mechanical energy

The sum of the kinetic energy and any potential energies of a system.

Motivation for Concept

It is a fundamental postulate of physics that energy is not created or destroyed. The total energy of the universe is fixed, but it can be transformed from one type of energy to another. In order to make wide use of this principle, many types of energy must be considered (e.g. nuclear, chemical/electrostatic, gravitational, kinetic, etc.). To give a first illustration of the principle, however, introductory physics introduces three energy types that are related to the topics of mechanics: kinetic energy, [gravitational potential energy], and [elastic potential energy]. Collectively, these three types of energy are classified as mechanical energy because of their role in the mechanics of macroscopic bodies.

Mathematical Definition of Mechanical Energy

The mechanical energy (E) of a system is the sum of the system's kinetic and potential energies:

In introductory mechanics, it is basically assumed that the possible constituents of the potential energy are [gravitational potential energy] (U_g)and [elastic potential energy] (U_e), so that the mechanical energy is essentially:

Generalized Work-Energy Theorem

The Work-Kinetic Energy Theorem states:

where W_net is the total work from all sources done on a point particle system.

We can now generalize this theorem slightly to allow for a system composed of rigid bodies that have only mechanical interactions (they do not emit radiation, transfer heat, etc.). We allow for the system to have conservative interactions, but we remove these interactions from the net work and instead account for them using potential energy. Essentially, we move the contribution of conservative forces from the right hand side of the Theorem to the left hand side. The result is:

so that the right hand side is the sum of the works arising from all forces that do not have an associated potential energy.

Conditions for Mechanical Energy Conservation

General Condition

From the generalized Work-Energy Theorem, we see that the mechanical energy will be constant (assuming only mechanical interactions) when the net non-conservative work done on the system is zero. Since [gravity] and [spring forces] are the only conservative forces commonly treated in introductory mechanics, this condition usually amounts to the constraint that the total work done by forces other than gravity or spring forces is zero.

Examples: Non-Conservative Forces Absent

One clear way to ensure that the work done by forces other than gravity and springs is zero is to design a system that exeperiences no other forces. Thus, for example, any system which involves pure freefall, or freefall after launch from a spring, or freefall onto a spring, etc, will clearly conserve mechanical energy (see figures below).

Examples: Movement Along Frictionless Surface

When a system is sliding along a (non-accelerating) surface, it is possible to include a normal force (in addition to springs and gravity) on the system without changing the mechanical energy. The reason is that an object moving along a surface will always be moving in a direction perpendicular to the normal force from the surface. Thus, the dot product of the normal force with the path will always be zero and the normal force will contribute zero work. Thus, assuming frictionless surfaces, systems like those sketched below will be compatible with mechanical energy conservation.

Diagrams and Mechanical Energy

Initial-State Final-State Diagrams

Because the Work-Energy Theorem and the principle Law of Change for the [Mechanical Energy and Non-Conservative Work] model involve only the initial and final energies of the system, it is useful to devote considerable attention to understanding the system's configuration at those times. It is customary to sketch the system in its initial and final configurations, labeling the quantities that are relevant for the kinetic and potential energies of the system.

Example – Vertical Launch

As an example of an initial-state final-state diagram, suppose we were asked to determine the speed of a 0.25 kg block launched vertically from rest by a spring of spring constant 500 N/m that was initially compressed 0.10 m from its natural length when the block reaches a height of 0.50 m above the natural position of the spring. We can neatly summarize these givens through a two-panel diagram: