Once the bubble diagram has been drawn with all its arrows, the calculation becomes almost mechanical. There’s a single balance rule that governs every single bubble, and from it follow, as special cases, all the theorems usually met separately.
Law — Energy balance for a bubble
The change in energy between the final and initial state of any bubble equals the algebraic sum of the energy transfers (arrow values), taken with a if the arrow enters, with a if it exits:
The power of this formulation lies in its universality: a single rule, applied bubble by bubble, replaces the entire arsenal of specific theorems. Just look at the arrows touching a bubble, add up the incoming ones and subtract the outgoing ones, to get the change in that type of energy.
This single rule includes as special cases:
- The kinetic-energy theorem: (algebraic sum of the works of the forces acting on the body).
- The potential-energy theorem: for the corresponding force.
The kinetic-energy theorem is simply the balance applied to the bubble: its change is the sum of the works of all the forces acting on the body. The potential-energy theorem is the balance applied to an bubble: the minus sign reflects the fact that, when the force does positive work (arrow leaving ), the potential reservoir empties out.
Warning — How to reason with unknown works
If a problem has works of unknown value, assume they are positive (arrow entering ) and solve the balance equations. If a work turns out negative at the end, it means the actual direction of the arrow was the opposite one: simply reverse it in the final diagram.
Links
Topics: Lavoro ed energia Concepts: Conservazione dell’energia meccanica · Lavoro di una forza Skills: Bilancio energetico a bolle
Related exercises: Problema — Lavoro della ciclista in salita · Esercizio svolto — mela e molla · Problema — Vero o falso su lavoro ed energia