Work

The physics term for the amount of energy required to move an object over a given path subject to a given force.

Formulas

For an object moving distance \(d\) with constant force \(F\) acting in the direction of motion,

\[{\rm{Work}} = Fd\]

If the force is a scalar that is not constant, and the motion runs from position \(x = a\) to \(x = b\) on the number line, then

\[\eqalign{{\rm{Work}} &= \int_a^b {f\left( x \right)dx} \\x &= {\rm{position}}\\f\left( x \right) &= {\rm{force \,at \,position\; }}x}\]

If the force \({{\bf{F}}}\) is a vector function and the object moves along curve \(C\), then

\[\eqalign{{\rm{Work}} &= \int\limits_C {{\bf{F}}\left( {{\bf{x}}} \right) \cdot {\bf{dx}}} \\{\bf{x}} &= {\rm{position \,vector}}\\{\bf{F}}\left( {{\bf{x}}} \right) &= {\rm{force \,vector \,at \,position \;}}{\bf{x}}}\]