm\})\; (\backslash text\{force\}).$
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:
- $\backslash tau\; =\; (\backslash text\{distance\; to\; centre\})\; (\backslash text\{force\}).$
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 exactly 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: 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, 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 your point of reference. If the net force $\backslash mathbf\{F\}$ is not zero, and $\backslash boldsymbol\{\backslash tau\}\_1$ is the torque measured from $\backslash mathbf\{r\}\_1$, then the torque measured from $\backslash mathbf\{r\}\_2$ is …
$\backslash boldsymbol\{\backslash tau\}\_2\; =\; \backslash boldsymbol\{\backslash tau\}\_1\; +\; (\backslash mathbf\{r\}\_1\; -\; \backslash mathbf\{r\}\_2)\; \backslash times\; \backslash mathbf\{F\}$
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 its rotational speed of the axis. 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.
and 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.
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. Mathematically, for rotation about a fixed axis through the center of mass, the work W can be expressed as
- $W\; =\; \backslash int\_\{\backslash theta\_1\}^\{\backslash theta\_2\}\; \backslash tau\backslash \; \backslash mathrm\{d\}\backslash theta,$
where τ is torque, and θ_{1} and θ_{2} represent (respectively) the initial and final of the body.
Proof
The work done by a variable force acting over a finite linear displacement $s$ is given by integrating the force with respect to an elemental linear displacement $\backslash mathrm\{d\}\backslash vec\{s\}$
- $W\; =\; \backslash int\_\{s\_1\}^\{s\_2\}\; \backslash vec\{F\}\backslash cdot\backslash mathrm\{d\}\backslash vec\{s\}$
However, the infinitesimal linear displacement $\backslash mathrm\{d\}\backslash vec\{s\}$ is related to a corresponding angular displacement $\backslash mathrm\{d\}\backslash vec\{\backslash theta\}$ and the radius vector $\backslash vec\{r\}$ as
- $\backslash mathrm\{d\}\backslash vec\{s\}\; =\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}\backslash times\backslash vec\{r\}$
Substitution in the above expression for work gives
- $W\; =\; \backslash int\_\{s\_1\}^\{s\_2\}\; \backslash vec\{F\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}\; \backslash times\; \backslash vec\{r\}$
The expression $\backslash vec\{F\}\backslash cdot\backslash mathrm\{d\}\backslash vec\{\backslash theta\}\backslash times\backslash vec\{r\}$ is a scalar triple product given by $\backslash left\backslash vec\{F\}\backslash ,\backslash mathrm\{d\}\backslash vec\{\backslash theta\}\backslash ,\backslash vec\{r\}\backslash right$. An alternate expression for the same scalar triple product is
- $\backslash left\backslash vec\{F\}\backslash ,\backslash mathrm\{d\}\backslash vec\{\backslash theta\}\backslash ,\backslash vec\{r\}\backslash right\; =\; \backslash vec\{r\}\; \backslash times\; \backslash vec\{F\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}$
But as per the definition of torque,
- $\backslash vec\{\backslash tau\}\; =\; \backslash vec\{r\}\backslash times\backslash vec\{F\}$
Corresponding substitution in the expression of work gives,
- $W\; =\; \backslash int\_\{s\_1\}^\{s\_2\}\; \backslash vec\{\backslash tau\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}$
Since the parameter of integration has been changed from linear displacement to angular displacement, the limits of the integration also change correspondingly, giving
- $W\; =\; \backslash int\_\{\backslash theta\; \_1\}^\{\backslash theta\; \_2\}\; \backslash vec\{\backslash tau\}\backslash cdot\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}$
If the torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., $\backslash vec\{\backslash tau\}\backslash cdot\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}\; =\; \backslash left|\backslash vec\{\backslash tau\}\backslash right|\backslash left|\; \backslash ,\; \backslash mathrm\{d\}\backslash vec\{\backslash theta\}\backslash right|\backslash cos\; 0\; =\; \backslash tau\; \backslash ,\; \backslash mathrm\{d\}\backslash theta$ giving
- $W\; =\; \backslash int\_\{\backslash theta\; \_1\}^\{\backslash theta\; \_2\}\; \backslash tau\; \backslash ,\; \backslash mathrm\{d\}\backslash theta$
It follows from the work-energy theorem that W also represents the change in the rotational kinetic energy E_{r} of the body, given by
- $E\_\{\backslash mathrm\{r\}\}\; =\; \backslash tfrac\{1\}\{2\}I\backslash omega^2,$
where I is the moment of inertia of the body and ω is its angular speed.[
]
Power is the work per unit time, given by
- $P\; =\; \backslash boldsymbol\{\backslash tau\}\; \backslash cdot\; \backslash boldsymbol\{\backslash omega\},$
where P is power, τ is torque, ω is the angular velocity, and $\backslash cdot$ represents the scalar product.
Algebraically, the equation may be rearranged to compute torque for a given angular speed and 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 : Bicycles are typically composed of two road wheels, front and rear gears (referred to as sprockets) meshing with a circular bicycle chain, and a derailleur gears if the bicycle's transmission system allows multiple gear ratios to be used (i.e. multi-speed bicycle), all of which attached to the bicycle 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 cadence (i.e. the number of pedal revolutions per minute) and the torque on Axle 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, rpm)_{input} pair is converted to a (torque, rpm)_{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.
Consistent units must be used. For metric SI units, power is , torque is and angular speed is per second (not rpm and not revolutions per second).
Also, the unit newton metre is dimensionally equivalent to the joule, which is the unit of energy. However, in the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar. This means that the dimensional equivalence of the newton metre and the joule may be applied in the former, but not in the latter case. This problem is addressed in which treats radians as a base unit rather than 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 (revolutions per time) is used in place of angular speed (radians per time), we multiply by a factor of 2 radians per revolution. In the following formulas, P is power, τ is torque, and ν (Greek letter nu) is rotational speed.
- $P\; =\; \backslash tau\; \backslash cdot\; 2\; \backslash pi\; \backslash cdot\; \backslash nu$
Showing units:
- $P\; (\{\backslash rm\; W\})\; =\; \backslash tau\; \{\backslash rm\; (N\; \backslash cdot\; m)\}\; \backslash cdot\; 2\; \backslash pi\; \{\backslash rm\; (rad/rev)\}\; \backslash cdot\; \backslash nu\; \{\backslash rm\; (rev/sec)\}$
Dividing by 60 seconds per minute gives us the following.
- $P\; (\{\backslash rm\; W\})\; =\; \backslash frac\{\; \backslash tau\; \{\backslash rm\; (N\; \backslash cdot\; m)\}\; \backslash cdot\; 2\; \backslash pi\; \{\backslash rm\; (rad/rev)\}\; \backslash cdot\; \backslash nu\; \{\backslash rm\; (rpm)\}\; \}\; \{60\}$
where rotational speed is in revolutions per minute (rpm).
Some people (e.g., American automotive engineers) use horsepower (mechanical) for power, foot-pounds (lbf⋅ft) for torque and rpm for rotational speed. This results in the formula changing to:
- $P\; (\{\backslash rm\; hp\})\; =\; \backslash frac\{\; \backslash tau\; \{\backslash rm\; (lbf\; \backslash cdot\; ft)\}\; \backslash cdot\; 2\; \backslash pi\; \{\backslash rm\; (rad/rev)\}\; \backslash cdot\; \backslash nu\; (\{\backslash rm\; rpm\})\}\; \{33,000\}.$
The constant below (in foot-pounds per minute) changes with the definition of the horsepower; for example, using metric horsepower, it becomes approximately 32,550.
The use of other units (e.g., BTU per hour for power) would require a different custom conversion factor.
Derivation
For a rotating object, the linear distance covered at the circumference of rotation is the product of the radius with the angle covered. That is: linear distance = radius × angular distance. And by definition, linear distance = linear speed × time = radius × angular speed × time.
By the definition of torque: torque = radius × force. We can rearrange this to determine force = torque ÷ radius. These two values can be substituted into the definition of power:
- $$
\begin{align}
\text{power} & = \frac{\text{force} \cdot \text{linear distance}}{\text{time}} \\6pt
& = \frac{\left(\dfrac{\text{torque}} r \right) \cdot (r \cdot \text{angular speed} \cdot t)} t \\6pt
& = \text{torque} \cdot \text{angular speed}.
\end{align}
The radius r and time t have dropped out of the equation. However, angular speed must be in radians, by the assumed direct relationship between linear speed and angular speed at the beginning of the derivation. If the rotational speed is measured in revolutions per unit of time, the linear speed and distance are increased proportionately by 2 in the above derivation to give:
- $\backslash text\{power\}\; =\; \backslash text\{torque\}\; \backslash cdot\; 2\; \backslash pi\; \backslash cdot\; \backslash text\{rotational\; speed\}.\; \backslash ,$
If torque is in newton metres and rotational speed in revolutions per second, the above equation gives power in newton metres per second or watts. If Imperial units are used, and if torque is in pounds-force feet and rotational speed in revolutions per minute, the above equation gives power in foot pounds-force per minute. The horsepower form of the equation is then derived by applying the conversion factor 33,000 ft⋅lbf/min per horsepower:
- $$
\begin{align}
\text{power} & = \text{torque} \cdot 2 \pi \cdot \text{rotational speed} \cdot \frac{\text{ft}\cdot\text{lbf}}{\text{min}} \cdot \frac{\text{horsepower}}{33,000 \cdot \frac{\text{ft}\cdot\text{lbf}}{\text{min}}} \\6pt
& \approx \frac {\text{torque} \cdot \text{RPM}}{5,252}
\end{align}
because $5252.113122\; \backslash approx\; \backslash frac\; \{33,000\}\; \{2\; \backslash pi\}.\; \backslash ,$
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 sum of torques due to several forces applied to a single point is equal to the torque due to the sum (resultant) of the forces. Mathematically, this follows from:
- $(\backslash mathbf\{r\}\backslash times\backslash mathbf\{F\}\_1)\; +\; (\backslash mathbf\{r\}\backslash times\backslash mathbf\{F\}\_2)\; +\; \backslash cdots\; =\; \backslash mathbf\{r\}\backslash times(\backslash mathbf\{F\}\_1+\backslash mathbf\{F\}\_2\; +\; \backslash cdots).$
From this it follows that if a pivoted beam of zero mass is balanced with two opposed forces then:
- $(\backslash mathbf\{r\}\backslash times\backslash mathbf\{F\}\_1)\; =\; (\backslash mathbf\{r\}\backslash times\backslash mathbf\{F\}\_2)\; .$
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.
See also
External links