2: Applying Models to Mechanical Phenomena
- Page ID
- 104089
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this Chapter we continue to work with the Energy-Interaction Model. We add all kinds of mechanical interactions to the thermal interactions we treated in Chapter 1. (Note: the term “mechanical” as in the phase “mechanical interactions” is typically used to imply everything other than thermal). Since the Energy-Interaction Model literally applies to every kind of interaction that scientists have ever encountered, we will be just scratching the surface of the realm of applications of this powerful model. We will, however, devote some attention to one area of application that occurs frequently in many phenomena–all kinds of things vibrate, from atoms and molecules to bridges and skyscrapers; that is, they move back and forth or oscillate in very predictable ways.
- 2.1: Where Are We Headed?
- In this second chapter we continue to work with the Energy-Interaction Model. We add all kinds of mechanical interactions to the thermal interactions we treated in Chapter 1. The term “mechanical” is typically used to imply everything other than thermal. We introduce the Intro Spring-Mass Oscillator Model in this chapter as an application of the Energy-Interaction Model. The Spring-Mass Oscillator Model will also play an important role in Chapter 3 for the particle model of matter.
- 2.2: Force
- In Chapter 1, we focused on energy transfers due to heat; here, we explore transfers due to work, requiring an understanding of force. Extending the Energy-Interaction Model to mechanical interactions, we define force as a vector with magnitude and direction. We distinguish between contact forces and long-range forces, like gravity. Understanding balanced versus unbalanced forces helps analyze how forces cause or prevent motion, setting up a foundation for studying work and energy change.
- 2.3: Work
- We now understand work as energy transfer between systems through force applied over a distance, considering only the force component parallel to displacement. Work, measured in joules, increases kinetic energy when force aligns with motion, as seen when pushing a box. Work also transfers energy to potential forms, such as gravitational potential energy in lifting objects or spring potential energy when stretching a spring. These examples illustrate how work drives energy changes in systems.
- 2.4: Mechanical Energy
- Mechanical energy relates to an object’s speed and position. We apply the Energy-Interaction Model to systems with changing speed and position, focusing on kinetic energy (mass and speed) and gravitational potential energy (height). Examples, like lifting and throwing, show energy transfers in mechanical systems. We also examine cases where mechanical energy converts to thermal energy through friction, highlighting interactions between mechanical and thermal energies.
- 2.5: Spring-Mass Oscillator
- We explore the spring-mass oscillator, introducing the restoring force and potential energy. The restoring force, following Hooke's Law, is proportional to displacement and returns the mass to equilibrium. Potential energy depends on displacement squared, increasing as the spring stretches or compresses. We explain that vertical spring-mass systems account for gravity if equilibrium is defined with the mass attached, making vertical and horizontal setups behave similarly.
- 2.6: Plotting Energies
- We explore energy conservation by plotting kinetic and potential energies. For a ball thrown upward, kinetic energy decreases as gravitational potential energy increases, keeping total energy constant. In oscillating spring-mass systems, kinetic and potential energies vary with displacement, but total energy remains constant. Time-averaged kinetic and potential energies are equal in systems with parabolic energy curves, showing oscillations’ universal nature.
- 2.7: Force and Potential Energy
- We explore the relationship between force and potential energy, where force is the slope of a potential energy curve. In a spring-mass system, the restoring force follows Hooke’s Law and points toward decreasing potential energy. More generally, force equals the negative slope of the potential energy graph, with steeper slopes indicating stronger forces. In multiple dimensions, force aligns with the steepest slope, or gradient, of the potential energy surface.
- 2.8: Looking Back and Ahead
- We’ve covered thermal, bond, kinetic, gravitational, and elastic energies, fitting into two categories: potential and kinetic. The Energy-Interaction Model treats thermal and mechanical interactions equally, even with friction. Microscopically, energy can convert between forms, though thermal energy has limits. Next, we’ll use particle models to understand matter and connect to thermodynamics. The model doesn’t cover interaction details, which Newton's Laws address.