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PowerPedia:Torque

Lasted edited by Andrew Munsey, updated on June 14, 2016 at 9:30 pm.

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In There was an error working with the wiki: Code[3]) can informally be thought of as "rotational force" or "angular force" which causes a change in rotational motion. This force is defined by linear force multiplied by a radius. The There was an error working with the wiki: Code[4], it is measured in pounds-feet (lb-ft). The symbol for torque is There was an error working with the wiki: Code[5], the There was an error working with the wiki: Code[6] tau. The concept of torque, also called There was an error working with the wiki: Code[7] or There was an error working with the wiki: Code[8], originated with the work of Archimedes on There was an error working with the wiki: Code[22]s. The rotational analogues of Force, Mass, and There was an error working with the wiki: Code[23] are torque, There was an error working with the wiki: Code[24], and There was an error working with the wiki: Code[25], respectively. The force applied to a lever, multiplied by its distance from the lever's There was an error working with the wiki: Code[26], is the torque. For example, a force of three There was an error working with the wiki: Code[27]s 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 There was an error working with the wiki: Code[28]s to the straight lever.

Use the right hand rule to show torque direction.

If you curl right hand fingers around the spin axis and in the direction of spin, then your thumb points in the direction of torque http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html.

Mathematically, the torque on a particle (which has the position r in some reference frame) can be defined as the There was an error working with the wiki: Code[29]:

:\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}

where

:r is the particle's There was an error working with the wiki: Code[30]

:F is the force acting on the particle,

or, more generally, torque can be defined as the rate of change of There was an error working with the wiki: Code[31],

:\boldsymbol{\tau}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}

where

:L is the angular momentum vector

:t stands for time.

As a consequence of either of these definitions, torque is a There was an error working with the wiki: Code[32], which points along the axis of the rotation it would tend to cause.

Units

Torque has dimensions of force times There was an error working with the wiki: Code[33] and the SI units of torque are stated as "There was an error working with the wiki: Code[34]s". Even though the order of "newton" and "metres" are mathematically interchangeable, the BIPM (There was an error working with the wiki: Code[35]) specifies that the order should be N·m not m·Nhttp://www1.bipm.org/en/si/derived_units/2-2-2.html.

The There was an error working with the wiki: Code[9], 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 dot distance", energy is always a scalar whereas torque is "force cross distance" and so is a There was an error working with the wiki: Code[10]-valued quantity. 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 2? joules. Mathematically,

:E= \tau \theta\

where

:E is the energy

:? is torque

:? is the angle moved, in There was an error working with the wiki: Code[36]s.

Other non-SI units of torque include "There was an error working with the wiki: Code[11]" or "foot-pounds-force" or "ounce-force-There was an error working with the wiki: Code[12]".

Machine torque

Torque is part of the basic specification of an There was an error working with the wiki: Code[13] output of an engine is expressed as its torque multiplied by its rotational speed. There was an error working with the wiki: Code[14] 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 There was an error working with the wiki: Code[37], 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 There was an error working with the wiki: Code[15] Power (physics) 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.

Torque is also the easiest way to explain There was an error working with the wiki: Code[38] in just about every There was an error working with the wiki: Code[39].

Relationship between torque, power and energy

If a There was an error working with the wiki: Code[16] is the work per unit There was an error working with the wiki: Code[40]. However, time and rotational distance are related by the There was an error working with the wiki: Code[41] where each revolution results in the There was an error working with the wiki: Code[42] of the circle being travelled by the force that is generating the torque. This means that torque that is causing the angular speed to increase is doing work and the generated power may be calculated as:

:\mbox{Power}=\mbox{torque} \times \mbox{angular speed} \,

On the right hand side, this is a There was an error working with the wiki: Code[17], 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. However in practice there is no direct way to measure power whereas torque and angular speed can be measured directly.

In practice, this relationship can be observed in power stations which are connected to a large electrical power There was an error working with the wiki: Code[18]'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 There was an error working with the wiki: Code[43]s and angular speed is There was an error working with the wiki: Code[44]s per second (not rpm and not revolutions per second).

Also, the unit newton-metre is There was an error working with the wiki: Code[19] to the There was an error working with the wiki: Code[20], whereas for Energy, it is assigned to a Scalar.

Conversion to other units

For different units of power, torque, or There was an error working with the wiki: Code[45], a conversion factor must be inserted into the equation. Also, if There was an error working with the wiki: Code[46] (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{Power} = \mbox{torque} \times 2 \pi \times \mbox{rotational speed} \,,

where rotational speed is in revolutions per unit time.

Useful formula in SI units:

: \mbox{Power (kW)} = \frac{ \mbox{torque (Nm)} \times \pi \times \mbox{rotational speed (rpm)}} {30000}

Some people (e.g. American automotive engineers) use There was an error working with the wiki: Code[47] (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:

: \mbox{Power (hp)} \approx \frac{ \mbox{torque(lbf}\cdot\mbox{ft)} \times \mbox{rotational speed (rpm)} }{5252}

This conversion factor is approximate because the transcendental number There was an error working with the wiki: Code[21] appears in it a more precise value is 5252.113&nbsp122&nbsp032&nbsp55... 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. There was an error working with the wiki: Code[48]/h for power) would require a different custom conversion factor.

Derivation

For a rotating object, the linear distance covered at the There was an error working with the wiki: Code[49] in a There was an error working with the wiki: Code[50] 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 (physics):

:\mbox{power} = \frac{\mbox{force} \times \mbox{linear distance}}{\mbox{time}}=\frac{\left(\frac{\mbox{torque}}{r}\right) \times (r \times \mbox{angular speed} \times t)} {t} = \mbox{torque} \times \mbox{angular speed}

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{power}=\mbox{torque} \times 2 \pi \times \mbox{rotational speed} \,

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{power} = \mbox{torque } \times\ 2 \pi\ \times \mbox{ rotational speed} \cdot \frac{\mbox{ft}\cdot\mbox{lbf}}{\mbox{min}} \times \frac{\mbox{horsepower}}{33000 \cdot \frac{\mbox{ft }\cdot\mbox{ lbf}}{\mbox{min}} } \approx \frac {\mbox{torque} \times \mbox{RPM}}{5252}

because 5252.113... = \frac {33,000} {2 \pi} \,.

References

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External links

Horsepower and Torque An article showing how power, torque, and gearing affect a vehicle's performance.

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

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