Work and potential energy

Potential energy is closely linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field.

If the work for an applied force is independent of the path, then the work done by the force is evaluated at the start and end of the trajectory of the point of application. This means that there is a function U(x), called a "potential," that can be evaluated at the two points x_A and x_B to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is

${\displaystyle W={\int <!--\; \int \; -->}_C\mathbf{F}\cdot <!--\; \cdot \; -->\mathrm{d}\mathbf{x}=U({\mathbf{x}}_A)-<!--\; -\; -->U({\mathbf{x}}_B)}$

where C is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, C, from A to B.

The function U(x) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces.

Derivable from a potential

In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve C takes a special form if the force F is related to a scalar field Φ(x) so that

${\displaystyle \mathbf{F}=\mathrm{\nabla <!--\; \nabla \; -->}\mathrm{\Phi <!--\; \Phi \; -->}=\left(\frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Phi <!--\; \Phi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}x},\frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Phi <!--\; \Phi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}y},\frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Phi <!--\; \Phi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}z}\right).}$

In this case, work along the curve is given by

${\displaystyle W={\int <!--\; \int \; -->}_C\mathbf{F}\cdot <!--\; \cdot \; -->\mathrm{d}\mathbf{x}={\int <!--\; \int \; -->}_C\mathrm{\nabla <!--\; \nabla \; -->}\mathrm{\Phi <!--\; \Phi \; -->}\cdot <!--\; \cdot \; -->\mathrm{d}\mathbf{x},}$

which can be evaluated using the gradient theorem to obtain

${\displaystyle W=\mathrm{\Phi <!--\; \Phi \; -->}({\mathbf{x}}_B)-<!--\; -\; -->\mathrm{\Phi <!--\; \Phi \; -->}({\mathbf{x}}_A).}$

This shows that when forces are derivable from a scalar field, the work of those forces along a curve C is computed by evaluating the scalar field at the start point A and the end point B of the curve. This means the work integral does not depend on the path between A and B and is said to be independent of the path.

Potential energy U=-Φ(x) is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is

${\displaystyle W=U({\mathbf{x}}_A)-<!--\; -\; -->U({\mathbf{x}}_B).}$

In this case, the application of the del operator to the work function yields,

${\displaystyle \mathrm{\nabla <!--\; \nabla \; -->}W=-<!--\; -\; -->\mathrm{\nabla <!--\; \nabla \; -->}U=-<!--\; -\; -->\left(\frac{\mathrm{\partial <!--\; \partial \; -->}U}{\mathrm{\partial <!--\; \partial \; -->}x},\frac{\mathrm{\partial <!--\; \partial \; -->}U}{\mathrm{\partial <!--\; \partial \; -->}y},\frac{\mathrm{\partial <!--\; \partial \; -->}U}{\mathrm{\partial <!--\; \partial \; -->}z}\right)=\mathbf{F},}$

and the force F is said to be "derivable from a potential." This also necessarily implies that F must be a conservative vector field. The potential U defines a force F at every point x in space, so the set of forces is called a force field.

Computing potential energy

Given a force field F(x), evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve γ(t)=r(t) from γ(a)=A to γ(b)=B, and computing,

For the force field F, let v= dr/dt, then the gradient theorem yields,

The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity v of the point of application, that is

${\displaystyle P(t)=-<!--\; -\; -->\mathrm{\nabla <!--\; \nabla \; -->}U\cdot <!--\; \cdot \; -->\mathbf{v}=\mathbf{F}\cdot <!--\; \cdot \; -->\mathbf{v}.}$

Examples of work that can be computed from potential functions are gravity and spring forces.