A torque (τ) in physics, also called a moment (of force), is a pseudo-vector that measures the tendency of a force to rotate an object about some axis (center). The magnitude of a torque is defined as the product of a force and the length of the lever arm (radius). Just as a force is a push or a pull, a torque can be thought of as a twist.
The SI unit for torque is the newton meter (N m). In Imperial and U.S. customary units, it is measured in foot pounds (ft·lbf) (also known as 'pound feet') and for smaller measurement of torque: inch pounds (in·lbf) or even inch ounces (in·ozf) . The symbol for torque is τ, the Greek letter tau. .
Mathematically, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:
The torque on a body determines the rate of change of its angular momentum,
As can be seen from either of these relationships, torque is a vector, which points along the axis of the rotation it would tend to cause.
where "×" indicates the vector cross product. The time-derivative of this is:
This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = mv, we can see that:
And by definition, torque τ = r×F.
Note that there is a hidden assumption that mass is constant — this is quite valid in non-relativistic mechanics. Also, total (summed) forces and torques have been used — it perhaps would have been more rigorous to write:
The joule, which is the SI unit for energy or 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 (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,
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.
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
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:
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.
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:
so if is constant,
Understanding the relationship between torque, power and engine speed is vital in automotive engineering, concerned as it is with transmitting 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 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). Therefore, these types of engines usually have quite different types of drivetrains from internal combustion engines.
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 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:
On the right hand side, this is a scalar product of two 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 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.
Also, the unit newton meter 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.
where rotational speed is in revolutions per unit time.
Useful formula in SI units:
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:
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.
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:
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 in the above derivation to give:
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: