torque

1. The moment of a force; the measure of a force's tendency to produce torsion and rotation about an axis, equal to the vector product of the radius vector from the axis of rotation to the point of application of the force and the force vector.
2. A turning or twisting force.

[From Latin torquēre, to twist.]

(tôrk)

[French, from Old French, from Latin torquēs, from torquēre, to twist.]

For more information on torque, visit Britannica.com.

Concept

Torque is the application of force where there is rotational motion. The most obvious example of torque in action is the operation of a crescent wrench loosening a lug nut, and a close second is a playground seesaw. But torque is also crucial to the operation of gyroscopes for navigation, and of various motors, both internal-combustion and electrical.

How It Works

Force, which may be defined as anything that causes an object to move or stop moving, is the linchpin of the three laws of motion formulated by Sir Isaac Newton (1642-1727.) The first law states that an object at rest will remain at rest, and an object in motion will remain in motion, unless or until outside forces act upon it. The second law defines force as the product of mass multiplied by acceleration. According to the third law, when one object exerts a force on another, the second object exerts on the first a force equal in magnitude but opposite in direction.

One way to envision the third law is in terms of an active event—for instance, two balls striking one another. As a result of the impact, each flies backward. Given the fact that the force on each is equal, and that force is the product of mass and acceleration (this is usually rendered with the formula F = ma), it is possible to make some predictions regarding the properties of mass and acceleration in this interchange. For instance, if the mass of one ball is relatively small compared to that of the other, its acceleration will be correspondingly greater, and it will thus be thrown backward faster.

On the other hand, the third law can be demonstrated when there is no apparent movement, as for instance, when a person is sitting on a chair, and the chair exerts an equal and opposite force upward. In such a situation, when all the forces acting on an object are in balance, that object is said to be in a state of equilibrium.

Physicists often discuss torque within the context of equilibrium, even though an object experiencing net torque is definitely not in equilibrium. In fact, torque provides a convenient means for testing and measuring the degree of rotational or circular acceleration experienced by an object, just as other means can be used to calculate the amount of linear acceleration. In equilibrium, the net sum of all forces acting on an object should be zero; thus in order to meet the standards of equilibrium, the sum of all torques on the object should also be zero.

Real-Life Applications

Seesaws and Wrenches

As for what torque is and how it works, it is best discuss it in relationship to actual objects in the physical world. Two in particular are favorites among physicists discussing torque: a seesaw and a wrench turning a lug nut. Both provide an easy means of illustrating the two ingredients of torque, force and moment arm.

In any object experiencing torque, there is a pivot point, which on the seesaw is the balance-point, and which in the wrench-and-lug nut combination is the lug nut itself. This is the area around which all the forces are directed. In each case, there is also a place where force is being applied. On the seesaw, it is the seats, each holding a child of differing weight. In the realm of physics, weight is actually a variety of force.

Whereas force is equal to mass multiplied by acceleration, weight is equal to mass multiplied by the acceleration due to gravity. The latter is equal to 32 ft (9.8 m)/sec². This means that for every second that an object experiencing gravitational force continues to fall, its velocity increases at the rate of 32 ft or 9.8 m per second. Thus, the formula for weight is essentially the same as that for force, with a more specific variety of acceleration substituted for the generalized term in the equation for force.

As for moment arm, this is the distance from the pivot point to the vector on which force is being applied. Moment arm is always perpendicular to the direction of force. Consider a wrench operating on a lug nut. The nut, as noted earlier, is the pivot point, and the moment arm is the distance from the lug nut to the place where the person operating the wrench has applied force. The torque that the lug nut experiences is the product of moment arm multiplied by force.

In English units, torque is measured in pound-feet, whereas the metric unit is Newtonmeters, or N·m. (One newton is the amount of force that, when applied to 1 kg of mass, will give it an acceleration of 1 m/sec²). Hence if a person were to a grip a wrench 9 in (23 cm) from the pivot point, the moment arm would be 0.75 ft (0.23 m.) If the person then applied 50 lb (11.24 N) of force, the lug nut would be experiencing 37.5 pound-feet (2.59 N·m) of torque.

The greater the amount of torque, the greater the tendency of the object to be put into rotation. In the case of a seesaw, its overall design, in particular the fact that it sits on the ground, means that its board can never undergo anything close to 360° rotation; nonetheless, the board does rotate within relatively narrow parameters. The effects of torque can be illustrated by imagining the clockwise rotational behavior of a seesaw viewed from the side, with a child sitting on the left and a teenager on the right.

Suppose the child weighs 50 lb (11.24 N) and sits 3 ft (0.91 m) from the pivot point, giving her side of the seesaw a torque of 150 pound-feet (10.28 N·m). On the other side, her teenage sister weighs 100 lb (22.48 N) and sits 6 ft (1.82 m) from the center, creating a torque of 600 pound-feet (40.91 N·m). As a result of the torque imbalance, the side holding the teenager will rotate clockwise, toward the ground, causing the child's side to also rotate clockwise—off the ground.

In order for the two to balance one another perfectly, the torque on each side has to be adjusted. One way would be by changing weight, but a more likely remedy is a change in position, and therefore, of moment arm. Since the teenager weighs exactly twice as much as the child, the moment arm on the child's side must be exactly twice as long as that on the teenager's.

Hence, a remedy would be for the two to switch positions with regard to the pivot point. The child would then move out an additional 3 ft (.91 m), to a distance of 6 ft (1.83 m) from the pivot, and the teenager would cut her distance from the pivot point in half, to just 3 ft (.91 m). In fact, however, any solution that gave the child a moment arm twice as long as that of the teenager would work: hence, if the teenager sat 1 ft (.3 m) from the pivot point, the child should be at 2 ft (.61 m) in order to maintain the balance, and so on.

On the other hand, there are many situations in which you may be unable to increase force, but can increase moment arm. Suppose you were trying to disengage a particularly stubborn lug nut, and after applying all your force, it still would not come loose. The solution would be to increase moment arm, either by grasping the wrench further from the pivot point, or by using a longer wrench.

For the same reason, on a door, the knob is placed as far as possible from the hinges. Here the hinge is the pivot point, and the door itself is the moment arm. In some situations of torque, however, moment arm may extend over "empty space," and for this reason, the handle of a wrench is not exactly the same as its moment arm. If one applies force on the wrench at a 90°-angle to the handle, then indeed handle and moment arm are identical; however, if that force were at a 45° angle, then the moment arm would be outside the handle, because moment arm and force are always perpendicular. And if one were to pull the wrench away from the lug nut, then there would be 0° difference between the direction of force and the pivot point—meaning that moment arm (and hence torque) would also be equal to zero.

Gyroscopes

A gyroscope consists of a wheel-like disk, called a flywheel, mounted on an axle, which in turn is mounted on a larger ring perpendicular to the plane of the wheel itself. An outer circle on the same plane as the flywheel provides structural stability, and indeed, the gyroscope may include several such concentric rings. Its focal point, however, is the flywheel and the axle. One end of the axle is typically attached to some outside object, while the other end is left free to float.

Once the flywheel is set spinning, gravity has a tendency to pull the unattached end of the axle downward, rotating it on an axis perpendicular to that of the flywheel. This should cause the gyroscope to fall over, but instead it begins to spin a third axis, a horizontal axis perpendicular both to the plane of the flywheel and to the direction of gravity. Thus, it is spinning on three axes, and as a result becomes very stable—that is, very resistant toward outside attempts to upset its balance.

This in turn makes the gyroscope a valued instrument for navigation: due to its high degree of gyroscopic inertia, it resists changes in orientation, and thus can guide a ship toward its destination. Gyroscopes, rather than magnets, are often the key element in a compass. A magnet will point to magnetic north, some distance from "true north" (that is, the North Pole.) But with a gyroscope whose axle has been aligned with true north before the flywheel is set spinning, it is possible to possess a much more accurate directional indicator. For this reason, gyroscopes are used on airplanes—particularly those flying over the poles—as well as submarines and even the Space Shuttle.

Torque, along with angular momentum, is the leading factor dictating the motion of a gyroscope. Think of angular momentum as the momentum (mass multiplied by velocity) that a turning object acquires. Due to a principle known as the conservation of angular momentum, a spinning object has a tendency to reach a constant level of angular momentum, and in order to do this, the sum of the external torques acting on the system must be reduced to zero. Thus angular momentum "wants" or "needs" to cancel out torque.

The "right-hand rule" can help you to understand the torque in a system such as the gyroscope. If you extend your right hand, palm downward, your fingers are analogous to the moment arm. Now if you curl your fingers downward, toward the ground, then your fingertips point in the direction of g—that is, gravitational force. At that point, your thumb (involuntarily, due to the bone structure of the hand) points in the direction of the torque vector.

When the gyroscope starts to spin, the vectors of angular momentum and torque are at odds with one another. Were this situation to persist, it would destabilize the gyroscope; instead, however, the two come into alignment. Using the right-hand rule, the torque vector on a gyroscope is horizontal in direction, and the vector of angular momentum eventually aligns with it. To achieve this, the gyroscope experiences what is known as gyroscopic precession, pivoting along its support post in an effort to bring angular momentum into alignment with torque. Once this happens, there is no net torque on the system, and the conservation of angular momentum is in effect.

Torque in Complex Machines

Torque is a factor in several complex machines such as the electric motor that—with variations—runs most household appliances. It is especially important to the operation of automobiles, playing a significant role in the engine and transmission.

An automobile engine produces energy, which the pistons or rotor convert into torque for transmission to the wheels. Though torque is greatest at high speeds, the amount of torque needed to operate a car does not always vary proportionately with speed. At moderate speeds and on level roads, the engine does not need to provide a great deal of torque. But when the car is starting, or climbing a steep hill, it is important that the engine supply enough torque to keep the car running; otherwise it will stall. To allocate torque and speed appropriately, the engine may decrease or increase the number of revolutions per minute to which the rotors are subjected.

Torque comes from the engine, but it has to be supplied to the transmission. In an automatic transmission, there are two principal components: the automatic gearbox and the torque converter. It is the job of the torque converter to transmit power from the flywheel of the engine to the gearbox, and it has to do so as smoothly as possible. The torque converter consists of three elements: an impeller, which is turned by the engine flywheel; a reactor that passes this motion on to a turbine; and the turbine itself, which turns the input shaft on the automatic gearbox. An infusion of oil to the converter assists the impeller and turbine in synchronizing movement, and this alignment of elements in the torque converter creates a smooth relationship between engine and gearbox. This also leads to an increase in the car's overall torque—that is, its turning force.

Torque is also important in the operation of electric motors, found in everything from vacuum cleaners and dishwashers to computer printers and videocassette recorders to subway systems and water-pumping stations. Torque in the context of electricity involves reference to a number of concepts beyond the scope of this discussion: current, conduction, magnetic field, and other topics relevant to electromagnetic force.

Where to Learn More

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Macaulay, David. The New Way Things Work. Boston: Houghton Mifflin, 1998.

"Rotational Motion." Physics Department, University of Guelph (Web site). <http://www.physics.uoguelph.ca/tutorials/torque/> (March 4, 2001).

"Rotational Motion—Torque." Lee College (Web site). <http://www.lee.edu/mathscience/physics/physics/Courses/LabManual/2b/2b.html> (March 4, 2001).

Schweiger, Peggy E. "Torque" (Web site). <http://www.cyberclassrooms.net/~pschweiger/rotmot.html> (March 4, 2001).

"Torque and Rotational Motion" (Web site). <http://online.cctt.org/curriculumguide/units/torque.asp> (March 4, 2001).

The product of a force and its perpendicular distance to a point of turning; also called the moment of the force. Torque produces torsion and tends to produce rotation. Torque arises from a force or forces acting tangentially to a cylinder or from any force or force system acting about a point. A couple, consisting of two equal, parallel, and oppositely directed forces, produces a torque or moment about the central point. A prime mover such as a turbine exerts a twisting effort on its output shaft, measured as torque. In structures, torque appears as the sum of moments of torsional shear forces acting on a transverse section of a shaft or beam. See also Couple; Torsion.

1. a force that produces or tends to produce rotation in a body. Such force applied to a tooth tends to cause rotation around its long axis. 2. force applied to a tooth to produce rotation of a tooth on a mesiodistal or buccolingual (labiolingual) axis. 3. a rotary force applied to a denture base.

That which tends to produce rotation; the product of a force and a lever arm which tends to twist a body, as the action of a wrench turning a nut on a bolt.

A force that produces a twisting or rotary movement in any plane about an axis of motion; it is the rotary effect of an eccentric force. Torque occurs when bones move around each other at joints which serve as the axes of movement. Thus, a muscle force when applied over a range of motion is measured as torque. Torque is a measure of the turning effect of a force on a lever, so that: torque of lever = force × lever arm (or moment arm) distance; also: torque = moment of inertia × angular acceleration. A torque is sometimes called a moment. The SI unit for torque is the newton-metre.

A rotatory force.

Harnessing engine power to produce twisting force
The twisting force needed to turn a propeller shaft is known as torque. Slow-turning engines are usually able to deliver the necessary torque at low revolutions per minute (rpm). Fast-revving engines produce less torque, so they need a reduction gear to convert their low-torque, high-rpm power into the high-torque, slow-revving power needed to turn a large propeller.Slow-turning engines have several advantages. They run more quietly and last longer than fast-revving engines. Unfortunately, their suitability for most pleasure-boat applications is limited because they’re also always big, heavy, and expensive. So the great majority of pleasure powerboats and sailboats are equipped with smaller, lighter, fast-revving engines and reduction gearboxes. Because of friction, adding a gearbox reduces the power available to turn the propeller by about 3 percent.The maximum torque output of an engine occurs well before peak rpm. For instance, a typical 70 hp marine diesel usually produces its maximum torque of about 150 foot-pounds at about 1,800 rpm, after which the figure declines to about 125 foot-pounds at the top speed of 3,200 rpm. Maximum torque is usually produced at between 50 and 70 percent of maximum rpm.This engine, therefore, will run most efficiently at 1,800 rpm; however, an increase in speed of 25 percent, to 2,250 rpm, will provide greatly increased performance for a relatively small increase in fuel consumption.See also Horsepower; Powerboat Engines; Sailboat Engines.

For other uses, see Torque (disambiguation).

Relationship between force F, torque τ, linear momentum p, and angular momentum L in a system which has rotation constrained in one plane only. (Forces and moments due to gravity and friction not considered.)

Torque, also called moment or moment of force (see the terminology below), is the tendency of a force to rotate an object about an axis,^[1] fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.

Loosely speaking, torque is a measure of the turning force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque (turning force) that loosens or tightens the nut or bolt.

The terminology for this concept is not straightforward: In physics, it is usually called "torque", and in mechanical engineering, it is called "moment".^[2] However, in mechanical engineering, the term "torque" means something different,^[3] described below. In this article, the word "torque" is always used in the physics sense, synonymous with "moment" in engineering.

The symbol for torque is typically τ, the Greek letter tau. When it is called moment, it is commonly denoted M.

The magnitude of torque depends on three quantities: First, the force applied; second, the length of the lever arm^[4] connecting the axis to the point of force application; and third, the angle between the two. In symbols:

where

τ is the torque vector and τ is the magnitude of the torque,

r is the displacement vector (a vector from the point from which torque is measured to the point where force is applied), and r is the length (or magnitude) of the lever arm vector,

F is the force vector, and F is the magnitude of the force,

× denotes the cross product,

θ is the angle between the force vector and the lever arm vector.

The length of the lever arm is particularly important; choosing this length appropriately lies behind the operation of levers, pulleys, gears, and most other simple machines involving a mechanical advantage.

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). For more on the units of torque, see below.

Contents 1 Terminology 2 History 3 Definition and relation to angular momentum 3.1 Proof of the equivalence of definitions 4 Units 5 Special cases and other facts 5.1 Moment arm formula 5.2 Static equilibrium 5.3 Net Force vs. Torque 6 Machine torque 7 Relationship between torque, power and energy 7.1 Conversion to other units 7.2 Derivation 8 Principle of Moments 9 See also 10 References 11 External links

Terminology

See also: Couple (mechanics)

In mechanical engineering (unlike physics), the terms "torque" and "moment" are not interchangeable. "Moment" is the general term for the tendency of one or more applied forces to rotate an object about an axis (the concept which in physics is called torque).^[3] "Torque" is a special case of this: If the applied force vectors add to zero (i.e., their "resultant" is zero), then the forces are called a "couple" and their moment is called a "torque".^[3]

For example, a rotational force down a shaft, such as a turning screw-driver, forms a couple, so the resulting moment is called a "torque". By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the net force is nonzero, this bending moment is not called a "torque".

This article follows physics terminology by calling all moments by the term "torque", whether or not they are associated with a couple.

History

The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers. The rotational analogues of force, mass, and acceleration are torque, moment of inertia, and angular acceleration, respectively.

Definition and relation to angular momentum

A particle is located at position r relative to its axis of rotation. When a force F is applied to the particle, only the perpendicular component F_⊥ produces a torque. This torque τ = r × F has magnitude τ = |r| |F_⊥| = |r| |F| sinθ and is directed outward from the page.

A force applied at a right angle to a lever multiplied by its distance from the lever's fulcrum (the length of the lever arm) is its torque. A force of three newtons applied two meters from the fulcrum, for example, exerts the same torque as a force of one newton applied six meters from the fulcrum. The direction of the torque can be determined by using the right hand grip rule: if the fingers of the right hand curl in the direction of rotation and the thumb points along the axis of rotation, then the thumb also points in the direction of the torque.^[5]

More generally, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:

where r is the particle's position vector relative to the fulcrum, and F is the force acting on the particle. The magnitude τ of the torque is given by

where r is the distance from the axis of rotation to the particle, F is the magnitude of the force applied, and θ is the angle between the position and force vectors. Alternatively,

where F_⊥ is the amount of force directed perpendicularly to the position of the particle. Any force directed parallel to the particle's position vector does not produce a torque.^[6]

It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors. It points along the axis of rotation, and its direction is determined by the right-hand rule.^[6]

The torque on a body determines the rate of change of the body's angular momentum,

where L is the angular momentum vector and t is time. If multiple torques are acting on the body, it is instead the net torque which determines the rate of change of the angular momentum:

For rotation about a fixed axis,

where I is the moment of inertia and ω is the angular velocity. It follows that

where α is the angular acceleration of the body, measured in rad s⁻².

Proof of the equivalence of definitions

The definition of angular momentum for a single particle is:

where "×" indicates the vector cross product and p is the particle's linear momentum. 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,

The cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma (Newton's 2nd law),

Then by definition, torque τ = r × F.

If multiple forces are applied, Newton's second law instead reads F_net = ma, and it follows that

The proof relies on the assumption that mass is constant; this is valid only in non-relativistic systems in which no mass is being ejected.

Units

Torque has dimensions of force times distance. Official SI literature suggests using the unit newton meter (N·m) or the unit joule per radian.^[7] The unit newton meter is properly denoted N·m or N m.^[8] This avoids ambiguity—for example, mN is the symbol for millinewton.

The joule, which is the SI unit for energy or work, is dimensionally equivalent to a newton meter, but it is not used for torque. Energy and torque are entirely different concepts, so the practice of using different unit names for them helps avoid mistakes and misunderstandings.^[7] 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,

where E is the energy, τ is magnitude of the torque, and θ is the angle moved (in radians). This equation motivates the alternate unit name joules per radian.^[7]

Other non-SI units of torque include "pound-force-feet", "foot-pounds-force", "inch-pounds-force", "ounce-force-inches", and "meter-kilograms-force". For all these units, the word "force" is often left out,^[9] for example abbreviating "pound-force-foot" to simply "pound-foot". (In this case, it would be implicit that the "pound" is pound-force and not pound-mass.)

Special cases and other facts

Moment arm formula

Moment arm diagram

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.

The torque caused by the two opposing forces F_g and −F_g causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.

Static equilibrium

For 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.

Net Force vs. Torque

When the net force on the system is zero, the torque measured from any point in space is the same. For example, the torque on a current-carrying loop in a uniform magnetic field is the same regardless of your point of reference.

Machine torque

Torque curve of a motorcycle ("BMW K 1200 R 2005"). The horizontal axis is the speed (in rpm) that the wheels are turning, and the vertical axis is the torque (in Newton metres) that the engine is capable of providing at that speed.

Torque 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 of the axis. 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 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. 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. Power at the drive wheels is equal to engine power less mechanical losses regardless of any gearing between the engine and drive wheels.

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). Reciprocating steam engines can start heavy loads from zero RPM without a clutch.

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. Mathematically, for rotation about a fixed axis through the center of mass,

where W is work, τ is torque, and θ₁ and θ₂ represent (respectively) the initial and final angular positions of the body.^[10] It follows from the work-energy theorem that W also represents the change in the rotational kinetic energy K_rot of the body, given by

where I is the moment of inertia of the body and ω is its angular speed.^[10]

Power is the work per unit time, given by

where P is power, τ is torque, ω is the angular velocity, and · represents the scalar product.

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 units

For 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 2π must be added because there are 2π radians in a revolution:

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.

Derivation

For 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 × angular speed. By definition, linear distance=linear speed × time=radius × angular speed × time.

By the definition of torque: torque=force × 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 2π 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 33000 ft·lbf/min per horsepower:

because

Principle of Moments

The Principle of Moments, also known as Varignon's theorem (not to be confused with the geometrical theorem of the same name) states that the sum of torques due to several forces applied to a single point is equal to the torque due to the sum (resultant) of the forces. Mathematically, this follows from:

See also

• Conversion of units
• Angular momentum
• Mechanical equilibrium
• Moment (physics)
• Rigid body dynamics
• Statics
• Torque converter
• Torque screwdriver
• Torque limiter
• Torque wrench
• Torque tester
• Torsion (mechanics)
• Couple (mechanics)

References

1. ^ Serway, R. A. and Jewett, Jr. J. W. (2003). Physics for Scientists and Engineers. 6th Ed. Brooks Cole. ISBN 0-53440-842-7.
2. ^ Physics for Engineering by Hendricks, Subramony, and Van Blerk, page 148, Web link
3. ^ ^{a b c} Dynamics, Theory and Applications by T.R. Kane and D.A. Levinson, 1985, pp. 90-99: Free download
4. ^ Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4. 
5. ^ "Right Hand Rule for Torque". http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html. Retrieved 2007-09-08. 
6. ^ ^{a b} Halliday, David; Resnick, Robert (1970). Fundamentals of Physics. John Wiley & Sons, Inc.. p. 184–85. 
7. ^ ^{a b c} From the official SI website: "...For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit newton metre, or it may be thought of as energy per angle, suggesting the unit joule per radian."
8. ^ "SI brochure Ed. 8, Section 5.1". Bureau International des Poids et Mesures. 2006. http://www1.bipm.org/en/si/si_brochure/chapter5/5-1.html. Retrieved 2007-04-01. 
9. ^ See, for example: "CNC Cookbook: Dictionary: N-Code to PWM". http://www.cnccookbook.com/MTCNCDictNtoPWM.htm. Retrieved 2008-12-17. 
10. ^ ^{a b} Kleppner, Daniel; Kolenkow, Robert (1973). An Introduction to Mechanics. McGraw-Hill. p. 267–68. 

External links

• 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 Unit Converter
• www.mechanismen.be-what is a moment (dutch).

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Dansk (Danish)
1.
n. - drejningsmoment, vridningsmoment
v. tr. - spænde, dreje

2.
n. - snoet halsring

Nederlands (Dutch)
torsie

Français (French)
1.
n. - (Phys) moment de torsion, (Aut) couple moteur, (Hist) torque
v. tr. - tourner, (Tech) serrer

2.
n. - bijou de métal (réalisé par torsion d'une bande de métal)

Deutsch (German)
1.
n. - Drehmoment
v. - eine Drehkraft ansetzen

2.
n. - Torques (Halsring)

Ελληνική (Greek)
n. - (μηχαν.) ροπή στρέψης
v. - εφαρμόζω ροπή στρέψης

Italiano (Italian)
collana, coppia, appaiare

Português (Portuguese)
n. - esforço de torção (m), rotação (f)
v. - colar

Русский (Russian)
крученное металлическое ожерелье/браслет, (физ.) вращающийся момент, реактивная штанга, (спец.) закручивать

Español (Spanish)
1.
n. - momento de torsión
v. tr. - medir el par de torsión, rotar, torcer

2.
n. - torques, collar

Svenska (Swedish)
n. - vridmoment (tekn.), propellermotstånd
v. - vrida, rotera, använda vridmoment

中文（简体）(Chinese (Simplified))
1. 扭矩, 转矩

2. 项链, 手镯, 金属领圈

中文（繁體）(Chinese (Traditional))
1.
n. - 項鏈, 手鐲, 金屬領圈

2.
n. - 扭矩, 轉矩
v. tr. - 轉矩

한국어 (Korean)
1.
n. - 회전 효과, 회전력, 염력
v. tr. - 회전시키다

2.
n. - (고대의) 목걸이

日本語 (Japanese)
n. - トルク, 回転効果, 首鎖, 回転力

العربيه (Arabic)
‏(الاسم) طوق معدني للعنق عندج قدامى الفرنسيين والجرمان (فعل) يطوق‏

עברית (Hebrew)
n. - ‮מומנט (כוח) הסיבוב‬
v. tr. - ‮סובב, העביר מומנט סיבוב ל-‬
n. - ‮ענק, קולר, אצעדה‬

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