Summary - Work

Work is a measure of the transfer of mechanical energy, i.e., the transfer of energy by a force rather than by heat conduction. When the force is constant, work can usually be calculated as

W = F_II|d| , [only if the force is constant]

where d is simply a less cumbersome notation for Δr, the vector from the initial position to the final position. Thus,

• A force in the same direction as the motion does positive work, i.e., transfers energy into the object on which it acts.

• A force in the opposite direction compared to the motion does negative work, i.e., transfers energy out of the object on which it acts.

• When there is no motion, no mechanical work is done. The human body burns calories when it exerts a force without moving, but this is an internal energy transfer of energy within the body, and thus does not fall within the scientific definition of work.

• A force perpendicular to the motion does no work.

When the force is not constant, the above equation should be generalized as the area under the graph of F_II versus d.

Machines such as pulleys, levers, and gears may increase or decrease a force, but they can never increase or decrease the amount of work done. That would violate conservation of energy unless the machine had some source of stored energy or some way to accept and store up energy.

There are some situations in which the equation W = Fk |d| is ambiguous or not true, and these issues are discussed rigorously in section 3.6. However, problems can usually be avoided by analyzing the types of energy being transferred before plunging into the math. In any case there is no substitute for a physical understanding of the processes involved.

The techniques developed for calculating work can also be applied to the calculation of potential energy. We fix some position as a reference position, and calculate the potential energy for some other position, x, as

PE_x = -W_ref→x .

The following two equations for potential energy have broader significance than might be suspected based on the limited situations in which they were derived:

[potential energy of a spring having spring constant k, when stretched or compressed from the equilibrium position xo; analogous equations apply for the twisting, bending, compression, or stretching of any object.]

[gravitational potential energy of objects of masses M and m, separated by a distance r; an analogous equation applies to the electrical potential energy of an electron in an atom.]