Mechanical Energy and Non-Conservative Work

Keys to Applicability

Can be applied to any system for which the change in mechanical energy can be attributed to work done by [non conservative forces] (as opposed to processes like heat transfer, radiative losses, etc.). The non-conservative forces can be external forces exerted on the system or internal forces resulting from the interactions between the elements inside the system. It is specially useful for systems where the non-conservative work is zero, in which case the mechanical energy of the system is constant.

Description

Assumed Knowledge

Prior Models

• Point Particle Dynamics

Vocabulary

• system
• internal force
• external force
• conservative force
• [non-conservative]
• kinetic energy
• [gravitational potential energy]
• [elastic potential energy]
• mechanical energy

Model Specification

System Structure

Constituents:  One or more point particles or rigid bodies. Technically, the system must be defined in such a way as to contain all objects that participate in any non-negligible conservative interactions that are present.

Interactions:  All forces that do [non-conservative] work on the system must be considered, including internal forces that perform such work. Conservative forces that are present should have their interaction represented by a potential energy rather than by work.

Descriptors

[Object Variables]:  None.

[State Variables]:  Mass (m^j) and possibly moment of inertia (I^j) for each object plus linear (v^j) and possibly rotational (ω^j) speeds for each object, or alternatively, the kinetic energy (K^j) may be specified directly. When a conservative interaction is present, some sort of position or separation is required for each object (h^j for near-earth gravity, r_jk for universal gravity, x_jk for an elastic interaction, etc.) unless the potential eenrgy (U^jk) is specified directly. Alternately, in place of separate kinetic and potential energies, the mechanical energy of the system (E) can be specified directly.

[Interaction Variables]:  Relevant non-conservative forces (F^NC_k) plus information about the objects' vector displacements (s^j) or the work done by the non-conservative forces (W^NC_k).

Model Equations

Relationships Among State Variables

Mathematical Statement of the Model

where W^NC_i,f is the work done by the all the non-conservative forces on the system between the initial state defined by E_i and the final state defined by E_f and is given by