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:

    “All too soon these feet must hide
    In the prison cells of pride,
    Lose the freedom of the sod,
    Like a colt’s for work be shod,”
    —John Greenleaf Whittier (1807–1892)

    “It can be fairly argued that the highest priority for mankind is to save itself from extinction. However, it can also be argued that a society that neglects its children and robs them of their human potential can extinguish itself without an external enemy.”
    —Selma Fraiberg (20th century)

    “Perhaps catastrophe is the natural human environment, and even though we spend a good deal of energy trying to get away from it, we are programmed for survival amid catastrophe.”
    —Germaine Greer (b. 1939)