第7章仕事と運動エネルギー

In physics, work is done on an object when energy is transferred to the object. In other words, work is done when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. We first define the increment of work dW done by a force $\overrightarrow{F}$ acting through an infinitesimal displacement $d\overrightarrow{r}$ as the dot product of these two vectors:

Then, we can add up the contributions for infinitesimal displacements, along a path between two positions, to get the total work.

The vectors involved in the definition of the work done by a force acting on a particle are illustrated in Figure 7.2. While in general, Equation 7.2 requires mathematics beyond the scope of this text, in many simple situations this integral becomes a familiar integral in one variable. We will examine several such examples and restrict our discussion to these cases.

We choose to express the dot product in terms of the magnitudes of the vectors and the cosine of the angle between them, because the meaning of the dot product for work can be put into words more directly in terms of magnitudes and angles. We could equally well have expressed the dot product in terms of the various components introduced in Vectors. In two dimensions, these were the x- and y-components in Cartesian coordinates, or the r- and $\phi$-components in polar coordinates; in three dimensions, it was just x-, y-, and z-components. Which choice is more convenient depends on the situation. In words, you can express Equation 7.1 for the work done by a force acting over a displacement as a product of one component acting parallel to the other component. From the properties of vectors, it doesn’t matter if you take the component of the force parallel to the displacement or the component of the displacement parallel to the force—you get the same result either way.

Recall that the magnitude of a force times the cosine of the angle the force makes with a given direction is the component of the force in the given direction. The components of a vector can be positive, negative, or zero, depending on whether the angle between the vector and the component-direction is between $0\text{°}$ and $90\text{°}$ or $90\text{°}$ and $180\text{°}$, or is equal to $90\text{°}$. As a result, the work done by a force can be positive, negative, or zero, depending on whether the force is generally in the direction of the displacement, generally opposite to the displacement, or perpendicular to the displacement. The maximum work is done by a given force when it is along the direction of the displacement ($\text{cos}\theta =\pm 1$), and zero work is done when the force is perpendicular to the displacement ($\text{cos}\theta =0$).

The units of work are units of force multiplied by units of length, which in the SI system is newtons times meters, $\text{N}·\text{m.}$ This combination is called a joule, for historical reasons that we will mention later, and is abbreviated as J. In the English system, still used in the United States, the unit of force is the pound (lb) and the unit of distance is the foot (ft), so the unit of work is the foot-pound $(\text{ft}·\text{lb})\text{.}$