In

_{1}) to x(''t''_{2}). This integral is computed along the trajectory of the particle, and is therefore said to be ''path dependent''.
If the force is always directed along this line, and the magnitude of the force is ''F'', then this integral simplifies to
:$W\; =\; \backslash int\_C\; F\backslash ,ds$
where ''s'' is displacement along the line. If F is constant, in addition to being directed along the line, then the integral simplifies further to
:$W\; =\; \backslash int\_C\; F\backslash ,ds\; =\; F\backslash int\_C\; ds\; =\; Fs$
where ''s'' is the displacement of the point along the line.
This calculation can be generalized for a constant force that is not directed along the line, followed by the particle. In this case the

_{1}) to ''φ''(''t''_{2}). This integral depends on the rotational trajectory ''φ''(''t''), and is therefore path-dependent.
If the torque T is aligned with the angular velocity vector so that,
:$\backslash mathbf\; =\; \backslash tau\; \backslash mathbf,$
and both the torque and angular velocity are constant, then the work takes the form,
:$W\; =\; \backslash int\_^\; \backslash tau\; \backslash dot\; dt\; =\; \backslash tau(\backslash phi\_2\; -\; \backslash phi\_1).$
This result can be understood more simply by considering the torque as arising from a force of constant magnitude ''F'', being applied perpendicularly to a lever arm at a distance ''r'', as shown in the figure. This force will act through the distance along the circular arc , so the work done is
:$W\; =\; F\; s\; =\; F\; r\; \backslash phi\; .$
Introduce the torque , to obtain
:$W\; =\; F\; r\; \backslash phi\; =\; \backslash tau\; \backslash phi\; ,$
as presented above.
Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.

_{1}) and x(''t''_{2}) to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is
:$W\; =\; \backslash int\_C\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; U(\backslash mathbf(t\_1))-U(\backslash mathbf(t\_2)).$
The function ''U''(x) is called the

_{g}'' is weight (pounds in imperial units, and newtons in SI units), and Δ''y'' is the change in height ''y''. Notice that the work done by gravity depends only on the vertical movement of the object. The presence of friction does not affect the work done on the object by its weight.

_{1}) to r(''t''_{2}) is given by
:$W=-\backslash int^\_\backslash frac\backslash mathbf\backslash cdot\; d\backslash mathbf=-\backslash int^\_\backslash frac\backslash mathbf\backslash cdot\backslash mathbfdt.$
Notice that the position and velocity of the mass ''m'' are given by
:$\backslash mathbf\; =\; r\backslash mathbf\_r,\; \backslash qquad\backslash mathbf\; =\; \backslash frac\; =\; \backslash dot\backslash mathbf\_r\; +\; r\backslash dot\backslash mathbf\_t,$
where e_{''r''} and e_{''t''} are the radial and tangential unit vectors directed relative to the vector from ''M'' to ''m'', and we use the fact that $d\; \backslash mathbf\_r/dt\; =\; \backslash dot\backslash mathbf\_t.$ Use this to simplify the formula for work of gravity to,
:$W=-\backslash int^\_\backslash frac(r\backslash mathbf\_r)\backslash cdot(\backslash dot\backslash mathbf\_r\; +\; r\backslash dot\backslash mathbf\_t)dt\; =\; -\backslash int^\_\backslash fracr\backslash dotdt\; =\; \backslash frac-\backslash frac.$
This calculation uses the fact that
:$\backslash fracr^=-r^\backslash dot=-\backslash frac.$
The function
:$U=-\backslash frac,$
is the gravitational potential function, also known as

_{x}''dt'', over time ''t'' is (1/2)''x''^{2}. The work is the product of the distance times the spring force, which is also dependent on distance; hence the ''x''^{2} result.

_{1}) to the point X(''t''_{2}) to obtain
:$\backslash int\_^\; \backslash mathbf\backslash cdot\backslash dot\; dt\; =\; m\backslash int\_^\backslash ddot\backslash cdot\backslash dotdt.$
The left side of this equation is the work of the applied force as it acts on the particle along the trajectory from time ''t''_{1} to time ''t''_{2}. This can also be written as
:$W\; =\; \backslash int\_^\; \backslash mathbf\backslash cdot\backslash dot\; dt\; =\; \backslash int\_^\; \backslash mathbf\backslash cdot\; d\backslash mathbf.$
This integral is computed along the trajectory X(''t'') of the particle and is therefore path dependent.
The right side of the first integral of Newton's equations can be simplified using the following identity
:$\backslash frac\backslash frac(\backslash dot\backslash cdot\; \backslash dot)\; =\; \backslash ddot\backslash cdot\backslash dot,$
(see

_{x} is the component of F along the X-axis, so
:$F\_x\; v\; =\; m\backslash dotv.$
Integration of both sides yields
:$\backslash int\_^F\_x\; v\; dt\; =\; \backslash frac\; v^2(t\_2)\; -\; \backslash frac\; v^2(t\_1).$
If ''F''_{x} is constant along the trajectory, then the integral of velocity is distance, so
:$F\_x\; (d(t\_2)-d(t\_1))\; =\; \backslash frac\; v^2(t\_2)\; -\; \backslash frac\; v^2(t\_1).$
As an example consider a car skidding to a stop, where ''k'' is the coefficient of friction and ''W'' is the weight of the car. Then the force along the trajectory is . The velocity ''v'' of the car can be determined from the length ''s'' of the skid using the work–energy principle,
:$kWs\; =\; \backslash frac\; v^2,\backslash quad\backslash mbox\backslash quad\; v\; =\; \backslash sqrt.$
Notice that this formula uses the fact that the mass of the vehicle is .

scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of this equation with the velocity, , yields
:$W\; v\_z\; =\; m\backslash dotV,$
where ''V'' is the magnitude of V. The constraint forces between the vehicle and the road cancel from this equation because , which means they do no work.
Integrate both sides to obtain
:$\backslash int\_^W\; v\_z\; dt\; =\; \backslash frac\; V^2(t\_2)\; -\; \backslash frac\; V^2\; (t\_1).$
The weight force ''W'' is constant along the trajectory and the integral of the vertical velocity is the vertical distance, therefore,
:$W\; \backslash Delta\; z\; =\; \backslash fracV^2.$
Recall that V(''t''_{1})=0. Notice that this result does not depend on the shape of the road followed by the vehicle.
In order to determine the distance along the road assume the downgrade is 6%, which is a steep road. This means the altitude decreases 6 feet for every 100 feet traveled—for angles this small the sin and tan functions are approximately equal. Therefore, the distance ''s'' in feet down a 6% grade to reach the velocity ''V'' is at least
:$s=\backslash frac=\; 8.3\backslash frac,\backslash quad\backslash mbox\backslash quad\; s=8.3\backslash frac\backslash approx\; 2000\backslash mbox.$
This formula uses the fact that the weight of the vehicle is .

_{1}, F_{2} ... F_{n} act on the points X_{1}, X_{2} ... X_{''n''} in a rigid body.
The trajectories of X_{''i''}, ''i'' = 1, ..., ''n'' are defined by the movement of the rigid body. This movement is given by the set of rotations 'A''(''t'')and the trajectory d(''t'') of a reference point in the body. Let the coordinates x_{''i''} ''i'' = 1, ..., ''n'' define these points in the moving rigid body's _{''i''} along their trajectories are
:$\backslash mathbf\_i\; =\; \backslash vec\backslash times(\backslash mathbf\_i-\backslash mathbf)\; +\; \backslash dot,$
where ω is the angular velocity vector obtained from the skew symmetric matrix
:$$_{''i''} can be determined by approximating the displacement by so
:$\backslash delta\; W\; =\; \backslash mathbf\_1\backslash cdot\backslash mathbf\_1\backslash delta\; t+\backslash mathbf\_2\backslash cdot\backslash mathbf\_2\backslash delta\; t\; +\; \backslash ldots\; +\; \backslash mathbf\_n\backslash cdot\backslash mathbf\_n\backslash delta\; t$
or
:$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\_i\backslash cdot\; (\backslash vec\backslash times(\backslash mathbf\_i-\backslash mathbf)\; +\; \backslash dot)\backslash delta\; t.$
This formula can be rewritten to obtain
:$\backslash delta\; W\; =\; \backslash left(\backslash sum\_^n\; \backslash mathbf\_i\backslash right)\backslash cdot\backslash dot\backslash delta\; t\; +\; \backslash left(\backslash sum\_^n\; \backslash left(\backslash mathbf\_i-\backslash mathbf\backslash right)\backslash times\backslash mathbf\_i\backslash right)\; \backslash cdot\; \backslash vec\backslash delta\; t\; =\; \backslash left(\backslash mathbf\backslash cdot\backslash dot+\; \backslash mathbf\backslash cdot\; \backslash vec\backslash right)\backslash delta\; t,$
where F and T are the applied at the reference point d of the moving frame ''M'' in the rigid body.

Work–energy principle

{{Authority control Energy (physics) Scalar physical quantities Mechanical engineering Mechanics Force Length

physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...

, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throu ...

and displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...

. 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.
For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). When the force is constant and the angle between the force and the displacement is , then the work done is given by:
:$W\; =\; F\; s\; \backslash cos$
Work is a scalar quantity, so it has only magnitude and no direction. Work transfers energy from one place to another, or one form to another. The SI unit of work is the joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on tha ...

(J), the same unit as for energy.
History

The ancient Greek understanding of physics was limited to thestatics
Statics is the branch of mechanics that is concerned with the analysis of (force and torque, torque, or "moment") acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with their enviro ...

of simple machines (the balance of forces), and did not include dynamics
Dynamics (from Greek language, Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power (disambiguation), power") or dynamic may refer to:
Physics and engineering
* Dynamics (mechanics)
** Aerodynamics, the study o ...

or the concept of work. During the Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in m ...

the dynamics of the ''Mechanical Powers'', as the simple machine
A simple machine is a mechanical device that changes the direction or magnitude of a force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature' ...

s were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (; 15 February 1564 – 8 January 1642) was an Italian astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the ...

in 1600 in ''Le Meccaniche'' (''On Mechanics''), in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain that simple machines do not create energy
In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform Work (thermodynamics), work on the body, or to heat it. En ...

, only transform it.
According to Jammer, the term ''work'' was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis (; 21 May 1792 – 19 September 1843) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of s ...

as "weight ''lifted'' through a height", which is based on the use of early steam engine
from Stott Park Bobbin Mill, Cumbria, England
A steam engine is a heat engine
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energ ...

s to lift buckets of water out of flooded ore mines. According to Rene Dugas, French engineer and historian, it is to "that we owe the term ''work'' in the sense that it is used in mechanics now". Although ''work'' was not formally used until 1826, similar concepts existed before then. In 1759, John Smeaton described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion." Smeaton continues that this quantity can be calculated if "the weight raised is multiplied by the height to which it can be raised in a given time," making this definition remarkably similar to Coriolis'.
Units

The SI unit of work is thejoule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on tha ...

(J), named after the 19th-century English physicist James Prescott Joule
James Prescott Joule (; 24 December 1818 11 October 1889) was an English physicist, mathematician and brewing (beer), brewer, born in Salford, Greater Manchester, Salford, Lancashire. Joule studied the nature of heat, and discovered its relat ...

, which is defined as the work required to exert a force of one newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* Newton (film), ''Newton'' (film), a 2017 Indian fil ...

through a displacement of one metre
The metre (British English, Commonwealth spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek noun , "measure", and ...

.
The dimensionally equivalent newton-metre
The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is a Physical unit, unit of torque (also called ) in the International System of Units, SI system. One newton-metre is equal to the torque resulting from a force of ...

(N⋅m) is sometimes used as the measuring unit for work, but this can be confused with the measurement unit of torque
In physics and mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Dis ...

. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton metres is a torque measurement, or a measurement of work.
Non-SI units of work include the newton-metre, erg
The erg is a unit of energy equal to 10−7joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object ...

, the foot-pound
The foot-pound force (symbol: ft⋅lbf, ft⋅lbf, or ft⋅lb ) is a unit of work or energy
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...

, the foot-poundal
The foot-poundal (symbol: ft-pdl) is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a thea ...

, the kilowatt hour
The kilowatt-hour ( SI symbol: kW⋅h or kW h; commonly written as kWh) is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, ...

, the litre-atmosphere, and the horsepower-hour
A horsepower-hour (symbol: hp⋅h) is an outdated unit of energy, not used in the International System of Units
International is an adjective (also used as a noun) meaning "between nations".
International may also refer to:
Music Albums
* Interna ...

. Due to work having the same physical dimension as heat
In thermodynamics, heat is energy in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work or transfer of matter. The various mechanisms of energy transfer that define heat are stated in the next section ...

, occasionally measurement units typically reserved for heat or energy content, such as therm
The therm (symbol, thm) is a non- SI unit of heat
In thermodynamics, heat is energy in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work or transfer of matter. The various mechanisms of energy tran ...

, BTU and calorie
The calorie is a unit of energy defined as the amount of heat needed to raise the temperature of a quantity of water by one degree.
For historical reasons, two main definitions of calorie are in wide use. The small calorie or gram calorie (usua ...

, are utilized as a measuring unit.
Work and energy

The work done by a constant force of magnitude on a point that moves a displacement in a straight line in the direction of the force is the product :$W\; =\; F\; s\; .$ For example, if a force of 10 newtons () acts along a point that travels 2 metres (), then . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Work is closely related toenergy
In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform Work (thermodynamics), work on the body, or to heat it. En ...

. The work–energy principle states that an increase in the kinetic energy of a rigid body
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force. Thus, if the net work is positive, then the particle’s kinetic energy increases by the amount of the work. If the net work done is negative, then the particle’s kinetic energy decreases by the amount of work.
From Newton's second law
In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon in which a ...

, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy corresponding to the linear velocity and angular velocity
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

of that body,
:$W\; =\; \backslash Delta\; E\_k.$
The work of forces generated by a potential function is known as potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potential ...

and the forces are said to be conservative
Conservatism is a Political philosophy, political and social philosophy promoting traditional social institutions. The central tenets of conservatism may vary in relation to the traditional values or practices of the culture and civilization ...

. Therefore, work on an object that is merely displaced in a conservative force field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, without change in velocity or rotation, is equal to ''minus'' the change of potential energy of the object,
:$W\; =\; -\backslash Delta\; E\_p.$
These formulas show that work is the energy associated with the action of a force, so work subsequently possesses the physical dimensions, and units, of energy.
The work/energy principles discussed here are identical to electric work/energy principles.
Constraint forces

Constraint forces determine the object's displacement in the system, limiting it within a range. For example, in the case of aslope
In mathematics, the slope or gradient of a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a ...

plus gravity, the object is ''stuck to'' the slope and, when attached to a taut string, it cannot move in an outwards direction to make the string any 'tauter'. It eliminates all displacements in that direction, that is, the velocity in the direction of the constraint is limited to 0, so that the constraint forces do not perform work on the system.
For a mechanical system
A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine that uses Power (physics), power to apply Force, forces and control Motion, movemen ...

, constraint forces eliminate movement in directions that characterize the constraint. Thus the virtual work done by the forces of constraint is zero, a result which is only true if friction forces are excluded.
Fixed, frictionless constraint forces do not perform work on the system, as the angle between the motion and the constraint forces is always . Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping.
For example, in a pulley system like the Atwood machine
The Atwood machine (or Atwood's machine) was invented in 1784 by the English mathematician George Atwood as a laboratory experiment to verify the Newton's laws of motion, mechanical laws of motion with constant acceleration. Atwood's machine is a ...

, the internal forces on the rope and at the supporting pulley do no work on the system. Therefore, work need only be computed for the gravitational forces acting on the bodies. Another example is the centripetal force
A centripetal force (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power ...

exerted ''inwards'' by a string on a ball in uniform circular motion
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...

''sideways'' constrains the ball to circular motion restricting its movement away from the centre of the circle. This force does zero work because it is perpendicular to the velocity of the ball.
The magnetic force
In physics (specifically in electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The elec ...

on a charged particle is , where ''q'' is the charge, v is the velocity of the particle, and B is the magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the ...

. The result of a cross product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is always perpendicular to both of the original vectors, so . The dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of two perpendicular vectors is always zero, so the work , and the magnetic force does not do work. It can change the direction of motion but never change the speed.
Mathematical calculation

For moving objects, the quantity of work/time (power) is integrated along the trajectory of the point of application of the force. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is thescalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of the force (a vector), and the velocity vector of the point of application. This scalar product of force and velocity is known as instantaneous power
Power typically refers to:
* Power (physics)
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, p ...

. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus
The fundamental theorem of calculus is a theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and c ...

, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), ''Physics'', Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527
Work is the result of a force on a point that follows a curve X, with a velocity v, at each instant. The small amount of work ''δW'' that occurs over an instant of time ''dt'' is calculated as
:$\backslash delta\; W\; =\; \backslash mathbf\backslash cdot\; d\backslash mathbf\; =\; \backslash mathbf\backslash cdot\backslash mathbfdt$
where the is the power over the instant ''dt''. The sum of these small amounts of work over the trajectory of the point yields the work,
:$W\; =\; \backslash int\_^\backslash mathbf\; \backslash cdot\; \backslash mathbfdt\; =\; \backslash int\_^\backslash mathbf\; \backslash cdot\; dt\; =\backslash int\_C\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf,$
where ''C'' is the trajectory from x(''t''dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

, where ''θ'' is the angle between the force vector and the direction of movement, that is
:$W\; =\; \backslash int\_C\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; Fs\backslash cos\backslash theta.$
When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force
In classical mechanics, a central force on an object is a force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science t ...

), no work is done, since the cosine of 90° is zero. Thus, no work can be performed by gravity on a planet with a circular orbit (this is ideal, as all orbits are slightly elliptical). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge.
Work done by a variable force

Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application pointvelocity
The velocity of an object is the rate of change of its position with respect to a frame of reference
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (ph ...

is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of force can be described by the scalar quantity called ''scalar tangential component'' (, where is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows:
:''Work of a force is the line integral of its scalar tangential component along the path of its application point.''
:If the force varies (e.g. compressing a spring) we need to use calculus to find the work done. If the force is given by (a function of ) then the work done by the force along the x-axis from to is:
:$W\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf$
Torque and rotation

A force couple results from equal and opposite forces, acting on two different points of a rigid body. The sum (resultant) of these forces may cancel, but their effect on the body is the couple or torque T. The work of the torque is calculated as :$dW\; =\; \backslash mathbf\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; dt,$ where the is the power over the instant ''δt''. The sum of these small amounts of work over the trajectory of the rigid body yields the work, :$W\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; dt.$ This integral is computed along the trajectory of the rigid body with an angular velocity ''ω'' that varies with time, and is therefore said to be ''path dependent''. If the angular velocity vector maintains a constant direction, then it takes the form, :$\backslash vec=\; \backslash dot\backslash mathbf,$ where φ is the angle of rotation about the constant unit vector S. In this case, the work of the torque becomes, :$W\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; dt\; =\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash frac\; dt\; =\; \backslash int\_C\backslash mathbf\backslash cdot\; \backslash mathbf\; \backslash ,\; d\backslash phi,$ where ''C'' is the trajectory from ''φ''(''t''Work and potential energy

The scalar product of a force F and the velocity v of its point of application defines thepower
Power typically refers to:
* Power (physics)
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, p ...

input to a system at an instant of time. Integration of this power over the trajectory of the point of application, , defines the work input to the system by the force.
Path dependence

Therefore, thework
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
* Work (physics), the product of ...

done by a force F on an object that travels along a curve ''C'' is given by the line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:
:$W\; =\; \backslash int\_C\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; \backslash int\_^\backslash mathbf\backslash cdot\; \backslash mathbfdt,$
where ''dx''(''t'') defines the trajectory ''C'' and v is the velocity along this trajectory.
In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent.
The time derivative of the integral for work yields the instantaneous power,
:$\backslash frac\; =\; P(t)\; =\; \backslash mathbf\backslash cdot\; \backslash mathbf\; .$
Path independence

If the work for an applied force is independent of the path, then the work done by the force, by thegradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...

, defines a potential function which is evaluated at the start and end of the trajectory of the point of application. This means that there is a potential function ''U''(x), that can be evaluated at the two points x(''t''potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potential ...

associated with the applied force. The force derived from such a potential function is said to be conservative
Conservatism is a Political philosophy, political and social philosophy promoting traditional social institutions. The central tenets of conservatism may vary in relation to the traditional values or practices of the culture and civilization ...

. Examples of forces that have potential energies are gravity and spring forces.
In this case, the gradient
In vector calculus, the gradient of a scalar-valued function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the Vec ...

of work yields
:$\backslash nabla\; W\; =\; -\backslash nabla\; U=\; -\backslash left(\backslash frac,\; \backslash frac,\; \backslash frac\backslash right)\; =\; \backslash mathbf,$
and the force F is said to be "derivable from a potential."
Because the potential ''U'' defines a force F at every point x in space, the set of forces is called a force field. The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity V of the body, that is
:$P(t)\; =\; -\backslash nabla\; U\; \backslash cdot\; \backslash mathbf\; =\; \backslash mathbf\backslash cdot\backslash mathbf.$
Work by gravity

In the absence of other forces, gravity results in a constant downward acceleration of every freely moving object. Near Earth's surface the acceleration due to gravity is and the gravitational force on an object of mass ''m'' is . It is convenient to imagine this gravitational force concentrated at thecenter of mass
In physics, the center of mass of a distribution of mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or with something, whether as an attribute or as a component of ...

of the object.
If an object with weight ''mg'' is displaced upwards or downwards a vertical distance , the work ''W'' done on the object is:
:$W\; =\; F\_g\; (y\_2\; -\; y\_1)\; =\; F\_g\backslash Delta\; y\; =\; mg\backslash Delta\; y$
where ''FWork by gravity in space

The force of gravity exerted by a mass ''M'' on another mass ''m'' is given by :$\backslash mathbf=-\backslash frac\backslash mathbf,$ where r is the position vector from ''M'' to ''m''. Let the mass ''m'' move at the velocity v; then the work of gravity on this mass as it moves from position r(''t''gravitational potential energy
Gravitational energy or gravitational potential energy is the potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or othe ...

. The negative sign follows the convention that work is gained from a loss of potential energy.
Work by a spring

Consider a spring that exerts a horizontal force that is proportional to its deflection in the ''x'' direction independent of how a body moves. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain :$W=\backslash int\_0^t\backslash mathbf\backslash cdot\backslash mathbfdt\; =-\backslash int\_0^tkx\; v\_x\; dt\; =\; -\backslash frackx^2.$ For convenience, consider contact with the spring occurs at , then the integral of the product of the distance ''x'' and the x-velocity, ''xv''Work by a gas

:$W=\backslash int\_a^bdV$ Where ''P'' is pressure, ''V'' is volume, and ''a'' and ''b'' are initial and final volumes.Work–energy principle

The principle of work andkinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion.
It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...

(also known as the work–energy principle) states that ''the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle.'' That is, the work ''W'' done by the resultant force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

on a particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or chemical , chemical properties ...

equals the change in the particle's kinetic energy $E\_k$,
:$W=\backslash Delta\; E\_k=\backslash tfrac\; 1\; 2\; mv\_2^2\; -\; \backslash tfrac\; 1\; 2\; mv\_1^2\; ,$
where $v\_1$ and $v\_2$ are the speed
In everyday use and in kinematics, the speed (commonly referred to as v) of an object is the magnitude (mathematics), magnitude of the change of its Position (vector), position; it is thus a Scalar (physics), scalar quantity. The average speed ...

s of the particle before and after the work is done, and ''m'' is its mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change of velocity) when a net force is applied. An object's mass ...

.
The derivation of the ''work–energy principle'' begins with Newton’s second law of motion and the resultant force on a particle. Computation of the scalar product of the forces with the velocity of the particle evaluates the instantaneous power added to the system.
Constraints define the direction of movement of the particle by ensuring there is no component of velocity in the direction of the constraint force. This also means the constraint forces do not add to the instantaneous power. The time integral of this scalar equation yields work from the instantaneous power, and kinetic energy from the scalar product of velocity and acceleration. The fact that the work–energy principle eliminates the constraint forces underlies Lagrangian mechanics
Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia

.
This section focuses on the work–energy principle as it applies to particle dynamics. In more general systems work can change the potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potential ...

of a mechanical device, the thermal energy in a thermal system, or the electrical energy
Electrical energy is energy derived from electric potential energy or kinetic energy. When used loosely, ''electrical energy'' refers to energy that has been converted ''from'' electric potential energy. This energy is supplied by the combination ...

in an electrical device. Work transfers energy from one place to another or one form to another.
Derivation for a particle moving along a straight line

In the case theresultant force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

F is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration ''a'' along a straight line. The relation between the net force and the acceleration is given by the equation ''F'' = ''ma'' (Newton's second law
In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon in which a ...

), and the particle displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...

''s'' can be expressed by the equation
:$s\; =\; \backslash frac$
which follows from $v\_2^2\; =\; v\_1^2\; +\; 2as$ (see Equations of motion
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

).
The work of the net force is calculated as the product of its magnitude and the particle displacement. Substituting the above equations, one obtains:
:$W\; =\; Fs\; =\; mas\; =\; ma\; \backslash left(\backslash frac\backslash right)\; =\; \backslash frac\; -\; \backslash frac\; =\; \backslash Delta$
Other derivation:
:$W\; =\; Fs\; =\; mas\; =\; m\; \backslash left(\backslash frac\backslash right)\; s=\; \backslash fracmv\_2\; ^2\; -\backslash frac\; mv\_1\; ^2\; =\; \backslash Delta$
In the general case of rectilinear motion, when the net force F is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle:
:$W\; =\; \backslash int\_^\; \backslash mathbf\backslash cdot\; \backslash mathbfdt\; =\; \backslash int\_^\; F\; \backslash ,v\; \backslash ,\; dt\; =\; \backslash int\_^\; ma\; \backslash ,v\; \backslash ,\; dt\; =\; m\; \backslash int\_^\; v\; \backslash ,\backslash frac\backslash ,dt\; =\; m\; \backslash int\_^\; v\backslash ,dv\; =\; \backslash tfrac12\; m\; \backslash left(v\_2^2\; -\; v\_1^2\backslash right)\; .$
General derivation of the work–energy theorem for a particle

For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. Some authors call this result ''work–energy principle'', but it is more widely known as the work–energy theorem: :$W\; =\; \backslash int\_^\; \backslash mathbf\backslash cdot\; \backslash mathbfdt\; =\; m\; \backslash int\_^\; \backslash mathbf\; \backslash cdot\; \backslash mathbfdt\; =\; \backslash frac\; \backslash int\_^\; \backslash frac\backslash ,dt\; =\; \backslash frac\; \backslash int\_^\; d\; v^2\; =\; \backslash frac\; -\; \backslash frac\; =\; \backslash Delta$ The identity $\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac\; \backslash frac$ requires some algebra. From the identity $v^2\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$ and definition $\backslash mathbf\; =\; \backslash frac$ it follows :$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash mathbf\; +\; \backslash mathbf\; \backslash cdot\; \backslash frac\; =\; 2\; \backslash frac\; \backslash cdot\; \backslash mathbf\; =\; 2\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; .$ The remaining part of the above derivation is just simple calculus, same as in the preceding rectilinear case.Derivation for a particle in constrained movement

In particle dynamics, a formula equating work applied to a system to its change in kinetic energy is obtained as a first integral ofNewton's second law of motion
In classical mechanics, Newton's laws of motion are three Scientific law, laws that describe the relationship between the motion of an object and the forces acting on it. The first law states that an object either remains at rest or continues ...

. It is useful to notice that the resultant force used in Newton's laws can be separated into forces that are applied to the particle and forces imposed by constraints on the movement of the particle. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle.
To see this, consider a particle P that follows the trajectory X(''t'') with a force F acting on it. Isolate the particle from its environment to expose constraint forces R, then Newton's Law takes the form
:$\backslash mathbf\; +\; \backslash mathbf\; =\; m\; \backslash ddot,$
where ''m'' is the mass of the particle.
Vector formulation

Note that n dots above a vector indicates its nthtime derivative
A time derivative is a derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

.
The scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of each side of Newton's law with the velocity vector yields
:$\backslash mathbf\backslash cdot\backslash dot\; =\; m\backslash ddot\backslash cdot\backslash dot,$
because the constraint forces are perpendicular to the particle velocity. Integrate this equation along its trajectory from the point X(''t''product rule
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero ...

for derivation). Now it is integrated explicitly to obtain the change in kinetic energy,
:$\backslash Delta\; K\; =\; m\backslash int\_^\backslash ddot\backslash cdot\backslash dotdt\; =\; \backslash frac\backslash int\_^\backslash frac\; (\backslash dot\; \backslash cdot\; \backslash dot)\; dt\; =\; \backslash frac\; \backslash dot\backslash cdot\; \backslash dot(t\_2)\; -\; \backslash frac\; \backslash dot\backslash cdot\; \backslash dot\; (t\_1)\; =\; \backslash fracm\; \backslash Delta\; \backslash mathbf^2\; ,$
where the kinetic energy of the particle is defined by the scalar quantity,
:$K\; =\; \backslash frac\; \backslash dot\; \backslash cdot\; \backslash dot\; =\backslash frac\; m$
Tangential and normal components

It is useful to resolve the velocity and acceleration vectors into tangential and normal components along the trajectory X(''t''), such that :$\backslash dot=v\; \backslash mathbf\backslash quad\backslash mbox\backslash quad\; \backslash ddot=\backslash dot\backslash mathbf\; +\; v^2\backslash kappa\; \backslash mathbf,$ where :$v=,\; \backslash dot,\; =\backslash sqrt.$ Then, thescalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of velocity with acceleration in Newton's second law takes the form
:$\backslash Delta\; K\; =\; m\backslash int\_^\backslash dotv\; \backslash ,\; dt\; =\; \backslash frac\; \backslash int\_^\; \backslash fracv^2\; \backslash ,\; dt\; =\; \backslash frac\; v^2(t\_2)\; -\; \backslash frac\; v^2(t\_1),$
where the kinetic energy of the particle is defined by the scalar quantity,
:$K\; =\; \backslash frac\; v^2\; =\; \backslash frac\; \backslash dot\; \backslash cdot\; \backslash dot.$
The result is the work–energy principle for particle dynamics,
:$W\; =\; \backslash Delta\; K.$
This derivation can be generalized to arbitrary rigid body systems.
Moving in a straight line (skid to a stop)

Consider the case of a vehicle moving along a straight horizontal trajectory under the action of a driving force and gravity that sum to F. The constraint forces between the vehicle and the road define R, and we have :$\backslash mathbf\; +\; \backslash mathbf\; =m\backslash ddot.$ For convenience let the trajectory be along the X-axis, so and the velocity is , then , and , where ''F''Coasting down a mountain road (gravity racing)

Consider the case of a vehicle that starts at rest and coasts down a mountain road, the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity ''V'', of say 60 mph (88 fps). Rolling resistance and air drag will slow the vehicle down so the actual distance will be greater than if these forces are neglected. Let the trajectory of the vehicle following the road be X(''t'') which is a curve in three-dimensional space. The force acting on the vehicle that pushes it down the road is the constant force of gravity , while the force of the road on the vehicle is the constraint force R. Newton's second law yields, :$\backslash mathbf\; +\; \backslash mathbf\; =\; m\; \backslash ddot.$ TheWork of forces acting on a rigid body

The work of forces acting at various points on a single rigid body can be calculated from the work of a . To see this, let the forces Freference frame
In physics, a frame of reference (or reference frame) consists of an abstract coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, wi ...

''M'', so that the trajectories traced in the fixed frame ''F'' are given by
:$\backslash mathbf\_i(t)=;\; href="/html/ALL/s/(t).html"\; ;"title="(t)">(t)$
The velocity of the points XOmega
Omega ( capital: Ω, lowercase: ω; Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the 24th and final letter in the Greek alphabet. In the Greek numeric system/ Isopsephy ( Gematria), it has a value of 800. The word literally ...

= \dotA^\mathrm,
known as the angular velocity matrix.
The small amount of work by the forces over the small displacements ''δ''rReferences

Bibliography

* *External links

Work–energy principle

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