Conservative forces

The key question, once the bubbles are drawn, is: how are they connected to each other? The answer depends on the type of force acting. Every force produces an energy transfer — an arrow in the diagram — with well-defined rules, and it’s the interaction diagram that tells us which arrows to draw. Conservative forces are the first of three families.

A force is called conservative if the work needed to carry a body from A to B doesn’t depend on the path taken, but only on the initial and final positions. Only conservative forces have an associated potential energy, and this is precisely why in the bubble diagram they give rise to an $E_\text{pot}$ bubble.

Principle — Rule for conservative forces

The work of a conservative force connects the $E_\text{cin}$ bubble of the body on which the force acts with the $E_\text{pot}$ bubble of that force.

The sign convention is: the work is positive if the arrow enters the $E_\text{cin}$ bubble, negative if it exits. In every case, the potential energy theorem guarantees that: $\Delta E_\text{pot} = -L$ that is, potential energy increases when the work is negative (the arrow exits $E_\text{cin}$, enters $E_\text{pot}$) and decreases when the work is positive (the arrow exits $E_\text{pot}$, enters $E_\text{cin}$).

Conservative forces: the work exchanges energy between the body’s $E_\text{cin}$ and the interaction’s $E_\text{pot}$. The “arrow” can go either way: positive if it enters $E_\text{cin}$, negative if it exits.

Summary

Conservative forces:

• Weight → $E_\text{pot,g} = mgh$
• Spring → $E_\text{pot,m} = \frac{1}{2}K(\ell-\ell_0)^2$
• Coulomb → $E_\text{pot,el} = k\dfrac{q_1q_2}{r}$
• Universal gravity → $E_\text{pot,G} = -\dfrac{Gm_1m_2}{d}$

Warning — An $E_\text{pot}$ bubble doesn't "belong to" a body

Potential energy is an energy of interaction between two bodies, not “of the body”. You don’t say “the apple’s $E_\text{pot}$”, but “the gravitational $E_\text{pot}$ of the apple–Earth interaction”. In the bubble diagram, however, we draw a single $E_\text{pot}$ bubble for each conservative interaction, not two.

Links

Topics: Lavoro ed energia Concepts: Forze conservative · Energia potenziale gravitazionale Methods: Diagramma a bolle di scambio energetico

Related exercises: Problema — Blocco su piano inclinato con molla (equilibrio e oscillazioni) · Esercizio svolto — mela e molla · Problema — Stima energia cinetica di un’auto