The Mechanical System

mechanics systems particles continuum
Last updated: 2025-01-18

The Mechanical System

So far we have referred to an object without actually specifying its shape or composition. Understanding what constitutes a “mechanical system” is essential for properly applying free-body analysis and Newton’s laws.

From Particles to Continua

Mechanical systems range from simple to complex:

  1. Single particle
  2. System of particles
  3. Continuum (cloud of particles)

Each level requires different analytical approaches while remaining grounded in the same fundamental principles.

The Particle

The simplest object is a particle, which is treated as:

  • A finite mass
  • Zero dimensions
  • Located at a single point in space

When this idealization works:

  • Object’s dimensions are negligible compared to distances traveled
  • Internal structure is irrelevant to the problem
  • Rotation can be ignored

Example: Planets in the solar system can often be treated as particles because distances between planets are much greater than planetary radii.

System of Particles

Next in complexity is a system of particles that move together in some cohesive way.

The Solar System Example

The solar system may, for many applications, be viewed as a system of particles:

  • Each planet treated as a point mass
  • Sun treated as a point mass
  • Distances between bodies much greater than their radii

Advantage of this view: It becomes possible to distinguish between:

  • Internal forces: Between planets and the sun
  • External forces: Acting on the entire solar system from outside (e.g., gravitational pull from other stars)

Air Molecules in a Jar

Similarly, a few air molecules in a glass jar can be treated as a system of particles:

  • Each molecule is a point mass
  • External force: Earth’s gravity acting on all molecules
  • Internal forces: Attractions, repulsions, or collisions between molecules

Internal Versus External Forces

Internal forces:

  • Act between components within the system
  • Occur in action-reaction pairs (Newton’s third law)
  • Sum to zero when considering the whole system
  • Affect relative motion within the system

External forces:

  • Act on the system from outside
  • Determine motion of the system as a whole
  • Do not cancel within system analysis

Continuum Systems

When the number of particles in the system becomes so large that it is no longer possible to keep track of each individual particle, mechanics shifts from a microscopic to a macroscopic system orientation.

The Statistical Approach

For enormous numbers of particles:

  • Particle motions are treated statistically
  • Matter is thought to be a homogeneous “cloud” of particles
  • Called a continuum

Examples:

  • The Milky Way: An entire cloud of stars
  • Air in a balloon: Approximately 10²³ molecules

Focus of Analysis

In continuum systems, the concern is with:

  • Deformation: Shape changes of the entire region
  • Bulk displacement: Movement of the system as a whole
  • Internal stress distribution: How forces are distributed through the material

Rather than tracking each particle’s intricate movement, we describe:

  • Velocity fields (velocity at each point in space)
  • Stress fields (stress at each point)
  • Strain fields (deformation at each point)

This leads to the basic principles of continuum mechanics, which will be developed in the next section.

Defining System Boundaries

Why Boundaries Matter

In isolating a free body, it’s important to understand the concept of a mechanical system. The choice of system boundary determines:

  • Which forces are internal (and thus cancel)
  • Which forces are external (and affect system motion)
  • What predictions can be made

Example: Vocal Fold Vibration

Different boundary choices give different insights:

System 1: Single Tissue Volume

  • Boundary: Small cubic volume of tissue
  • Internal forces: Molecular bonds within the volume
  • External forces: Forces from adjacent tissue, air pressure
  • Reveals: Local stress and strain

System 2: Entire Vocal Fold

  • Boundary: One complete vocal fold
  • Internal forces: Forces between tissue regions within fold
  • External forces: Muscle forces, aerodynamic forces, forces from opposite fold
  • Reveals: Gross motion patterns

System 3: Both Vocal Folds Together

  • Boundary: Both folds as a unit
  • Internal forces: Forces between the two folds
  • External forces: Muscle forces, aerodynamic forces from below and above
  • Reveals: Symmetry or asymmetry of vibration

Choosing the Right System

The optimal system definition depends on the question:

For understanding phonation threshold:

  • System: Both vocal folds
  • Focus: Aerodynamic forces needed to initiate vibration

For understanding mucosal wave:

  • System: Vertical slice through one fold
  • Focus: Vertical phase differences in tissue motion

For understanding tissue stress:

  • System: Small tissue volume
  • Focus: Local deformation and stress distribution

From Discrete to Continuous

The transition from particle systems to continuum systems represents a fundamental shift in perspective:

Particle System View

  • Track individual entities
  • Count discrete interactions
  • Solve equations for each particle
  • Suitable when numbers are manageable

Continuum View

  • Describe fields (continuous functions of position)
  • Use partial differential equations
  • Apply to regions of space
  • Necessary when particles are too numerous to track

Vocal Fold as Continuum

Vocal fold tissue is clearly a continuum:

  • Contains vast numbers of molecules
  • Impossible to track each molecule
  • Behavior described by fields: velocity field, stress field, strain field
  • Requires continuum mechanics for analysis

However, vocal fold tissue is not a simple continuum:

  • Layered structure (cover, ligament, body)
  • Anisotropic (different properties in different directions)
  • Viscoelastic (time-dependent behavior)
  • Inhomogeneous (properties vary with location)

Summary

A mechanical system can be defined at various levels of description:

  • Single particle: Simplest idealization for point masses
  • System of particles: Collection of discrete masses with internal and external forces
  • Continuum: Matter distributed continuously through space

The choice of system definition determines:

  • Which forces are considered internal versus external
  • What level of detail is captured
  • What analytical methods are appropriate

For voice science:

  • Macroscopic level: Treat whole vocal folds as masses on springs
  • Mesoscopic level: Consider layered structure and regional variations
  • Microscopic level: Examine molecular structure and tissue composition

Each level provides different insights, and complete understanding requires integration across scales. The continuum mechanics framework provides the mathematical tools for analyzing distributed matter like vocal fold tissue, which we explore in the next major section.

Understanding mechanical systems—from particles to continua—establishes the conceptual foundation for applying Newton’s laws to complex biological structures like the larynx.

Key Takeaways

  • ✅ A particle is the simplest mechanical system: finite mass at a single point in space
  • ✅ Systems of particles allow distinction between internal forces (between components) and external forces (on the whole system)
  • ✅ Continuum systems treat matter as distributed continuously when particle numbers are too large to track individually
  • ✅ System boundary definition determines which forces are internal versus external
  • ✅ Vocal fold tissue is best analyzed as a continuum due to vast numbers of molecules
  • ✅ Different system definitions (single volume, whole fold, both folds) provide different insights

Further Reading

  1. Halliday, D., & Resnick, R. (1988). Fundamentals of physics. New York: Wiley.
  2. Fung, Y. C. (1981). Biomechanics: Mechanical properties of living tissues. New York: Springer-Verlag.
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