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Introduction to the Model

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.

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.

Relevant Definitions

\begin{large}\[E = K + U\]\end{large}

\begin{large}\[ K = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\]\end{large}

\begin{large}\[W_{fi} = \int_{\rm path} \vec{F}(\vec{s}) \cdot d\vec{s} = \int_{t_{i}}^{t_{f}} \vec{F}(t) \cdot \vec{v}(t)\:dt\]\end{large}

S.I.M. Structure of the 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.

Relevant Interactions

Any external force that performs that perform work on the system must be considered, and also any internal non-conservative forces that perform work. Any internal conservative forces that are present should have their interaction represented by the associated potential energy rather than by the work.

Law of Change

Mathematical Representation

\begin{large}\[ \frac{dE}{dt} = \sum \left(\vec{F}^{\rm ext} + \vec{F}^{\rm NC}\right)\cdot \vec{v} \]\end{large}

\begin{large}\[ E_{f} = E_{i} + \sum W^{\rm ext}_{fi} + \sum W^{\rm NC}_{fi} \] \end{large}

Diagrammatic Representations

• Initial-state final-state diagram.
• Energy bar graph.

Relevant Examples

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id cons

Examples Involving Constant Mechanical Energy

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id noncons

Examples Involving Non-Conservative Work

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id grav

Examples Involving Gravitational Potential Energy

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id elas

Examples Involving Elastic (Spring) Potential Energy

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id rot

Examples Involving Rotational Kinetic Energy

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id all

All Examples Using this Model

Pictures courtesy:

• Wikimedia Commons user Boris23
• Wikimedia Commons user Ellywa
• Wikimedia Commons user Evanherk