mechanical work

TheInfoList

In
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: :$W = F s \cos$ 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 the
statics 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 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), 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 to
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 ...

. 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, :$W = \Delta E_k.$ 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, :$W = -\Delta E_p.$ 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 a
slope 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 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 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 :$\delta W = \mathbf\cdot d\mathbf = \mathbf\cdot\mathbfdt$ 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, :$W = \int_^\mathbf \cdot \mathbfdt = \int_^\mathbf \cdot dt =\int_C \mathbf \cdot d\mathbf,$ where ''C'' is the trajectory from x(''t''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 :$W = \int_C F\,ds$ 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 :$W = \int_C F\,ds = F\int_C ds = Fs$ 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
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 :$W = \int_C \mathbf \cdot d\mathbf = Fs\cos\theta.$ 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 point
velocity 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: :$W = \int_^ \mathbf \cdot d\mathbf$

## 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 :$dW = \mathbf \cdot \boldsymbol \, dt,$ 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, :$W = \int_^ \mathbf \cdot \boldsymbol \, dt.$ 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, :$\vec= \dot\mathbf,$ where φ is the angle of rotation about the constant unit vector S. In this case, the work of the torque becomes, :$W = \int_^ \mathbf \cdot \boldsymbol \, dt = \int_^ \mathbf \cdot \mathbf \frac dt = \int_C\mathbf\cdot \mathbf \, d\phi,$ where ''C'' is the trajectory from ''φ''(''t''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, :$\mathbf = \tau \mathbf,$ and both the torque and angular velocity are constant, then the work takes the form, :$W = \int_^ \tau \dot dt = \tau\left(\phi_2 - \phi_1\right).$ 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 :$W = F s = F r \phi .$ Introduce the torque , to obtain :$W = F r \phi = \tau \phi ,$ as presented above. Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.

# Work and potential energy

The scalar product of a force F and the velocity v of its point of application defines the
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 ...
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, the
work 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 ...
: :$W = \int_C \mathbf \cdot d\mathbf = \int_^\mathbf\cdot \mathbfdt,$ 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, :$\frac = P\left(t\right) = \mathbf\cdot \mathbf .$

## Path independence

If the work for an applied force is independent of the path, then the work done by the force, by the
gradient 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''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 :$W = \int_C \mathbf \cdot \mathrm\mathbf = \int_^ \mathbf \cdot \mathrm\mathbf = U\left(\mathbf\left(t_1\right)\right)-U\left(\mathbf\left(t_2\right)\right).$ The function ''U''(x) is called 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 ...

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 :$\nabla W = -\nabla U= -\left(\frac, \frac, \frac\right) = \mathbf,$ 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 :$P(t) = -\nabla U \cdot \mathbf = \mathbf\cdot\mathbf.$

## 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 the
center 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: :$W = F_g \left(y_2 - y_1\right) = F_g\Delta y = mg\Delta y$ where ''Fg'' 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.

## Work by gravity in space

The force of gravity exerted by a mass ''M'' on another mass ''m'' is given by :$\mathbf=-\frac\mathbf,$ 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''1) to r(''t''2) is given by :$W=-\int^_\frac\mathbf\cdot d\mathbf=-\int^_\frac\mathbf\cdot\mathbfdt.$ Notice that the position and velocity of the mass ''m'' are given by :$\mathbf = r\mathbf_r, \qquad\mathbf = \frac = \dot\mathbf_r + r\dot\mathbf_t,$ 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 $d \mathbf_r/dt = \dot\mathbf_t.$ Use this to simplify the formula for work of gravity to, :$W=-\int^_\frac\left(r\mathbf_r\right)\cdot\left(\dot\mathbf_r + r\dot\mathbf_t\right)dt = -\int^_\fracr\dotdt = \frac-\frac.$ This calculation uses the fact that :$\fracr^=-r^\dot=-\frac.$ The function :$U=-\frac,$ is the gravitational potential function, also known as
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 :$W=\int_0^t\mathbf\cdot\mathbfdt =-\int_0^tkx v_x dt = -\frackx^2.$ For convenience, consider contact with the spring occurs at , then the integral of the product of the distance ''x'' and the x-velocity, ''xv''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.

## Work by a gas

:$W=\int_a^bdV$ Where ''P'' is pressure, ''V'' is volume, and ''a'' and ''b'' are initial and final volumes.

# Work–energy principle

The principle of work and
kinetic 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 $E_k$, :$W=\Delta E_k=\tfrac 1 2 mv_2^2 - \tfrac 1 2 mv_1^2 ,$ 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 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 ...

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 :$s = \frac$ 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: :$W = Fs = mas = ma \left\left(\frac\right\right) = \frac - \frac = \Delta$ Other derivation: :$W = Fs = mas = m \left\left(\frac\right\right) s= \fracmv_2 ^2 -\frac mv_1 ^2 = \Delta$ 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: :$W = \int_^ \mathbf\cdot \mathbfdt = \int_^ F \,v \, dt = \int_^ ma \,v \, dt = m \int_^ v \,\frac\,dt = m \int_^ v\,dv = \tfrac12 m \left\left(v_2^2 - v_1^2\right\right) .$

## 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: :$W = \int_^ \mathbf\cdot \mathbfdt = m \int_^ \mathbf \cdot \mathbfdt = \frac \int_^ \frac\,dt = \frac \int_^ d v^2 = \frac - \frac = \Delta$ The identity $\mathbf \cdot \mathbf = \frac \frac$ requires some algebra. From the identity $v^2 = \mathbf \cdot \mathbf$ and definition $\mathbf = \frac$ it follows :$\frac = \frac = \frac \cdot \mathbf + \mathbf \cdot \frac = 2 \frac \cdot \mathbf = 2 \mathbf \cdot \mathbf .$ 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 of
Newton'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 :$\mathbf + \mathbf = m \ddot,$ where ''m'' is the mass of the particle.

### Vector formulation

Note that n dots above a vector indicates its nth
time 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 :$\mathbf\cdot\dot = m\ddot\cdot\dot,$ because the constraint forces are perpendicular to the particle velocity. Integrate this equation along its trajectory from the point X(''t''1) to the point X(''t''2) to obtain :$\int_^ \mathbf\cdot\dot dt = m\int_^\ddot\cdot\dotdt.$ 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 :$W = \int_^ \mathbf\cdot\dot dt = \int_^ \mathbf\cdot d\mathbf.$ 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 :$\frac\frac\left(\dot\cdot \dot\right) = \ddot\cdot\dot,$ (see
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, :$\Delta K = m\int_^\ddot\cdot\dotdt = \frac\int_^\frac \left(\dot \cdot \dot\right) dt = \frac \dot\cdot \dot\left(t_2\right) - \frac \dot\cdot \dot \left(t_1\right) = \fracm \Delta \mathbf^2 ,$ where the kinetic energy of the particle is defined by the scalar quantity, :$K = \frac \dot \cdot \dot =\frac m$

### 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 :$\dot=v \mathbf\quad\mbox\quad \ddot=\dot\mathbf + v^2\kappa \mathbf,$ where :$v=, \dot, =\sqrt.$ Then, 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 velocity with acceleration in Newton's second law takes the form :$\Delta K = m\int_^\dotv \, dt = \frac \int_^ \fracv^2 \, dt = \frac v^2\left(t_2\right) - \frac v^2\left(t_1\right),$ where the kinetic energy of the particle is defined by the scalar quantity, :$K = \frac v^2 = \frac \dot \cdot \dot.$ The result is the work–energy principle for particle dynamics, :$W = \Delta K.$ 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 :$\mathbf + \mathbf =m\ddot.$ For convenience let the trajectory be along the X-axis, so and the velocity is , then , and , where ''F''x is the component of F along the X-axis, so :$F_x v = m\dotv.$ Integration of both sides yields :$\int_^F_x v dt = \frac v^2\left(t_2\right) - \frac v^2\left(t_1\right).$ If ''F''x is constant along the trajectory, then the integral of velocity is distance, so :$F_x \left(d\left(t_2\right)-d\left(t_1\right)\right) = \frac v^2\left(t_2\right) - \frac v^2\left(t_1\right).$ 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, :$kWs = \frac v^2,\quad\mbox\quad v = \sqrt.$ Notice that this formula uses the fact that the mass of the vehicle is .

## 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, :$\mathbf + \mathbf = m \ddot.$ 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 this equation with the velocity, , yields :$W v_z = m\dotV,$ 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 :$\int_^W v_z dt = \frac V^2\left(t_2\right) - \frac V^2 \left(t_1\right).$ The weight force ''W'' is constant along the trajectory and the integral of the vertical velocity is the vertical distance, therefore, :$W \Delta z = \fracV^2.$ 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 :$s=\frac= 8.3\frac,\quad\mbox\quad s=8.3\frac\approx 2000\mbox.$ This formula uses the fact that the weight of the vehicle is .

# Work 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 F1, F2 ... Fn act on the points X1, X2 ... 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
reference 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 : The velocity of the points X''i'' along their trajectories are :$\mathbf_i = \vec\times\left(\mathbf_i-\mathbf\right) + \dot,$ where ω is the angular velocity vector obtained from the skew symmetric matrix :
Omega 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''r''i'' can be determined by approximating the displacement by so :$\delta W = \mathbf_1\cdot\mathbf_1\delta t+\mathbf_2\cdot\mathbf_2\delta t + \ldots + \mathbf_n\cdot\mathbf_n\delta t$ or :$\delta W = \sum_^n \mathbf_i\cdot \left(\vec\times\left(\mathbf_i-\mathbf\right) + \dot\right)\delta t.$ This formula can be rewritten to obtain :$\delta W = \left\left(\sum_^n \mathbf_i\right\right)\cdot\dot\delta t + \left\left(\sum_^n \left\left(\mathbf_i-\mathbf\right\right)\times\mathbf_i\right\right) \cdot \vec\delta t = \left\left(\mathbf\cdot\dot+ \mathbf\cdot \vec\right\right)\delta t,$ where F and T are the applied at the reference point d of the moving frame ''M'' in the rigid body.

# Bibliography

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Work–energy principle
{{Authority control Energy (physics) Scalar physical quantities Mechanical engineering Mechanics Force Length