Mechanical Energy

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

MotivationforConcept"> Motivation for Concept

MathematicalDefinitionofMechanicalEnergy"> Mathematical Definition of Mechanical Energy

GeneralizedWork-EnergyTheorem"> Generalized Work-Energy Theorem

ConditionsforMechanicalEnergyConservation"> Conditions for Mechanical Energy Conservation

GeneralCondition"> 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 [gravitation] 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-ConservativeForcesAbsent"> 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:MovementAlongFrictionlessSurface"> 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.

DiagramsandMechanicalEnergy"> Diagrams and Mechanical Energy

Initial-StateFinal-StateDiagrams"> 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--VerticalLaunch"> 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:

Initial StateFinal State

The act of drawing the diagram will often clarify the givens, and it will also remind us to choose a zero height for the gravitational potential energy.