In Physics, torque can be thought of informally as "rotational force". Torque is measured in units of newton metres. The concept of Torque, also called moment or couple, originated with the work of Archimedes on levers. The rotational analogues of Force, Mass and Acceleration are torque, Moment of inertia and Angular acceleration respectively. 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:
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.
The Joule, the SI unit for Energy or work, is also defined as 1 N·m, but this unit is not used for torque. Of course, the dimensional equivalence of these units 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,
where
E is the energy
T is torque
θ is the angle moved, in radians.
Other non-SI units of torque include "Pound-force-feet" or "foot-pounds-force" or "ounce-force-inches or meter-kilograms-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. 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 which is 0.5m long, the torque will be 5 N·m, assuming that the person pulls the spanner in the direction best suited to turning bolts.
| Σ T = | dL ––– dt |
where L is angular momentum.
Angular momentum on a rigid body can be written in terms of its Moment of inertia
so if
| T = I | d ω ––––– dt | = I α |
where α is Angular acceleration, a quantity usually measured in rad/s².
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. 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 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.
Torque is also the easiest way to explain Mechanical advantage in just about every Simple machine.
Mathematically, the equation may be rearranged to compute torque for a given power output. However in practice there is no direct way to measure power whereas torque and angular speed can be measured directly.
Consistent units must be used. For metric SI units power is watts, torque is newton-metres and angular speed is radians per second (not rpm and not even revolutions per second).
Some people (e.g. American automotive engineers) use Horsepower (imperial mechanical) for power, foot-pounds (lbf·ft) for torque and rpm's (revolutions per minute) for angular speed. This results in the formula changing to:
| { | { ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 5252 |
This conversion factor is approximate because the transcendental number π appears in it; a more precise value is 5252.113 122 032 55... 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.
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:
| { | { –––––––––––––––––––––––––––––––––––––––– | = | (( –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– t | = { |
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
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:
| { | { ––––––––––––––––––– | × | ––––––––––––––––––––––––––––––––––––––––––– 33,000 ( { | = | { ––––––––––––––––––––––––– 5252 |
Because
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