Work by Variable Force — Definition

Definition

Imagine pushing a box across a smooth floor. If you push with a steady, unchanging strength and in a straight line, that's a constant force. Calculating the work done is straightforward: just multiply your pushing force by the distance the box moved in the direction of your push.

This is the simple formula $W = F \times d$. However, what if your push isn't steady? What if you push harder at the beginning, then ease up, or perhaps push from different angles as the box moves? This is where the concept of 'Work by Variable Force' comes into play.

A variable force is one whose magnitude, direction, or both, change as the object moves from one point to another. Think about stretching a spring: the more you stretch it, the harder it pulls back. The force required to stretch it isn't constant; it increases with the extension.

Another example is the gravitational force between two celestial bodies; it changes with the distance between them. In such scenarios, we cannot simply multiply the force by the total displacement because the force itself is not a single value throughout the motion.

To tackle this, we use a powerful mathematical tool called integration. The idea is to break down the object's entire path into tiny, infinitesimally small segments. Over each tiny segment, the force can be considered almost constant.

For each tiny segment, say $dvec{r}$, the work done is also tiny, $dW$. We calculate this tiny work as the dot product of the force vector $vec{F}$ at that point and the tiny displacement vector $dvec{r}$, so $dW = vec{F} cdot dvec{r}$.

The dot product ensures we only consider the component of the force that is parallel to the displacement. Once we have calculated the work done for all these tiny segments, we 'sum' them up to find the total work.

This 'summing up' process for infinitesimally small quantities is precisely what integration does. So, the total work done by a variable force from an initial position $vec{r_1}$ to a final position $vec{r_2}$ is given by the integral: $W = \int_{\vec{r_1}}^{\vec{r_2}} \vec{F} \cdot d\vec{r}$.

Graphically, if you plot the force (or the component of force parallel to displacement) against the displacement, the work done by a variable force is represented by the area under the force-displacement curve.

This visual representation is incredibly useful for understanding and solving problems, especially in one dimension. Understanding work by variable force is crucial for many areas of physics, including understanding springs, gravitational potential energy, and electric potential energy, all of which involve forces that change with position.