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# work done

 Definition/Summary Work is a measure of change of energy. The net work done equals the change in Kinetic Energy (this is the Work-Energy Theorem). Work is the integral of the scalar product (dot-product) of two vectors: Force and Displacement. "Displacement" means the change in position of the point at which the force is applied. So Work is a scalar (an ordinary number), with dimensions of mass times distance-squared over time-squared. The SI unit is the amount of Work done by a Force of one newton acting over a displacement of one metre, and is called the joule (J), or newton-metre (N-m). The SI unit of Power, which is the rate of Work done, is one joule per second, and is called the watt (W).

 Equations Work is the integral of the dot product of force and displacement. $$W\,=\,\int_{\mathbf{a}}^{\mathbf{a}+\mathbf{d}} \mathbf{F} \cdot d\mathbf{r}$$ For a constant force, work is the dot product of the force with the total displacement. $$W\,=\,\mathbf{F}\cdot\mathbf{d}\;.$$ The above work equals the magnitude of the force times the magnitude of the displacement times the cosine of the angle between the force and displacement: $$W\,=\,Fd\cos(\theta)\;.$$ In a uniform gravitational field, Work done by gravity on a body moving along any path C starting at a height $h_1$ and ending at a height $h_2$ is: $$W\,=\,\int_{(x,y,h_1)}^{(x',y',h_2)} (-mg\hat z)\cdot d\mathbf{r}\,=\,mg(h_1 - h_2)$$ In an inverse-square gravitational field, Work done by gravity on a body moving along any path C starting at a height $r_1$ and ending at a height $r_2$ is: $$W\,=\,\int_{r_1}^{r_2} -GmM\frac{\mathbf{r}}{r^3}\cdot d\mathbf{r}\,=\,GmM\left(\frac{1}{r_2} - \frac{1}{r_1}\right)$$ which, if $r_1$ is very close to $r_2$, is approximately the same as the previous formula, with $$g\,=\,\frac{GM}{r_1^2}$$

 Scientists Sadi Carnot (1796-1832) James Joule (1818-1889)

 Breakdown Physics > Classical Mechanics >> Newtonian Dynamics

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 Extended explanation Conservative Force: If the work done after a total displacement of zero is zero (so the change in Kinetic Energy is zero), then the force is said to be conservative (for example, friction is not conservative, because a body moving in a full circle under friction loses Kinetic Energy, but gravity is conservative). For a completely non-conservative force, work equals loss of mechanical energy (the energy lost generally becomes radiation or heat). Potential energy: For a conservative force, work done depends only on position and not on the path taken. Potential energy is another name for work done by a conservative force. Potential energy depends only on position, and is the work done relative to some arbitrarily-chosen position (the position of zero potential energy, chosen so as to make calculations easy). For example, in a uniform gravitational field of strength g, when a mass m is moved by any path through a height h, the work done is mgh. Gearing: A machine has gearing G if the force out is G times the force in: $F_1\,=\,G F_0$ If no energy is lost, then the work out equals the work in. Since work equals force times displacement (strictly, the inner product of force and displacement), that means that the displacement of the point of application of the force out is 1/G times the displacement of the point of application of the force in: $$d_1\,=\,\frac{d_0}{G}$$ Conversely, if a system has $d_1\,\neq\,d_0$, then $$F_1\,=\,\frac{d_0}{d_1} F_0$$ For example, the gearing of a lever is the ratio of the lengths of its two "lever arms". So a lever, or a pulley system, in which the displacement out is less than the displacement in, can lift a heavy object with a force less than its weight. Derivation of Work-Energy Theorem: $$\Delta\,W\ =\ \int d\mathbf{x} \cdot \mathbf{F}_{\rm net}\ =\ \int (\mathbf{v}\,dt) \cdot \left( \frac{d(m\mathbf{v})}{dt} \right)\ =\ \int d \left( \frac{1}{2} m\mathbf{v}^2 \right) \ =\ \Delta\,KE$$ Relativistic version: $$\Delta\,W\ =\ \int d\mathbf{x} \cdot \mathbf{F}_{\rm net}\ =\ \int (\mathbf{v}\,dt)\cdot\left(\frac{d(m\mathbf{v}/\sqrt{1 - \mathbf{v}^2/c^2})}{dt}\right)$$ $$=\ \int d\left(mc^2\sqrt{1 - \mathbf{v}^2/c^2}\right)\ \ +\ \int d\left(\frac{m\mathbf{v}^2}{\sqrt{1 - \mathbf{v}^2/c^2}}\right)\ =\ \int d\left(\frac{mc^2}{\sqrt{1 - \mathbf{v}^2/c^2}}\right)\ =\ \Delta\,E$$

Commentary

 Aterioluwa @ 08:05 AM Nov17-13 I'm so confused guys. the work i know of is just force * distance

 Govind.A.S @ 06:54 AM Sep5-12 Why do you say "Potential energy depends only on position,"?. What about "configuration"? or elastic potential energy of a spring? Microscopically, it depends on positin of atoms, but when we say "position of a spring" we mean entire spring. A compressed spring in gravitational field has more potential energy than a relaxed on at the same "position".

 christpinus @ 07:16 AM Aug4-12 I AGREE WITH THE INVERSE-SQUARE GRAVITATION FIELD EQUATION BUT WHAT IF AN OBJECT IS MOVING FROM INFINITY TO A POSITION ABOVE THE EARTH'S SURFACE SAY 'P'? THANKS ~edit(tiny-tim): put r1 = ∞, so 1/r1 = 0

 @ 12:44 PM Aug12-11 force comes with change in inertia????????

 @ 09:54 AM Mar22-11 by KARAMAWORLD explanation suppost be with examples

 MechaMZ @ 02:34 AM Sep30-09 the "conservative force" link under "see also" at the side bar is broken ~EDIT(tiny-tim): fixed: thanks, MechaMZ!

 tiny-tim @ 05:44 PM Mar20-09 Added potential energy is another name for work done by a conservative force.

 tiny-tim @ 02:31 AM Dec2-08 Sorry, but I don't see how "net force" can do work, since in the general case the displacements (of the points of application) are different, so there's no such thing as "net displacement" to multiply the "net force" by. Reference to "net force" removed.

 robphy @ 10:37 PM Dec1-08 Added "net work" and "Work done by the net force" to the sentence for the Work-Energy theorem in definition/summary section.

 tiny-tim @ 08:23 AM Oct17-08 Added relativistic version and its derivation.

 tiny-tim @ 12:18 PM Oct13-08 Changed to "Work-Energy Theorem", since that's what everyone on PF seems to call it.

 olgranpappy @ 08:36 PM Oct12-08 changed "derivation" in extexpl to "derivation of work kinetic energy theorem".

 olgranpappy @ 08:32 PM Oct12-08 corrected inverse square-law force in Equation to actually be inverse *square* law. corrected misstatement regarding size of "heights" at end of Equation section (don't both have to be "small" rather one must be near the other). Corrected definition of Force and following Equations.

 tiny-tim @ 03:44 AM Oct12-08 Added derivation, based on suggestion by olgranpappy .

 tiny-tim @ 05:34 AM Jul24-08 Ping! Had the bright idea of changing the title from "work (mechanics)" to "work done". "work (mechanics)" was getting no autolinking at all, but "work done" will get plenty, with virtually no false positives.