In physics and mechanics, torque is the rotational correspondent of linear force.Serway, R. A. and Jewett, J. W. Jr. (2003). Physics for Scientists and Engineers. 6th ed. Brooks Cole. . It is also referred to as the moment of force, or simply the moment. Just as a linear force is a push or a pull applied to a body, a torque can be thought of as a twist applied to an object with respect to a chosen axis. For example, when driving a screw, a screwdriver applies torque to the screw, causing it to tend to rotate around its axis.
Torque is generally referred to using different vocabulary depending on geographical location and field of study, with torque generally being associated with physics and moment being associated with engineering. This article follows the definition used in US physics in its usage of the word torque.Physics for Engineering by Hendricks, Subramony, and Van Blerk, Chinappi page 148, Web link
Torque is typically represented mathematically using the lowercase Greek letter tau (𝜏). When being referred to as moment of force, it is commonly denoted by .
Historical terminology
The term torque (from Latin , '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:
In mechanical engineering in the UK and the US, torque is generally referred to as moment of force, usually shortened to moment.Kane, T.R. Kane and D.A. Levinson (1985). Dynamics, Theory and Applications pp. 90–99: Free download . This terminology can be traced back to at least 1811 in Siméon Denis Poisson's . 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:
where is the torque vector and is the magnitude of the torque; is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied), and r is the magnitude of the position vector; is the force vector, F is the magnitude of the force vector, and F⊥ is the amount of force directed perpendicularly to the position of the particle; denotes the cross product, which produces a vector that is perpendicular both to and to following the right-hand rule; is the angle between the force vector and the lever arm vector.
The SI unit for torque is the newton-meter (N⋅m). For more on the units of torque, see .
Relationship with the angular momentum The net torque on a body determines the rate of change of the body's angular momentum,
where is the angular momentum vector and is time. For the motion of a point particle,
where is the moment of inertia and is the orbital angular velocity pseudovector. It follows that
using the derivative of a vector is
This equation is the rotational analogue of Newton's second law for point particles, and is valid for any type of trajectory. In some simple cases like a rotating disc, where only the moment of inertia on rotating axis is, the rotational Newton's second law can bewhere .
Proof of the equivalence of definitions The definition of angular momentum for a single point particle is:
where p is the particle's linear momentum and r is the position vector from the origin. The time-derivative of this is:
This result can easily be proven by splitting the vectors into components and applying the product rule. But because the rate of change of linear momentum is force and the rate of change of position is velocity ,
The cross product of momentum with its associated velocity is zero because velocity and momentum are parallel, so the second term vanishes. Therefore, torque on a particle is equal to the first derivative of its angular momentum with respect to time. If multiple forces are applied, according Newton's second law it follows that
This is a general proof for point particles, but it 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.
Derivatives of torque In physics, rotatum is the derivative of torque with respect to timewhere τ is torque.
This word is derived from the Latin word 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:
Because the rate of change of force is yank and the rate of change of position is velocity , the expression can be further simplified to:
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
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
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
where P is power, τ is torque, ω is the angular velocity, and represents the scalar product.
Algebraically, the equation may be rearranged to compute torque for a given angular speed and power output. 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).
Proof The work done by a variable force acting over a finite linear displacement is given by integrating the force with respect to an elemental linear displacement
However, the infinitesimal linear displacement is related to a corresponding angular displacement and the radius vector as
Substitution in the above expression for work, gives
The expression inside the integral is a scalar triple product , but as per the definition of torque, and since the parameter of integration has been changed from linear displacement to angular displacement, the equation becomes
If the torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., giving
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:
From this it follows that the torques resulting from N number of forces acting around a pivot on an object are balanced when
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.From the official SI website , The International System of Units – 9th edition – Text in English Section 2.3.4: "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." 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:
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. Torque has the dimension of force times distance, symbolically , and those fundamental dimensions are the same as that for energy or work.
The traditional imperial units for torque are the pound foot (lbf-ft), or, for small values, the pound inch (lbf-in). In the US, 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).
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 2 radians per revolution. In the following formulas, P is power, τ is torque, and ν (Greek letter nu) is rotational speed.
Showing units:
Dividing by 60 seconds per minute gives us the following.
where rotational speed is in revolutions per minute (rpm, rev/min).