Work and the Work-Energy Theorem

The Core Idea

Imagine pushing a heavy box across a rough floor. You apply a force over a certain distance. The "effort" you expend depends not just on how hard you push, but also on how far the box moves in the direction of your push. If you push sideways while the box moves forward, you do no work on its forward motion. Work is the measure of energy transferred to an object by a force acting through a displacement. It is not a new kind of energy, but rather the mechanism by which energy is changed. If the net force on an object does positive work, the object speeds up; if it does negative work, the object slows down. This theorem connects the cause (force acting over distance) to the effect (change in kinetic energy) in a single, elegant equation.

Defining Work

Work is defined as the dot product of the force vector and the displacement vector. For a constant force \(\vec{F}\) acting over a straight-line displacement \(\Delta \vec{r}\), the work \(W\) is given by:

where \(\theta\) is the angle between the force and the displacement. The unit of work is the joule (\(\mathrm{J}\)), equivalent to one newton-metre (\(\mathrm{N \cdot m}\)). Note that work is a scalar quantity; it has magnitude but no direction. If the force is perpendicular to the motion (\(\theta = 90^\circ\)), the work done is zero. This explains why the normal force from a horizontal table does no work on a sliding block, as the force is vertical while the displacement is horizontal.

For a variable force \(\vec{F}(\vec{r})\) acting along a curved path from point \(A\) to point \(B\), we must sum the infinitesimal work done along each small segment \(d\vec{r}\). This leads to the line integral definition:

This integral depends on the path taken if the force is non-conservative, such as friction. For the net force acting along the path the particle actually follows, however, the result is always the change in kinetic energy, as the theorem below shows.

The Work-Energy Theorem

The Work-Energy Theorem states that the net work done on a particle by all forces acting on it equals the change in its kinetic energy. This result is derived directly from Newton's Second Law and the definition of kinetic energy. The theorem is valid for any net force, whether constant or variable, and for any trajectory.

Here, \(K = \frac{1}{2}mv^2\) is the kinetic energy. The theorem implies that if the net work is zero, the speed of the object remains constant, even if its direction changes (as in uniform circular motion, where the centripetal force does no work). Conversely, if the net work is positive, the kinetic energy increases; if negative, it decreases. This scalar equation is often easier to solve than the vector equations of motion, especially when time is not a variable of interest.

Applying the Theorem

To apply the work-energy theorem, follow these steps: First, identify the initial and final states of the system. Second, draw a free-body diagram to identify all forces acting on the object. Third, determine the work done by each force. Forces that are perpendicular to the displacement do zero work. Fourth, sum the work done by all forces to find the net work. Finally, equate the net work to the change in kinetic energy and solve for the unknown quantity, usually the final speed.

Consider a block sliding down a frictionless incline. The forces are gravity and the normal force. The normal force is perpendicular to the displacement, so it does no work. Gravity does positive work. The net work is equal to the work done by gravity. By equating this to the change in kinetic energy, we can find the speed at the bottom of the incline without needing to know the time taken or the acceleration directly. This method is robust and avoids the complexities of vector decomposition in non-Cartesian coordinates.