torque n : a twisting force [syn: torsion]
- Rhymes: -ɔː(r)k
a rotational or twisting force
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
- \tau = (\textrm) \cdot \textrm
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:
- \tau = (\textrm) \cdot \textrm
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 by applying force perpendicular to the spanner.
Force at an angle
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:
- \tau=rF \sin\theta
Static equilibriumFor 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, we use three equations.
Torque as a function of timeTorque is the time-derivative of angular momentum, just as force is the time derivative of linear momentum:
- \boldsymbol = \,\!
- L is angular momentum.
Angular momentum on a rigid body can be written in terms of its moment of inertia \boldsymbol I \,\! and its angular velocity \boldsymbol:
- \mathbf=I\,\boldsymbol \,\!
so if \boldsymbol I \,\! is constant,
- \boldsymbol=I=I\boldsymbol \,\!
Machine torqueTorque is part of the basic specification of an engine: the power output of an engine is 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 1,000–6,000 rpm 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 somewhat 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, concerned as it is with transmitting power from the engine through the drive train to the wheels. Typically power is 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.
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. 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:
- \mbox=\mbox \cdot \mbox \,
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.
Consistent units must be used. For metric SI units power is watts, torque is newton meters and angular speed is radians per second (not rpm and not revolutions per second).
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.
Conversion to other unitsFor different units of power, torque, or angular speed, a conversion factor must be inserted into the equation. Also, if rotational speed (revolutions per time) is used in place of angular speed (radians per time), a conversion factor of 2 \pi must be added because there are 2 \pi radians in a revolution:
- \mbox = \mbox \times 2 \pi \times \mbox \,,
where rotational speed is in revolutions per unit time.
Useful formula in SI units:
- \mbox = \frac
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:
- \mbox \approx \frac.
This conversion factor is approximate because the transcendental number π appears in it; a more precise value is 5252.113 122 032 55... It comes from 33,000 (ft·lbf./min) / 2π (radians/revolution). It also changes with the definition of the horsepower, of course; for example, using the metric horsepower, it becomes ~5180.
Use of other units (e.g. BTU/h for power) would require a different custom conversion factor.
DerivationFor a rotating object, the linear distance covered at the circumference in a radian of rotation is the product of the radius with the angular speed. That is: linear speed = radius x angular speed. By definition, linear distance=linear speed x time=radius x angular speed x time.
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:
- \mbox = \frac=\frac = \mbox \times \mbox
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 \pi in the above derivation to give:
- \mbox=\mbox \times 2 \pi \times \mbox \,
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:
- \mbox = \mbox \times\ 2 \pi\ \times \mbox \cdot \frac \times \frac \approx \frac
because 5252.113555... = \frac \,.
- Power and Torque Explained A clear explanation of the relationship between Power and Torque, and how they relate to engine performance.
- "Horsepower and Torque" An article showing how power, torque, and gearing affect a vehicle's performance.
- "Torque vs. Horsepower: Yet Another Argument" An automotive perspective
- a discussion of torque and angular momentum in an online textbook
- Torque and Angular Momentum in Circular Motion on Project PHYSNET.
- An interactive simulation of torque
torque in Arabic: عزم الدوران
torque in Bosnian: Moment
torque in Catalan: Moment de força
torque in Czech: Kroutící moment
torque in Danish: Drejningsmoment
torque in German: Drehmoment
torque in Estonian: Jõumoment
torque in Spanish: Par de giro
torque in French: Couple (mécanique)
torque in Galician: Torque (magnitude)
torque in Korean: 돌림힘
torque in Croatian: Moment
torque in Ido: Momento
torque in Indonesian: Torsi
torque in Italian: Momento torcente
torque in Hebrew: מומנט כוח
torque in Latvian: Spēka moments
torque in Hungarian: Nyomaték
torque in Malay (macrolanguage): Tork
torque in Dutch: Koppel (natuurkunde)
torque in Japanese: トルク
torque in Norwegian Nynorsk: Dreiemoment
torque in Norwegian: Dreiemoment
torque in Polish: Moment siły
torque in Portuguese: Torque
torque in Russian: Момент силы
torque in Simple English: Torque
torque in Slovenian: Navor
torque in Serbian: Момент силе
torque in Finnish: Vääntömomentti
torque in Swedish: Vridmoment
torque in Ukrainian: Момент сили
torque in Vietnamese: Mô men lực
torque in Turkish: Tork
torque in Yiddish: טוירק
torque in Chinese: 力矩
torque in Tamil: கோண விசை