Mechanical Energy and Non-Conservative Work
Description and Assumptions
If we ignore non-mechanical processes like heat transfer, radiative losses, etc., then we arrive at a model involving only mechanical energy which changes due to the application (or extraction) of the work done by non-conservative forces 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.
Problem Cues
The model is especially useful for systems where the non-conservative work is zero, in which case the mechanical energy of the system is constant. The most important cue for mechanical energy conservation is the dominance of gravity or spring forces (both conservative forces) in a problem. Since friction is a common source of non-conservative work, another important cue for problems in which mechancial energy is conserved is an explicit statement such as "frictionless surface" or "smooth track".
Learning Objectives
Students will be assumed to understand this model who can:
- Compute the translational kinetic energy of an object.
- Compute the rotational kinetic energy of a rigid body rotating about an axis.
- Apply the constraint of rolling without slipping.
- Define the term non-conservative.
- Calculate the work done by a force acting on a moving object.
- State the Work-Kinetic Energy Theorem.
- Name the conservative forces commonly encountered in mechanics problems.
- Explain why the zero point of the (near-earth) gravitational potential energy is arbitrary.
- Define the variables appearing in the expression for elastic potential energy.
- Calculate the total mechanical energy of a system containing any number of rotating and translating rigid bodies near the surface of the earth that interact via springs.
- Construct intitial-state final-state diagrams to summarize the mechanical energy of a system.
- Describe the conditions under which mechanical energy is conserved.
Model
Compatible Systems
One or more point particles or rigid bodies, plus any conservative interactitons that can be accounted for as potential energies of the system.
In introductory mechanics, the only commonly encountered conservative interactions are gravity and springs.
Relevant Interactions
All non-conservative forces that perform 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 the associated potential energy rather than by the work.
Occasionally it is easier to consider the work of conservative forces directly, omitting their potential energy.
Relevant Definitions
\begin
\begin
& E = \sum_
K + \sum_
U
& K = \frac
mv^
+ \frac
I\omega^
&W = \int_
\vec
\cdot d\vec
\end
\end
The system potential energy is the sum of all the potential energies produced by interactions between system constituents. Even when there are two system constituents involved (for example in a double star) each interaction produces only one potential energy.
Law of Change
\begin
[ E_
= E_
+ \sum_
W ] \end
Diagrammatic Representations
- Initial-state final-state diagram.
- [Energy bar graph].
Relevant Examples
ExamplesInvolvingConstantMechanicalEnergy"> Examples Involving Constant Mechanical Energy
ExamplesInvolvingNon-ConservativeWork"> Examples Involving Non-Conservative Work
ExamplesInvolvingGravitationalPotentialEnergy"> Examples Involving Gravitational Potential Energy
ExamplesInvolvingElastic(Spring)PotentialEnergy"> Examples Involving Elastic (Spring) Potential Energy
ExamplesInvolvingRotationalKineticEnergy"> Examples Involving Rotational Kinetic Energy
AllExamplesUsingthisModel"> All Examples Using this Model
Pictures courtesy:
- Wikimedia Commons
user Boris23 - Wikimedia Commons user Ellywa
- Wikimedia Commons user Evanherk