Potential Energy - Work and Potential Energy

Work and Potential Energy

The work of a force acting on a moving body yields a difference in potential energy when the integration of the work is path independent. The scalar product of a force F and the velocity v of its point of application defines the power input to a system at an instant of time. Integration of this power over the trajectory of the point of application, C=x(t), defines the work input to the system by the force.

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(t1) and x(t2) 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

W = \int_C \bold{F} \cdot \mathrm{d}\bold{x} = \int_{\mathbf{x}(t_1)}^{\mathbf{x}(t_2)} \bold{F} \cdot \mathrm{d}\bold{x} = U(\mathbf{x}(t_1))-U(\mathbf{x}(t_2)).

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.

In this case, the partial derivative of work yields

and the force F is said to be "derivable from a potential."

Because the potential U defines a force F at every point x in space, the set of forces is called a force field. 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 body, that is

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

Read more about this topic:  Potential Energy

Famous quotes containing the words work, potential and/or energy:

    “And now my work is done, which neither the anger of
    Jove, nor fire, nor sword, nor the gnawing tooth of time
    shall ever be able to undo.... Wherever Rome’s power
    extends over the conquered world, I shall have mention
    on men’s lips.”
    —Ovid (Publius Ovidius Naso)

    “The real community of man ... is the community of those who seek the truth, of the potential knowers.”
    —Allan Bloom (1930–1992)

    “Violence among young people ... is an aspect of their desire to create. They don’t know how to use their energy creatively so they do the opposite and destroy.”
    —Anthony Burgess (b. 1917)