torque

Other Definitions
torque (dict)

This article is about the physical concept. For another meaning see Torque (jewellery)

The concept of torque in physics, also called moment or couple, originated with the work of Archimedes on levers. Informally, torque can be thought of as "rotational force". The rotational analogues of force, mass and acceleration are torque, moment of inertia and angular acceleration. The force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque. For example, a force of three newtons applied two metres from the fulcrum exerts the same torque as one newton applied six metres from the fulcrum. This assumes the force is in a direction at right angles to the straight lever. More generally, one may define torque as the cross product:

\boldsymbol{T} = \mathbf{r} \times \mathbf{F}

where r is the vector from the axis of rotation to the point on which the force is acting F is the vector of force.

Units

Torque has dimensions of distance × force; the same as energy. However, the SI units of torque are usually stated as "newton-metres" rather than joules. Of course this 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,

E=\boldsymbol{T}\times \theta

where E is the energy θ is the angle moved, in radians. Other non-SI units of torque include "pound-force-feet" A very useful special case, often given as the definition of torque in fields other than physics, is as follows:

\boldsymbol{T} = (\textrm{moment\ arm}) \times \textrm{force}

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. Note that 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:

\boldsymbol{T} = (\textrm{distance\ to\ centre}) \times \textrm{force}

For example, if a person places a force of 10 N on a spanner which is 0.5 m long, the torque will be 5 N·m, assuming that the person pulls the spanner in the direction best suited to turning bolts. If the force is at an angle θ from the perpendicular then, from the definition of cross product, the magnitude of the torque arising is:

\boldsymbol{T} = (\textrm{distance\ to\ centre}) \times \cos ( \theta ) \times \textrm{force}

For an object to be at static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments). 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, we use three equations. Torque is the time-derivative of angular momentum, just as force is the time derivative of linear momentum. For multiple torques acting simultaneously:

\sum\boldsymbol{T} ={d\mathbf{L} \over dt}

where L is angular momentum. Torque on a rigid body can be written in terms of its moment of inertia

\boldsymbol I

and its angular velocity

\boldsymbol{\omega}

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

so if

\boldsymbol I

is constant,

\boldsymbol{T}=I{d\boldsymbol{\omega} \over dt}=I\boldsymbol{\alpha}

where α is angular acceleration, a quantity usually measured in rad/s².

Machine torque

Torque is part of the basic specification of an engine: the power output of an engine is simply expressed as its torque multiplied by its rotational speed. Internal-combustion engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000-6,000rpm 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 some way 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, being concerned with the transmission of power from the engine through the drive train to the wheels of a vehicle. The gearing of the drive train must be chosen appropriately to make the most of the torque characteristics. The following expression relates torque to horsepower.

\mathbf{Torque} = {\mathbf{HP} * 5252 \over \mathbf{RPM}}

Steam engines and electric motors tend to produce maximum torque at or around 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. A torque wrench is used where the tightness of screws and bolts is crucial. It limits the amount of torque applied to a fastener. Torque is also the easiest way to explain mechanical advantage in just about every simple machine.

Torque

Units

Machine torque

See also