- Mechanical work
In

physics ,**mechanical work**is the amount ofenergy transferred by aforce . Like energy, it is a scalar quantity, with SI units ofjoules . The term "work" was first coined in the 1830s by the French mathematicianGaspard-Gustave Coriolis . [*cite book | last = Jammer | first = Max | title = Concepts of Force | publisher = Dover Publications, Inc. | year = 1957 | id = ISBN 0-486-40689-X*]According to the "work-energy theorem" if an external force acts upon an object, causing its

kinetic energy to change from "E_{k1}" to "E_{k2}", then the mechanical work ("W") is given by: [*Tipler (1991), page 138.*]:$W\; =\; Delta\; E\_k\; =\; E\_\{k2\}\; -\; E\_\{k1\}\; =\; frac\{1\}\{2\}m\; Delta\; (v^2)\; ,!$

where "m" is the

mass of the object and "v" is the object'sspeed .The mechanical work applied to an object can be calculated from the scalar multiplication of the applied

force (F) and thedisplacement (d) of the object. This is given by:: $W\; =\; mathbf\{F\}cdot\; mathbf\{d\}=Fd\; cos\; heta\; ,!$

**Introduction**Work can be zero even when there is a force. The

centripetal force incircular motion , for example, does zero work because thekinetic energy of the moving object doesn't change. Likewise when a book sits on a table, the table does no work on the book despite exerting a force equivalent to $mg$ upwards, because no energy is transferred into or out of the book.Heat conduction is not considered to be a form of work, since the energy gets transferred into atomic vibrations rather than a macroscopic displacement.

**Units**The SI unit of work is the

joule (J), which is defined as the work done by a force of onenewton acting over a distance of onemeter . 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, through a gravitational height, out of flooded ore mines. The dimensionally equivalentnewton-meter (N·m) is sometimes used instead; however, it is also sometimes reserved fortorque to distinguish its units from work or energy.Non-SI units of work include the

erg , thefoot-pound , thefoot-poundal , and theliter-atmosphere .**Mathematical calculation****Force and displacement**Force and displacement are both vector quantities and they are combined using the

dot product to evaluate the mechanical work, a scalar quantity:: $W\; =\; old\{F\}\; cdot\; old\{d\}\; =\; F\; d\; cosphi$ (1)

where $phi$ is the angle between the force and the displacement vector.

In order for this formula to be valid, the force and angle must remain constant. The object's path must always remain on a single, straight line, though it may change directions while moving along the line.

In situations where the force changes over

time , or the path deviates from a straight line, equation (1) is not generally applicable although 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.The general definition of mechanical work is given by the following

line integral :: $W\_C\; :=\; int\_\{C\}\; old\{F\}\; cdot\; mathrm\{d\}old\{s\}$ (2)

where::"C" is the path or

curve traversed by the object;:**F**is theforce vector; :**s**is theposition vector .The expression "δW"=

**F**·d**s**is aninexact differential which means that the calculation of "W_{C}" is path-dependent and cannot be differentiated to give**F**·d**s**.Equation (2) explains how a non-zero 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. This is what happens during 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.The possibility of a nonzero force doing zero work illustrates the difference between work and a related quantity,

impulse , which is the integral of force over time. Impulse measures change in a body'smomentum , 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.**Mechanical energy**The "mechanical energy" of a body is that part of its total

energy which is subject to change by mechanical work. It includeskinetic energy andpotential energy . Some notable forms of energy that it does not include arethermal energy (which can be increased byfriction al work, but not easily decreased) andrest energy (which is constant as long as therest mass remains the same).If an external force "F" acts upon a body, causing its

kinetic energy to change from "E_{k1}" to "E_{k2}", then: [*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*] :$W\; =\; Delta\; E\_k\; =\; E\_\{k2\}\; -\; E\_\{k1\}\; =\; Delta\; E\_k\; =\; frac\{1\}\{2\}\; mv\_2\; ^2\; -\; frac\{1\}\{2\}\; mv\_1\; ^2\; =\; frac\{1\}\{2\}\; m\; Delta\; (v^2)$Thus we have derived the result, that the mechanical work done by an external force acting upon a body is proportional to the difference in the squares of the speeds. (It should be observed that the last term in the equation above is $Delta\; v^2$ rather than $(Delta\; v)^2$.)

The principle of "conservation of mechanical energy" states that, if a system is subject only to

conservative force s (e.g. only to agravitational force ), or if the sum of the work of all the other forces is zero, 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.: $(E\_k\; +\; E\_p)\_1\; =\; (E\_k\; +\; E\_p)\_2\; ,!$where

* $E\_k$ is thekinetic energy , and

* $E\_p$ is thepotential 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.**Frame of reference**The work done by a force acting on an object depends on the

inertial frame of reference , because the distance covered while applying the force does. Due to there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.**References****Bibliography***cite book | author=Serway, Raymond A.; Jewett, John W. | title=Physics for Scientists and Engineers | edition=6th ed. | publisher=Brooks/Cole | year=2004 | id=ISBN 0-534-40842-7

*cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Mechanics| edition=3rd ed., extended version | publisher=W. H. Freeman | year=1991 | id=ISBN 0-87901-432-6**External links*** [

*http://www.lightandmatter.com/html_books/2cl/ch03/ch03.html Work*] - a chapter from an online textbook

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