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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}
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".
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<br /> -<br /> \mathbf{v}^2/c^2})}{dt}\right)
=\ \int d\left(mc^2\sqrt{1<br /> -<br /> \mathbf{v}^2/c^2}\right)\ \ +\ \int d\left(\frac{m\mathbf{v}^2}{\sqrt{1<br /> -<br /> \mathbf{v}^2/c^2}}\right)\ =\ \int d\left(\frac{mc^2}{\sqrt{1<br /> -<br /> \mathbf{v}^2/c^2}}\right)\ =\ \Delta\,E
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
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}
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<br /> -<br /> \mathbf{v}^2/c^2})}{dt}\right)
=\ \int d\left(mc^2\sqrt{1<br /> -<br /> \mathbf{v}^2/c^2}\right)\ \ +\ \int d\left(\frac{m\mathbf{v}^2}{\sqrt{1<br /> -<br /> \mathbf{v}^2/c^2}}\right)\ =\ \int d\left(\frac{mc^2}{\sqrt{1<br /> -<br /> \mathbf{v}^2/c^2}}\right)\ =\ \Delta\,E
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
