Torque

Last updated
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

In physics and mechanics, torque is the rotational analogue of linear force. [1] It is also referred to as the moment of force (also abbreviated to moment). It describes the rate of change of angular momentum that would be imparted to an isolated body.

Contents

The concept originated with the studies by Archimedes of the usage of levers, which is reflected in his famous quote: "Give me a lever and a place to stand and I will move the Earth". Just as a linear force is a push or a pull applied to a body, a torque can be thought of as a twist applied to an object with respect to a chosen point. Torque is defined as the product of the magnitude of the perpendicular component of the force and the distance of the line of action of a force from the point around which it is being determined. The law of conservation of energy can also be used to understand torque. 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.

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

where

The SI unit for torque is the newton-metre (N⋅m). For more on the units of torque, see § Units.

History

The term torque (from Latin torquēre , 'to twist') is said to have been suggested by James Thomson and appeared in print in April, 1884. [3] [4] [5] Usage is attested the same year by Silvanus P. Thompson in the first edition of Dynamo-Electric Machinery. [5] Thompson motivates the term as follows: [4]

Just as the Newtonian definition of force is that which produces or tends to produce motion (along a line), so torque may be defined as that which produces or tends to produce torsion (around an axis). It is better to use a term which treats this action as a single definite entity than to use terms like "couple" and "moment", which suggest more complex ideas. The single notion of a twist applied to turn a shaft is better than the more complex notion of applying a linear force (or a pair of forces) with a certain leverage.

Today, torque is referred to using different vocabulary depending on geographical location and field of study. This article follows the definition used in US physics in its usage of the word torque. [6]

In the UK and in US mechanical engineering, torque is referred to as moment of force, usually shortened to moment. [7] This terminology can be traced back to at least 1811 in Siméon Denis Poisson's Traité de mécanique. [8] An English translation of Poisson's work appears in 1842.

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| sin θ and is directed outward from the page.

A force applied perpendicularly 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. [9]

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

where F is the force acting on the particle. The magnitude τ of the torque is given by

where 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. [10] [11]

It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors. Conversely, the torque vector defines the plane in which the position and force vectors lie. The resulting torque vector direction is determined by the right-hand rule. [10]

The net 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.

For the motion of a point particle,

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

using the derivative of a vector is

This equation is the rotational analogue of Newton's second law for point particles, and is valid for any type of trajectory. In some simple cases like a rotating disc, where only the moment of inertia on rotating axis is, the rotational Newton's second law can be

where and .

Proof of the equivalence of definitions

The definition of angular momentum for a single point particle is:

where p is the particle's linear momentum and r is the position vector from the origin. 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 for point particles.

The proof can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass.

Units

Torque has the dimension of force times distance, symbolically T−2L2M. Although those fundamental dimensions are the same as that for energy or work, official SI literature suggests using the unit newton-metre (N⋅m) and never the joule. [12] [13] The unit newton-metre is properly denoted N⋅m. [13]

The traditional imperial and U.S. customary units for torque are the pound foot (lbf-ft), or for small values the pound inch (lbf-in). In the US, torque is most commonly referred to as the foot-pound (denoted as either lb-ft or ft-lb) and the inch-pound (denoted as in-lb). [14] [15] Practitioners depend on context and the hyphen in the abbreviation to know that these refer to torque and not to energy or moment of mass (as the symbolism ft-lb would properly imply).

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 acting 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 the 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 rotational 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 rotational 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 the angular speed of the drive shaft. 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 an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass, the work W can be expressed as

where τ is torque, and θ1 and θ2 represent (respectively) the initial and final angular positions of the body. [16]

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 principle 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. [16]

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. 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 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 angular speed (i.e. the number of pedal revolutions per minute times 2π) and the torque at the 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, angular speed)input pair is converted to a (torque, angular speed)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.

For SI units, the unit of power is the watt, the unit of torque is the newton-metre and the unit of angular speed is the radian per second (not rpm and not revolutions per second).

The unit newton-metre is dimensionally equivalent to the joule, which is the unit of energy. In the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar. This means that the dimensional equivalence of the newton-metre and the joule may be applied in the former, but not in the latter case. This problem is addressed in orientational analysis, which treats the radian as a base unit rather than as a dimensionless unit. [17]

Conversion to other units

A conversion factor may be necessary when using different units of power or torque. For example, if rotational speed (unit: revolution per minute or second) is used in place of angular speed (unit: radian per second), we must multiply by 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, rev/min).

Some people (e.g., American automotive engineers) use horsepower (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.

The 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 per unit of time, 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 resultant torques due to several forces applied to about a point is equal to the sum of the contributing torques:

From this it follows that the torques resulting from two forces acting around a pivot on an object are balanced when

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

<span class="mw-page-title-main">Angular momentum</span> Conserved physical quantity; rotational analogue of linear momentum

In physics, angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

<span class="mw-page-title-main">Centripetal force</span> Force directed to the center of rotation

A centripetal force is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In the theory of Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

<span class="mw-page-title-main">Jerk (physics)</span> Rate of change of acceleration with time

In physics, jerk (also known as jolt) is the rate of change of an object's acceleration over time. It is a vector quantity (having both magnitude and direction). Jerk is most commonly denoted by the symbol j and expressed in m/s3 (SI units) or standard gravities per second (g0/s).

In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. Power is a scalar quantity.

<span class="mw-page-title-main">Precession</span> Periodic change in the direction of a rotation axis

Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

<span class="mw-page-title-main">Equations of motion</span> Equations that describe the behavior of a physical system

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 behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. 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.

Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

<span class="mw-page-title-main">Angular velocity</span> Pseudovector representing an objects change in orientation with respect to time

In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.

In physics, angular acceleration is the time rate of change of angular velocity. Following the two types of angular velocity, spin angular velocity and orbital angular velocity, the respective types of angular acceleration are: spin angular acceleration, involving a rigid body about an axis of rotation intersecting the body's centroid; and orbital angular acceleration, involving a point particle and an external axis.

<span class="mw-page-title-main">Work (physics)</span> Process of energy transfer to an object via force application through displacement

In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do positive work if when applied it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.

<span class="mw-page-title-main">Moment of inertia</span> Scalar measure of the rotational inertia with respect to a fixed axis of rotation

The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin 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 rate of rotation by a given amount.

<span class="mw-page-title-main">Rigid body dynamics</span> Study of the effects of forces on undeformable bodies

In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, which remains at a constant distance from the axis of rotation. In circular motion, the distance between the body and a fixed point on its surface remains the same, i.e., the body is assumed rigid.

<span class="mw-page-title-main">Rotating reference frame</span> Concept in classical mechanics

A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

<span class="mw-page-title-main">Larmor formula</span> Gives the total power radiated by an accelerating, nonrelativistic point charge

In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.

<span class="mw-page-title-main">Rotation around a fixed axis</span> Type of motion

Rotation around a fixed axis or axial rotation is a special case of rotational motion around an axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result.

<span class="mw-page-title-main">Pendulum (mechanics)</span> Free swinging suspended body

A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back towards the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

<span class="mw-page-title-main">Axis–angle representation</span> Parameterization of a rotation into a unit vector and angle

In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction (geometry) of an axis of rotation, and an angle of rotation θ describing the magnitude and sense of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained. For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame.

References

  1. Serway, R. A. and Jewett, J. W. Jr. (2003). Physics for Scientists and Engineers. 6th ed. Brooks Cole. ISBN   0-534-40842-7.
  2. Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN   0-7167-0809-4.
  3. Thomson, James; Larmor, Joseph (1912). Collected Papers in Physics and Engineering. University Press. p. civ.
  4. 1 2 Thompson, Silvanus Phillips (1893). Dynamo-electric machinery: A Manual For Students Of Electrotechnics (4th ed.). New York, Harvard publishing co. p. 108.
  5. 1 2 "torque". Oxford English Dictionary. 1933.
  6. Physics for Engineering by Hendricks, Subramony, and Van Blerk, Chinappi page 148, Web link Archived 2017-07-11 at the Wayback Machine
  7. Kane, T.R. Kane and D.A. Levinson (1985). Dynamics, Theory and Applications pp. 90–99: Free download Archived 2015-06-19 at the Wayback Machine .
  8. Poisson, Siméon-Denis (1811). Traité de mécanique, tome premier. p.  67.
  9. "Right Hand Rule for Torque". Archived from the original on 2007-08-19. Retrieved 2007-09-08.
  10. 1 2 Halliday, David; Resnick, Robert (1970). Fundamentals of Physics. John Wiley & Sons. pp. 184–85.
  11. Knight, Randall; Jones, Brian; Field, Stuart (2016). College Physics: A Strategic Approach (3rd technology update ed.). Boston: Pearson. p. 199. ISBN   9780134143323. OCLC   922464227.
  12. From the official SI website Archived 2021-04-19 at the Wayback Machine , The International System of Units – 9th edition – Text in English Section 2.3.4: "For example, the quantity torque is the cross product of a position vector and a force vector. The SI unit is newton-metre. Even though torque has the same dimension as energy (SI unit joule), the joule is never used for expressing torque."
  13. 1 2 "SI brochure Ed. 9, Section 2.3.4" (PDF). Bureau International des Poids et Mesures. 2019. Archived (PDF) from the original on 2020-07-26. Retrieved 2020-05-29.
  14. "Dial Torque Wrenches from Grainger". Grainger. 2020. Demonstration that, as in most US industrial settings, the torque ranges are given in ft-lb rather than lbf-ft.
  15. Erjavec, Jack (22 January 2010). Manual Transmissions & Transaxles: Classroom manual. Cengage Learning. p. 38. ISBN   978-1-4354-3933-7.
  16. 1 2 Kleppner, Daniel; Kolenkow, Robert (1973). An Introduction to Mechanics . McGraw-Hill. pp.  267–268. ISBN   9780070350489.
  17. Page, Chester H. (1979). "Rebuttal to de Boer's 'Group properties of quantities and units'". American Journal of Physics. 47 (9): 820. Bibcode:1979AmJPh..47..820P. doi:10.1119/1.11704.