Torque

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Torque
Torque animation.gif
Relationship between force F, torque τ, linear momentum p, and angular momentum L in a system which has rotation constrained to only one plane (forces and moments due to gravity and friction not considered).
Common symbols
, M
SI unit N⋅m
Other units
pound-force-feet, lbf⋅inch, ozf⋅in
In SI base units kg⋅m2⋅s−2
Dimension ML2T−2

Torque, moment, or moment of force is the rotational equivalent of linear force. [1] The concept originated with the studies of Archimedes on the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. The symbol for torque is typically , the lowercase Greek letter tau . When being referred to as moment of force, it is commonly denoted by M.

Force Any interaction that, when unopposed, will change the motion of an object

In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

Archimedes Greek mathematician, physicist, engineer, inventor, and astronomer

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.

Lever one of the six simple machines

A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. A lever is a rigid body capable of rotating on a point on itself. On the basis of the location of fulcrum, load and effort, the lever is divided into three types. It is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide leverage. The ratio of the output force to the input force is the mechanical advantage of the lever.

Contents

In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the position vector (distance vector) and the force vector. The magnitude of torque of a rigid body depends on three quantities: the force applied, the lever arm vector [2] connecting the origin to the point of force application, and the angle between the force and lever arm vectors. In symbols:

Pseudovector Physical quantity that changes sign with improper rotation

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true vector, also known, in this context, as a polar vector, which on reflection matches its mirror image.

Cross product mathematical operation on two vectors

In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol . Given two linearly independent vectors and , the cross product, , is a vector that is perpendicular to both and and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.

Euclidean vector geometric object that has magnitude (or length) and direction

In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial pointA with a terminal pointB, and denoted by

where

is the torque vector and is the magnitude of the torque,
r is the position vector (a vector from the origin of the coordinate system defined to the point where the force is applied)
F is the force vector,
× denotes the cross product, which is defined as magnitudes of the respective vectors times .
is the angle between the force vector and the lever arm vector.

The SI unit for torque is N⋅m. For more on the units of torque, see Units.

Defining terminology

The term torque was introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. [3] However, torque is referred to using different vocabulary depending on geographical location and field of study. This article refers to the definition used in US physics in its usage of the word torque. [4] In the UK and in US mechanical engineering, torque is referred to as moment of force, usually shortened to moment. [5] In US physics [4] and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, where the term torque is used for the closely related "resultant moment of a couple". [5]

James Thomson (engineer) British engineer

Professor James Thomson FRS FRSE LLD was an engineer and physicist whose reputation is substantial though it is overshadowed by that of his younger brother William Thomson.

William Thomson, 1st Baron Kelvin British physicist and engineer

William Thomson, 1st Baron Kelvin, was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging discipline of physics in its modern form. He worked closely with mathematics professor Hugh Blackburn in his work. He also had a career as an electric telegraph engineer and inventor, which propelled him into the public eye and ensured his wealth, fame and honour. For his work on the transatlantic telegraph project he was knighted in 1866 by Queen Victoria, becoming Sir William Thomson. He had extensive maritime interests and was most noted for his work on the mariner's compass, which previously had limited reliability.

Mechanical engineering discipline of engineering

Mechanical engineering is the discipline that applies engineering, physics, engineering mathematics, and materials science principles to design, analyze, manufacture, and maintain mechanical systems. It is one of the oldest and broadest of the engineering disciplines.

Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the general term used for the tendency of one or more applied forces to rotate an object about an axis, but not necessarily to change the angular momentum of the object (the concept which is called torque in physics). [5] For example, a rotational force applied to a shaft causing acceleration, such as a drill bit accelerating from rest, results in a moment called a torque. By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the angular momentum of the beam is not changing, this bending moment is not called a torque. Similarly with any force couple on an object that has no change to its angular momentum, such moment is also not called a torque.

Angular momentum measure of the extent to which an object will continue to rotate in the absence of an applied torque

In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a system remains constant unless acted on by an external torque.

In physics, the angular velocity of a particle is the rate at which its angular position about a chosen center point changes: that is, the time rate of change of its angular displacement relative to the origin. It is measured in angle per unit time, radians per second in SI units, and is usually represented by the symbol omega. By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.

Moment of inertia moment of inertia

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate. It is an extensive (additive) property: for a point mass the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems. Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

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 t = r x F has magnitude t = |r| |F[?]| = |r| |F| sin th and is directed outward from the page. Torque, position, and force.svg
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 metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres 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 are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque. [6]

Newton (unit) SI unit of force

The newton is the International System of Units (SI) derived unit of force. It is named after Isaac Newton in recognition of his work on classical mechanics, specifically Newton's second law of motion.

The metre or meter is the base unit of length in the International System of Units (SI). The SI unit symbol is m. The metre is defined as the length of the path travelled by light in a vacuum in 1/299 792 458 second.

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. [7] [8]

It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors. The torque vector points along the axis of the rotation that the force vector (starting from rest) would initiate. The resulting torque vector direction is determined by the right-hand rule. [7]

The unbalanced torque on a body along axis of rotation 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 the motion of a point particle,

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

where α is the angular acceleration of the particle, and p|| is the radial component of its linear momentum. This equation is the rotational analogue of Newton's Second Law for point particles, and is valid for any type of trajectory. Note that although force and acceleration are always parallel and directly proportional, the torque τ need not be parallel or directly proportional to the angular acceleration α. This arises from the fact that although mass is always conserved, the moment of inertia in general is not.

Proof of the equivalence of definitions

The definition of angular momentum for a single particle is:

where "×" indicates the vector cross product, p is the particle's linear momentum, and r is the displacement vector from the origin (the origin is assumed to be a fixed location anywhere in space). 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 definition of force (whether or not mass is constant) and the definition of velocity

The cross product of momentum with its associated velocity is zero because velocity and momentum are parallel, so the second term vanishes.

By definition, torque τ = r × F. Therefore, torque on a particle is equal to the first derivative of its angular momentum with respect to time.

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

This is a general proof.

Units

Torque has dimension force times distance, symbolically L2MT−2. Official SI literature suggests using the unit newton metre (N⋅m) or the unit joule per radian . [9] The unit newton metre is properly denoted N⋅m or N m. [10] This avoids ambiguity with mN, millinewtons.

The SI unit for energy or work is the joule. It is dimensionally equivalent to a force of one newton acting over a distance of one metre, but it is not used for torque. Energy and torque are entirely different concepts, so the practice of using different unit names (i.e., reserving newton metres for torque and using only joules for energy) helps avoid mistakes and misunderstandings. [9] 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, τ is magnitude of the torque, and θ is the angle moved (in radians). This equation motivates the alternate unit name joules per radian. [9]

In Imperial units, "pound-force-feet" (lbf⋅ft), "foot-pounds-force", "inch-pounds-force", "ounce-force-inches" (ozf⋅in) [ citation needed ] are used, and other non-SI units of torque includes "metre-kilograms-force". For all these units, the word "force" is often left out. [11] 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).

Torque is sometimes listed with units that do not make dimensional sense, such as the gram-centimeter. In this case, "gram" should be understood as the force given by the weight of 1 gram on the surface of the Earth (i.e. 0.00980665 N). The surface of the Earth has a standard gravitational field strength of 9.80665 N/kg.

Special cases and other facts

Moment arm formula

Moment arm diagram Moment arm.svg
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 to the right, 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 at the terminal end of a wrench that is 0.5 m long (or a force of 10 N exactly 0.5 m from the twist point of a wrench of any length), the torque will be 5 Nm – assuming that the person moves the wrench by applying force in the plane of movement and perpendicular to the wrench.

The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque. This causes the top to precess. PrecessionOfATop.svg
The torque caused by the two opposing forces Fg and −Fg 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, three equations are used.

Net force versus 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. If the net force is not zero, and is the torque measured from , then the torque measured from is …

Machine torque

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

Torque forms 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). One can measure the varying torque output over that range with a dynamometer, and show it as a torque curve.

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 and electric motors 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. [12]

Proof

The work done by a variable force acting over a finite linear displacement is given by integrating the force with respect to an elemental linear displacement

However, the infinitesimal linear displacement is related to a corresponding angular displacement and the radius vector as

Substitution in the above expression for work gives

The expression is a scalar triple product given by . An alternate expression for the same scalar triple product is

But as per the definition of torque,

Corresponding substitution in the expression of work gives,

Since the parameter of integration has been changed from linear displacement to angular displacement, the limits of the integration also change correspondingly, giving

If the torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., giving

It follows from the work-energy theorem that W also represents the change in the rotational kinetic energy Er of the body, given by

where I is the moment of inertia of the body and ω is its angular speed. [12]

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.

Algebraically, the equation may be rearranged to compute torque for a given angular speed and 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 bicycles: Bicycles are typically composed of two road wheels, front and rear gears (referred to as sprockets) meshing with a circular chain, and a derailleur mechanism if the bicycle's transmission system allows multiple gear ratios to be used (i.e. multi-speed bicycle), all of which attached to the frame. A cyclist, the person who rides the bicycle, provides the input power by turning pedals, thereby cranking the front sprocket (commonly referred to as chainring). The input power provided by the cyclist is equal to the product of cadence (i.e. the number of pedal revolutions per minute) and the torque on spindle of the bicycle's crankset. The bicycle's drivetrain transmits the input power to the road wheel, which in turn conveys the received power to the road as the output power of the bicycle. Depending on the gear ratio of the bicycle, a (torque, rpm)input pair is converted to a (torque, rpm)output pair. By using a larger rear gear, or by switching to a lower gear in multi-speed bicycles, angular speed of the road wheels is decreased while the torque is increased, product of which (i.e. power) does not change.

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 revolutions per second).

Also, the unit newton metre 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

A conversion factor may be necessary when using different units of power or torque. For example, if rotational speed (revolutions per time) is used in place of angular speed (radians per time), we multiply by a factor of 2π radians per revolution. In the following formulas, P is power, τ is torque, and ν (Greek letter nu) is rotational speed.

Showing units:

Dividing by 60 seconds per minute gives us the following.

where rotational speed is in revolutions per minute (rpm).

Some people (e.g., American automotive engineers) use horsepower (imperial mechanical) for power, foot-pounds (lbf⋅ft) for torque and rpm for rotational speed. This results in the formula changing to:

The constant below (in foot pounds per minute) changes with the definition of the horsepower; for example, using metric horsepower, it becomes approximately 32,550.

Use of other units (e.g., BTU per hour for power) would require a different custom conversion factor.

Derivation

For a rotating object, the linear distance covered at the circumference of rotation is the product of the radius with the angle covered. That is: linear distance = radius × angular distance. And by definition, linear distance = linear speed × time = radius × angular speed × time.

By the definition of torque: torque = radius × force. 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 newton metres and rotational speed in revolutions per second, the above equation gives power in newton metres per second or watts. If Imperial units are used, and if torque is in pounds-force feet and rotational speed in revolutions per minute, the above equation gives power in foot pounds-force per minute. The horsepower form of the equation is then derived by applying the conversion factor 33,000 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:

Torque multiplier

Torque can be multiplied via three methods: by locating the fulcrum such that the length of a lever is increased; by using a longer lever; or by the use of a speed reducing gearset or gear box. Such a mechanism multiplies torque, as rotation rate is reduced.

See also

Related Research Articles

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In physics, power is the rate of doing work or of transferring heat, i.e. the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second (J/s), known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written:

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

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In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the work done on the ball as it falls is equal to the weight of the ball multiplied by the distance to the ground. When the force is constant and the angle between the force and the displacement is θ, then the work done is given by W = Fs cos θ.

Unit vector vector of length one

In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

Rigid body dynamics studies the movement of systems of interconnected bodies under the action of external forces; described by the laws of kinematics and by the application of Newtons second law (kinetics) or their derivative form Lagrangian mechanics

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Larmor formula

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Rotation around a fixed axis

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Axis–angle representation

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In physics, magnetization dynamics is the branch of solid-state physics that describes the evolution of the magnetization of a material.

In physics, rotatum is the derivative of torque with respect to time. Expressed as an equation, rotatum Ρ is:

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