In more complex problems the motion passes through several states (A, B, C, D…), each with different conditions: friction present or absent, spring compressed or relaxed, body in flight or in contact with the surface. A single balance written between the start and the end of the motion would risk mixing together situations too different from each other, in which elastic-potential-energy terms, friction contributions, and changes in height appear or disappear. The winning strategy is to break the path into homogeneous stretches.
The central idea is to apply conservation of energy state by state: the final values of one state become the initial values of the next. Each transition between two consecutive states is a small conservation problem in its own right, with its own forces and its own active terms; the link between the pieces is guaranteed by the fact that the body leaves one transition with the same speed, the same height and the same energy with which it enters the next. So conservation of energy is not applied just once, but propagates link by link along the whole chain.
Concretely this means that the energy table changes shape at every transition, because the terms in play change at every transition. The passage from a state in which the spring is acting to one in which it is relaxed makes elastic potential energy appear or disappear; entering a stretch with friction adds the thermal-energy column, which from that point on can only grow; a stretch of free flight conserves only kinetic and gravitational potential energy. Keeping a separate table for each transition, and faithfully carrying over the final values of one as the initial values of the next, is what makes it possible to tackle otherwise unmanageable problems without losing control of the overall energy balance.
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Topics: Lavoro ed energia Concepts: Conservazione dell’energia meccanica · Energia termica da attrito Skills: Uso della tabella energetica · Conservazione dell’energia Methods: Tabella energetica a stati
Related exercises: Esercizio svolto — blocco su piano inclinato con molla e attrito · Problema — Proiettile verticale con resistenza · Problema — Conservazione dell’energia con attrito