Difference between revisions of "Mechanical work" - New World Encyclopedia

Latest revision as of 03:50, 9 November 2022

The baseball pitcher does work on the ball by transferring energy into it.

In physics, mechanical work is the amount of energy transferred by a force. Like energy, it is a scalar quantity, with SI units of joules. Heat conduction is not considered to be a form of work, since there is no macroscopically measurable force, only microscopic forces occurring in atomic collisions. In the 1830s, French mathematician Gaspard-Gustave Coriolis coined the term work for the product of force and distance.^[1]

Positive and negative signs of work indicate whether the object exerting the force is transferring energy to some other object, or receiving it. A baseball pitcher, for example, does positive work on the ball, but the catcher does negative work on it. Work can be zero even when there is a force. The centripetal force in uniform circular motion, for example, does zero work because the kinetic energy of the moving object doesn't change. Likewise, when a book sits on a table, the table does no work on the book, because no energy is transferred into or out of the book.

Calculation

When the force is constant and along the same line as the motion, the work can be calculated by multiplying the force by the distance, $W=Fd$ (letting both F and d have positive or negative signs, according to the coordinate system chosen). When the force does not lie along the same line as the motion, this can be generalized to the scalar product of force and displacement vectors.

In the simplest case, that of a body moving in a steady direction, and acted on by a constant force parallel to that direction, the work is given by these formulas:

W=FD\,\!

(1)

W=(1/2)mv_{2}^{2}-(1/2)mv_{1}^{2}

(derived simply from the above equation)

where

F is the portion of the force acting in the same direction as the motion, and

D is the distance traveled by the object. Note that distance is a scalar quantity and so, too, is work.

m is the mass of the object

v₂ is the final velocity

v₁ is the initial velocity

The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector quantities, and combined using the dot product:

W={\vec {F}}\cdot {\vec {D}}=|{\vec {F}}||{\vec {D}}|\cos \phi

(2)

where $\phi \,$ is the angle between the force and the displacement vector. This formula holds true even when the object changes its direction of travel throughout the motion.

In situations in which the force changes over time, and/or the direction of motion changes over time, equation (1) is not directly applicable. However, under mild restrictions, it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps. This is formalized by the following line integral, which can be taken as a rather general definition of work:

W_{C}:=\int _{C}{\vec {F}}\cdot \mathrm {d} {\vec {s}}

(3)

where:

C is the path or curve traversed by the object

{\vec {F}}

is the force vector

{\vec {s}}

is the position vector

It must be emphasized that $W_{C}\,$ is explicitly a function of the path $C\!$ . In general the work $W_{C}\,$ depends on the total displacement of the path $C\!$ (The longer the path traveled by the object, the more work will be done).

Using vector notation, equation (3) readily explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero (as is the case in circular motion).

The possibility of a nonzero force doing zero work exemplifies the difference between work and a related quantity, impulse, which is the integral of force over time. Impulse measures change in a body's momentum, a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity. For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.

Units

The SI unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one meter. This definition is based on Sadi Carnot's 1824 definition of work as "weight lifted through a height," which is based on the fact that early steam engines were principally used to lift buckets of water, though a gravitational height, out of flooded ore mines. The dimensionally equivalent newton-meter (N•m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy.

Non-SI units of work include the erg, the foot-pound, the foot-poundal, and the liter-atmosphere.

Types of work

Forms of work that are not evidently mechanical in fact represent special cases of this principle. For instance, in the case of "electrical work," an electric field does work on charged particles as they move through a medium.

One mechanism of heat conduction is collisions between fast-moving atoms in a warm body with slow-moving atoms in a cold body. Although colliding atoms do work on each other, the force averages to nearly zero in bulk, so conduction is not considered to be mechanical work.

Work due to volume change

Work is done when the volume of a fluid changes. Work in such circumstances is represented by the following equation:

W_{C}=-\int _{C}P\,\mathrm {d} V

(4)

where:

W = work done on the system
P = external pressure
V = volume

Like all work functions, PV work is dependent on the path $C\!$ . PV work is often measured in the (non-SI) units of liter-atmospheres, where 1 L•atm = 101.3 J.

Mechanical energy

The mechanical energy of a body is that part of its total energy which is subject to change by mechanical work. It includes kinetic energy and potential energy. Some notable forms of energy that it does not include are thermal energy (which can be increased by frictional work, but not easily decreased) and rest energy (which is constant as long as the rest mass remains the same).

The relation between work and kinetic energy

If an external work W acts upon a body, causing its kinetic energy to change from E_k1 to E_k2, then:

W=\Delta E_{k}=E_{k2}-E_{k1}\,\!

Also, if we substitute the equation for kinetic energy that states $E_{k}=(1/2)mv^{2}$ , we then get:

W=\Delta ((1/2)mv^{2})=(1/2)mv_{2}^{2}-(1/2)mv_{1}^{2}

^[2]

Conservation of mechanical energy

The principle of conservation of mechanical energy states that, if a system is subject only to conservative forces (such as a gravitational force), its total mechanical energy remains constant.

For instance, if an object with constant mass is in free fall, the total energy of position 1 will equal that of position 2.

(K_{E}+P_{E})_{1}=(K_{E}+P_{E})_{2}\,\!

where

$K_{E}$ is the kinetic energy, and
$P_{E}$ is the potential energy.

The external work will usually be done by the friction force between the system on the motion or the internal, non-conservative force in the system, or loss of energy due to heat.

Notes

↑ Jammer, Max. Concepts of Force. Cambridge, MA: Harvard University Press, 1957. ISBN 0-486-40689-X
↑ Zitzewitz, Paul et al. Physics: Principles and Problems. New York: Glencoe/McGraw-Hill, 2005. ISBN 0-07-845813-7.

References
ISBN links support NWE through referral fees

Young, Hugh D, and Freedman, Roger A. Physics for Scientists and Engineers. 11th ed. San Fransisco, CA: Pearson, 2003. ISBN 080538684X
Parker, Sybil P. McGraw-Hill encyclopedia of physics. New York: McGraw-Hill, 1983. ISBN 0070452539
Serway, Raymond A. Physics for scientists and engineers with modern physics. Fort Worth, Tex: Saunders College, 2004. ISBN 0534409490
Tipler, Paul Allen. Physics for scientists and engineers. New York: W.H. Freeman/Worth Publishers, 1999. ISBN 9781572598140

External links

All links retrieved November 8, 2022.

Work hyperphysics. Georgia State University.

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[1] Jammer, Max. Concepts of Force. Cambridge, MA: Harvard University Press, 1957. ISBN 0-486-40689-X

[2] Zitzewitz, Paul et al. Physics: Principles and Problems. New York: Glencoe/McGraw-Hill, 2005. ISBN 0-07-845813-7.

[1]

[2]

@@ Line 1: / Line 1: @@
-In [[physics]], '''mechanical work''' is the amount of [[energy]] transferred by a [[force]]. Like energy, it is a [[scalar (physics)|scalar]] quantity, with [[SI]] units of [[joules]]. Heat conduction is not considered to be a form of work, since there is no macroscopically measurable force, only microscopic forces occurring in atomic collisions.  In the 1830s, the French mathematician [[Gaspard-Gustave Coriolis]] coined the term '''work''' for the product of force and distance.<ref>{{cite book | last = Jammer | first = Max  | title = Concepts of Force | publisher = Dover Publications, Inc. | year = 1957 | id = ISBN 0-486-40689-X}}</ref>
+{{Edboard}}{{Copyedited}}{{Images OK}}{{Submitted}}{{Approved}}{{Paid}}
+[[Image:Baseball pitching motion 2004.jpg|right|thumb|500px|The baseball pitcher does work on the ball by transferring energy into it.]]
+In [[physics]], '''mechanical work''' is the amount of [[energy]] transferred by a [[force]]. Like energy, it is a [[scalar (physics)|scalar]] quantity, with [[SI]] units of [[joules]]. Heat conduction is not considered to be a form of work, since there is no macroscopically measurable force, only microscopic forces occurring in atomic collisions. In the 1830s, [[France|French]] [[mathematics|mathematician]] [[Gaspard-Gustave Coriolis]] coined the term ''work'' for the product of force and distance.<ref>Jammer, Max. ''Concepts of Force''. Cambridge, MA: Harvard University Press, 1957. ISBN 0-486-40689-X</ref>
+{{toc}}
+Positive and negative signs of work indicate whether the object exerting the force is transferring energy to some other object, or receiving it. A baseball pitcher, for example, does positive work on the ball, but the catcher does negative work on it. Work can be zero even when there is a force. The [[centripetal force]] in uniform [[circular motion]], for example, does zero work because the kinetic energy of the moving object doesn't change. Likewise, when a book sits on a table, the table does no work on the book, because no energy is transferred into or out of the book.
-Positive and negative signs of work indicate whether the object exerting the force is transferring energy to some other object, or receiving it.  A baseball pitcher, for example, does positive work on the ball, but the catcher does negative work on it. Work can be zero even when there is a force.  The [[centripetal force]] in uniform [[circular motion]], for example, does zero work because the kinetic energy of the moving object doesn't change. Likewise, when a book sits on a table, the table does no work on the book, because no energy is transferred into or out of the book.
+== Calculation ==
-When the force is constant and along the same line as the motion, the work can be calculated by multiplying the force by the distance, <math>W=Fd</math> (letting both F and d have positive or negative signs, according to the coordinate system chosen). When the force does not lie along the same line as the motion, this can be generalized to the [[scalar product]] of [[force (physics)|force]] and displacement vectors.
+When the [[force]] is constant and along the same line as the motion, the work can be calculated by multiplying the force by the [[distance]], <math>W=Fd</math> (letting both F and d have positive or negative signs, according to the coordinate system chosen). When the force does not lie along the same line as the motion, this can be generalized to the [[scalar product]] of [[force]] and displacement vectors.
-[[Image:Baseball pitching motion 2004.jpg|right|thumb|500px|The baseball pitcher does work on the ball by transferring energy into it.]]
-== Calculation ==
+In the simplest case, that of a body moving in a steady direction, and acted on by a constant force parallel to that direction, the work is given by these formulas:
-In the simplest case, that of a body moving in a steady direction, and acted on by a constant force parallel to that direction, the work is given by these formulas
-: <math>W = F D  \,\!</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (1)
+: <math>W = F D \,\!</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ''(1)''
-: <math>W = (m v^2)/2  \,\!</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (derived simply from the above equation)
+: <math>W = (1/2) mv_2 ^2 - (1/2) mv_1 ^2</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (derived simply from the above equation)
 where
 : ''F'' is the portion of the force acting in the same direction as the motion, and
 : ''D'' is the distance traveled by the object. Note that distance is a scalar quantity and so, too, is work.
+: ''m'' is the [[mass]] of the object
+: ''v<sub>2</sub>'' is the final [[velocity]]
+: ''v<sub>1</sub>'' is the initial velocity
-The work is taken to be negative when the force opposes the motion.  More generally, the force and distance are taken to be [[vector (spatial)|vector]] quantities, and combined using the [[dot product]]:
+The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be [[vector (spatial)|vector]] quantities, and combined using the [[dot product]]:
-: <math>W = \vec{F} \cdot \vec{D} = |\vec{F}| |\vec{D}| \cos\phi  </math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (2)
+: <math>W = \vec{F} \cdot \vec{D} = |\vec{F}| |\vec{D}| \cos\phi </math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ''(2)
+''
 where <math>\phi \,</math> is the angle between the force and the displacement vector. This formula holds true even when the object changes its direction of travel throughout the motion.
-In situations in which the force changes over time, and/or the direction of motion changes over time, equation (1) is not directly applicable.  However, under mild restrictions, it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps.  This is formalized by the following [[line integral]], which can be taken as a rather general definition of work:
+In situations in which the force changes over time, and/or the direction of motion changes over time, equation (1) is not directly applicable. However, under mild restrictions, it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps. This is formalized by the following [[line integral]], which can be taken as a rather general definition of work:
-: <math>W_C := \int_{C} \vec{F} \cdot \mathrm{d}\vec{s} </math>  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)
+: <math>W_C := \int_{C} \vec{F} \cdot \mathrm{d}\vec{s} </math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ''(3)''
 where:
-:''C'' is the path or [[curve]] traversed by the object;
+:''C'' is the path or [[curve]] traversed by the object
-:<math> \vec F</math> is the [[force]] vector;
+:<math> \vec F</math> is the [[force]] vector
-:<math>\vec s</math> is the [[position vector]].
+:<math>\vec s</math> is the [[position vector]]
-It must be emphasized that <math>W_C\,</math> is explicitly a function of the path <math>C\!</math>.  If work were a ''potential'' it would depend only on the endpoints of the path, but this is not the case; in general the work <math>W_C\,</math> depends on every detail of the path <math>C\!</math>.  As a related matter, it is not proper to write <math>\mathrm{d}W = \vec{F} \cdot \mathrm{d}\vec{s}</math> nor <math>\mathrm{d}W=\!</math> anything (except perhaps in trivial cases, which we exclude from further consideration).  That's because d(anything) is guaranteed to be an exact differential, whereas the correct expression <math>\vec{F} \cdot \mathrm{d}\vec{s}</math> is an [[inexact differential]].  It is fairly common to see "<math>\mathrm{d}W\,</math>" used as shorthand for <math>\vec{F} \cdot \mathrm{d}\vec{s}</math>, but this must be considered highly informal and mathematically unjustifiable.  Certainly there is no function "<math>W\,</math>" that can be differentiated to give <math>\vec{F} \cdot \mathrm{d}\vec{s}</math>.
+It must be emphasized that <math>W_C\,</math> is explicitly a function of the path <math>C\!</math>. In general the work <math>W_C\,</math> depends on the total displacement of the path <math>C\!</math> (The longer the path traveled by the object, the more work will be done).
-Equation (3) readily explains how a nonzero force can do zero work.  The simplest case is where the force is always perpendicular to the direction of motion, making the [[integrand]] always zero (viz. circular motion).  However, even if the integrand sometimes takes nonzero values, it can still integrate to zero if it is sometimes negative and sometimes positive.
+Using [[vector]] notation, equation (3) readily explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the [[integrand]] always zero (as is the case in circular motion).
-The possibility of a nonzero force doing zero work exemplifies the difference between work and a related quantity: [[impulse]] (the integral of force over time).  Impulse measures change in a body's [[momentum]], a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity.  For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.
+The possibility of a nonzero force doing zero work exemplifies the difference between work and a related quantity, [[impulse]], which is the integral of force over [[time]]. Impulse measures change in a body's [[momentum]], a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity. For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.
 == Units ==
-{{main|work (thermodynamics)}}
-The SI unit of work is the [[joule]] (J), which is defined as the work done by a force of one [[newton]] acting over a distance of one [[meter]].  This definition is based on [[Nicolas Léonard Sadi Carnot|Sadi Carnot]]'s 1824 definition of work as "weight ''lifted'' through a height", which is based on the fact that early steam engines were principally used to lift buckets of water, though a gravitational height, out of flooded ore mines.  The dimensionally equivalent [[newton-meter]] (N·m) is sometimes used instead; however, it is also sometimes reserved for [[torque]] to distinguish its units from work or energy.
+The SI unit of work is the [[joule]] (J), which is defined as the work done by a force of one [[newton]] acting over a distance of one [[meter]]. This definition is based on [[Nicolas Léonard Sadi Carnot|Sadi Carnot]]'s 1824 definition of work as "weight ''lifted'' through a height," which is based on the fact that early steam engines were principally used to lift buckets of [[water]], though a gravitational height, out of flooded ore mines. The dimensionally equivalent [[newton-meter]] (N•m) is sometimes used instead; however, it is also sometimes reserved for [[torque]] to distinguish its units from work or energy.
 Non-SI units of work include the [[erg]], the [[foot-pound]], the [[foot-poundal]], and the [[liter-atmosphere]].
 ==Types of work==
-Forms of work that are not evidently mechanical in fact represent special cases of this principle.  For instance, in the case of "electrical work", an [[electric field]] does work on [[electric charge|charge]]d particles as they move through a medium.
+Forms of work that are not evidently mechanical in fact represent special cases of this principle. For instance, in the case of "electrical work," an [[electric field]] does work on [[electric charge|charge]]d particles as they move through a medium.
-One mechanism of [[heat conduction]] is collisions between fast-moving [[atom]]s in a warm body with slow-moving atoms in a cold body.  Although colliding atoms do work on each other, the force averages to nearly zero in bulk, so conduction is not considered to be mechanical work.
+One mechanism of [[heat conduction]] is collisions between fast-moving [[atom]]s in a warm body with slow-moving atoms in a cold body. Although colliding atoms do work on each other, the force averages to nearly zero in bulk, so conduction is not considered to be mechanical work.
-=== PV work ===
+=== Work due to volume change===
-[[Chemical thermodynamics]] studies ''PV work'', which occurs when the volume of a fluid changes.  PV work is represented by the following equation:
+Work is done when the volume of a fluid changes. Work in such circumstances is represented by the following equation:
 :<math>W_C =-\int_C P\,\mathrm{d}V</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (4)
@@ Line 58: / Line 63: @@
 * ''V'' = volume
-Like all work functions, PV work is [[Path dependent (mathematics)|dependent]] on the path <math>C\!</math>.  (The path in question is a curve in the [[Euclidean space]] specified by the fluid's [[pressure]] and [[volume]], and infinitely many such curves are possible.)  From a thermodynamic perspective, this fact implies that ''PV'' work is not a [[state function]].  This means that the differential <math>P\,\mathrm{d}V</math> is an [[inexact differential]]. Some prefer to write the 'd' with a line through or use <math>\delta W</math> instead to signal this condition.
+Like all work functions, PV work is [[Path dependent (mathematics)|dependent]] on the path <math>C\!</math>.
+PV work is often measured in the (non-SI) units of liter-atmospheres, where 1 L•atm = 101.3 J.
-From a mathematical point of view, that is to say, <math>\mathrm{d}W</math> is not an [[exact form|exact]] [[one-form]]. The use of a different symbol for the differential warns there is actually no function ([[0-form]]) <math>W</math> which is the [[potential]] of <math>\mathrm{d}W</math>. If there were, indeed, this function <math>W</math>, we should be able to use the [[Stokes Theorem]], and calculate the above integral by just evaluating this putative function, the potential of <math>\mathrm{d}W</math>, at the [[Boundary (topology)|boundary]] of the path, that is, the initial and final points, and therefore the work would be a state function. This impossibility is consistent with the fact that it does not make sense to refer to ''the work on a point''; work presupposes a path.
-PV work is often measured in the (non-SI) units of liter-atmospheres, where 1 L·atm = 101.3 J.
 ==Mechanical energy==
-{{main|Mechanical energy}}
-The ''mechanical energy'' of a body is that part of its total [[energy]] which is subject to change by mechanical work.  It includes [[kinetic energy]] and [[potential energy]].  Some notable forms of energy that it does not include are [[thermal energy]] (which can be increased by [[friction]]al work, but not easily decreased) and [[rest energy]] (which is constant as long as the [[rest mass]] remains the same).
+The ''mechanical energy'' of a body is that part of its total [[energy]] which is subject to change by mechanical work. It includes [[kinetic energy]] and [[potential energy]]. Some notable forms of energy that it does not include are [[thermal energy]] (which can be increased by [[friction]]al work, but not easily decreased) and [[rest energy]] (which is constant as long as the [[rest mass]] remains the same).
 ===The relation between work and kinetic energy===
 If an external work ''W'' acts upon a body, causing its [[kinetic energy]] to change from ''E<sub>k1</sub>'' to ''E<sub>k2</sub>'', then:
 :<math>W = \Delta E_k = E_{k2} - E_{k1}\,\!</math>
 Also, if we substitute the equation for kinetic energy that states <math>E_k = (1/2) mv^2 </math>, we then get:
-:<math>W = \Delta ((1/2)  mv^2) = (1/2)  mv_2 ^2 - (1/2)  mv_1 ^2</math><ref>{{cite book | last = Zitzewitz,Elliott, Haase, Harper, Herzog, Nelson, Nelson, Schuler, Zorn | title = Physics: Principles and Problems | publisher = McGraw-Hill Glencoe, The McGraw-Hill Companies, Inc. | year = 2005 | id = ISBN 0-07-845813-7}}</ref>
+:<math>W = \Delta ((1/2) mv^2) = (1/2) mv_2 ^2 - (1/2) mv_1 ^2</math><ref>Zitzewitz, Paul et al. ''Physics: Principles and Problems''. New York: Glencoe/McGraw-Hill, 2005. ISBN 0-07-845813-7.</ref>
 ===Conservation of mechanical energy===
-The principle of ''conservation of mechanical energy'' states that, if a system is subject only to [[conservative force]]s (e.g. only to a [[gravitational force]]), its total mechanical energy remains constant.
-For instance, if an object with constant mass is in free fall, the total energy of position 1 will equal that of position 2.
+The principle of ''conservation of mechanical energy'' states that, if a system is subject only to [[conservative force]]s (such as a [[gravitational force]]), its total mechanical energy remains constant.
-: <math>(E_k + E_p)_1 = (E_k + E_p)_2 \,\!</math>
+For instance, if an object with constant [[mass]] is in free fall, the total energy of position 1 will equal that of position 2.
+: <math>(K_E + P_E)_1 = (K_E + P_E)_2 \,\!</math>
 where
-* <math>E_k</math> is the [[kinetic energy]], and
+* <math>K_E</math> is the [[kinetic energy]], and
-* <math>E_p</math> is the [[potential energy]].
+* <math>P_E</math> is the [[potential energy]].
-The external work will usually be done by the friction force between the system on the motion or the internal-non conservative force in the system or loss of energy due to heat.
+The external work will usually be done by the friction force between the system on the motion or the internal, non-conservative force in the system, or loss of energy due to [[heat]].
+== See also ==
+* [[Force]]
+* [[Electricity]]
+* [[Energy]]
+* [[Heat]]
+== Notes ==
+<references />
 ==References==
-<references />
+* Young, Hugh D, and Freedman, Roger A. ''Physics for Scientists and Engineers''. 11th ed. San Fransisco, CA: Pearson, 2003. ISBN 080538684X
+* Parker, Sybil P. ''McGraw-Hill encyclopedia of physics.'' New York: McGraw-Hill, 1983. ISBN 0070452539
+* Serway, Raymond A. ''Physics for scientists and engineers with modern physics.'' Fort Worth, Tex: Saunders College, 2004. ISBN 0534409490
+* Tipler, Paul Allen. ''Physics for scientists and engineers.'' New York: W.H. Freeman/Worth Publishers, 1999. ISBN 9781572598140
 == External links ==
-* [http://www.lightandmatter.com/html_books/2cl/ch03/ch03.html Work] - a chapter from an online textbook
+All links retrieved November 8, 2022.
-* [http://physnet.org/home/modules/pdf_modules/m20.pdf ''Work, Power, Kinetic Energy''] on [http://www.physnet.org Project PHYSNET]
+* [http://hyperphysics.phy-astr.gsu.edu/hbase/wcon.html ''Work''] hyperphysics. Georgia State University.
 [[Category:Physical sciences]]