Definition and formula of work

Bodies exchange energy through the forces with which they interact. The transfer of energy via a force has a precise name: it is called work. Work is thus the bridge between the language of forces and that of energy: whenever a force acts along a displacement, energy passes from one side to the other, and the amount of energy transferred is exactly the work done by that force.

Principle — Work of a constant force

$\ev{L = \vec{F} \cdot \Delta\vec{s} = |\vec{F}|\,|\Delta\vec{s}|\,\cos\alpha}$ where $\alpha$ is the angle between the force and the displacement.

The presence of $\cos\alpha$ is the heart of the formula: what matters is not only the magnitude of the force and the length of the displacement, but also their alignment. It is the component of the force along the direction of motion that does work; the perpendicular component transfers no energy. Hence the three possible cases, all governed by the sign of the cosine.

Key formula

$L > 0: \text{the force accelerates the body}$ $L < 0: \text{the force slows the body}$ $L = 0: \text{force} \perp \text{displacement}$

• If the force has the same direction as the motion ($\alpha = 0$, $\cos\alpha = 1$): $L > 0$, the body accelerates and receives energy.
• If the force is opposite to the motion ($\alpha = 180^\circ$, $\cos\alpha = -1$): $L < 0$, the body slows down and gives up energy.
• If the force is perpendicular to the motion ($\alpha = 90^\circ$, $\cos\alpha = 0$): $L = 0$, no energy transfer.

The sign of the work is therefore not an arbitrary convention, but tells the direction of the energy flow: positive when the force gives energy to the body, negative when it takes energy away.

Links

Topics: Lavoro ed energia Concepts: Lavoro di una forza Skills: Impostazione simbolica

Related exercises: Problem — True or false on work and energy · Problem — Power of a lift · Problem — Work from a triangular force graph