  We analyze a category of problems that is of interest in many physical
situations, including those encountered in introductory physics classes:
systems with two well-delineated parts that exchange energy, eventually
reaching a shared equilibrium with a loss of mechanical or electrical energy.
Such systems can be constrained by a constant of the system (e.g., mass,
charge, momentum, or angular momentum) that uniquely determines the mechanical
or electrical energy of the equilibrium state, regardless of the dissipation
mechanism. A representative example would be a perfectly inelastic collision
between two objects in one dimension, for which momentum conservation requires
that some of the initial kinetic energy is dissipated by conversion to thermal
or other forms as the two objects reach a common final velocity. We discuss how
this feature manifests in a suite of four well-known and disparate problems
that all share a common mathematical formalism. These examples, in which the
energy dissipated during the process can be difficult to solve directly from
dissipation rates, can be approached by students in a first-year physics class
by considering conservation laws and can therefore be useful for teaching about
energy transformations and conserved quantities. We then illustrate how to
extend this method by applying it to a final example.
