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4.10. Classical Mechanics (CME)
Classical Mechanics provides the foundational principles governing the motion of macroscopic objects, from particles to rigid bodies and continuous media. It encompasses Newtonian dynamics, conservation laws, Lagrangian and Hamiltonian formalisms, oscillatory systems, and celestial mechanics.
4.10.1. CME/Kinematics and Dynamics of Particles (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Mathematical description of motion and the study of forces that cause or modify motion for point particles, forming the basis of mechanical and structural engineering analysis.
Topics:
Core
- Vector description of position, velocity, and acceleration in Cartesian, polar, and spherical coordinates
- Newton's three laws of motion and inertial reference frames
- Formulation and solution of equations of motion for constant and variable forces
- Static and kinetic friction forces, including angle of repose
- Numerical integration of equations of motion for engineering dynamics problems
Learning Outcomes:
Core:
- Formulate the equations of motion for a particle in one, two, and three dimensions given a force law [Usage]
- Identify all forces acting on a particle and draw a correct free-body diagram [Familiarity]
- Calculate the terminal velocity of an object falling through a fluid with drag resistance [Assessment]
- Implement a numerical algorithm to simulate the dynamics of an engineering mechanical system [Usage]
- Construct and interpret phase space diagrams for simple engineering oscillatory systems [Assessment]
4.10.2. CME/Work, Energy, and Conservative Forces (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Energy concepts in mechanics: work-energy theorem, kinetic and potential energy, conservation of mechanical energy, and properties of conservative force fields.
Topics:
Core
- Definition of work as a line integral \(W = \int \vec{F} \cdot d\vec{r}\)
- Work-energy theorem: \(W_{total} = \Delta K\)
- Potential energy function and the force relation \(\vec{F} = -\nabla U\)
- Conservation of total mechanical energy \(E = K + U\)
- Energy diagrams, turning points, and equilibrium stability
- Work done by non-conservative forces and energy dissipation in engineering systems
Learning Outcomes:
Core:
- Calculate the work done by a force along a specified engineering path [Usage]
- Apply conservation of mechanical energy to solve for speeds and positions in engineering dynamics [Usage]
- Analyze an energy diagram to identify equilibrium positions and stability [Assessment]
- Estimate energy dissipated by friction and damping forces in mechanical engineering systems [Assessment]
4.10.3. CME/Rotational Motion and Angular Momentum (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Rotational kinematics and dynamics, torque, moment of inertia, rotational energy, and conservation of angular momentum for engineering machines and structures.
Topics:
Core
- Rotational kinematics: angular displacement, velocity, and acceleration
- Torque: \(\vec{\tau} = \vec{r} \times \vec{F}\) and its computation
- Moment of inertia \(I = \sum m_i r_i^2\) and the parallel axis theorem
- Rotational kinetic energy: \(K_{rot} = \frac{1}{2}I\omega^2\)
- Conservation of angular momentum in engineering machines and rotating structures
- Rolling without slipping dynamics and applications to wheels and shafts
Learning Outcomes:
Core:
- Calculate torque about a given axis for an engineering power transmission system [Usage]
- Determine the moment of inertia for machine components such as rods, disks, and shafts [Usage]
- Apply conservation of angular momentum to analyze speed changes in rotating engineering systems [Assessment]
- Analyze combined translational and rotational motion in wheeled engineering systems [Assessment]
4.10.4. CME/Systems of Particles and Linear Momentum (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Dynamics of multi-particle systems: center of mass motion, linear momentum conservation, and collision mechanics relevant to structural impacts and machinery.
Topics:
Core
- Definition and computation of center of mass for particle systems and distributed bodies
- Linear momentum of a particle and system: \(\vec{P} = M\vec{V}_{CM}\)
- Conservation of linear momentum for isolated engineering systems
- Two-body collisions: elastic, inelastic, and the coefficient of restitution
- Variable mass systems such as rockets and conveyor belts
Learning Outcomes:
Core:
- Locate the center of mass for discrete particle systems and distributed engineering objects [Usage]
- Apply conservation of linear momentum to analyze impact and collision problems [Usage]
- Classify collisions as elastic or inelastic and determine energy loss [Familiarity]
- Solve variable-mass system equations to determine velocity gain in propulsion problems [Usage]
4.10.5. CME/Non-Inertial Reference Frames and Fictitious Forces (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Physics in accelerating reference frames: centrifugal, Coriolis, and Euler forces, with applications to Earth's rotation.
Topics:
Core
- Transformations between inertial and accelerating reference frames
- Centrifugal force defined as \(F_{cent} = -m\vec{\omega} \times (\vec{\omega} \times \vec{r}')\)
- Coriolis force defined as \(F_{Cor} = -2m\vec{\omega} \times \vec{v}'\)
- Euler force for non-constant rotation: \(F_{Euler} = -m\frac{d\vec{\omega}}{dt} \times \vec{r}'\)
- Effective gravity on the rotating Earth: \(g_{eff} = g - \vec{\omega} \times (\vec{\omega} \times \vec{R})\)
- Foucault pendulum as a demonstration of the Earth's rotation
- Coriolis effect on global weather patterns and ocean currents
Elective
- Larmor precession in magnetic fields viewed as a rotating frame effect
- Tidal forces as differential gravitational effects in co-rotating frames
- Connection to General Relativity's equivalence principle
Learning Outcomes:
Core:
- Transform Newton's second law to a uniformly rotating reference frame [Usage]
- Calculate centrifugal and Coriolis forces for given positions and velocities in a rotating frame [Usage]
- Predict the deflection direction of projectiles due to the Earth's rotation through the Coriolis effect [Assessment]
- Explain the precession of a Foucault pendulum in terms of the Earth's rotation [Familiarity]
- Compute effective gravitational acceleration at different latitudes on Earth [Usage]
Elective:
- Analyze Larmor precession using the concept of rotating reference frames [Assessment]
- Compare tidal forces to centrifugal forces in co-rotating systems [Assessment]
4.10.6. CME/Lagrangian and Hamiltonian Mechanics (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Advanced formulations of mechanics: principle of least action, Lagrangian formulation, Hamiltonian mechanics, and canonical transformations.
Topics:
Core
- Principle of least action and Hamilton's principle of stationary action
- Lagrangian definition: \(L = T - V\) for conservative systems
- Euler-Lagrange equations of motion: \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) = \frac{\partial L}{\partial q_i}\)
- Generalized coordinates and the treatment of holonomic constraints
- Cyclic coordinates and the associated conservation laws
- Legendre transform from the Lagrangian to the Hamiltonian: \(H = \sum p_i\dot{q}_i - L\)
- Hamilton's equations: \(\dot{q}_i = \frac{\partial H}{\partial p_i}\) and \(\dot{p}_i = -\frac{\partial H}{\partial q_i}\)
Elective
- Poisson brackets: \(\{f,g\} = \sum\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)\)
- Canonical transformations and their generating functions
- Action-angle variables for periodic Hamiltonian systems
Learning Outcomes:
Core:
- Construct the Lagrangian for simple mechanical systems in appropriate generalized coordinates [Usage]
- Derive equations of motion from the Euler-Lagrange equations [Usage]
- Identify cyclic coordinates and their associated conserved quantities in a given Lagrangian [Assessment]
- Formulate the Hamiltonian from a given Lagrangian system [Usage]
- Solve Hamilton's equations for simple systems such as the harmonic oscillator [Usage]
Elective:
- Compute Poisson brackets for pairs of dynamical variables [Usage]
- Apply canonical transformations to simplify the description of Hamiltonian systems [Usage]
4.10.7. CME/Continuum Mechanics Foundations (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Introduction to continuous media: stress and strain tensors, elasticity, fluid mechanics, and conservation laws.
Topics:
Elective
- Continuum hypothesis and the definition of density fields
- Stress tensor \(\sigma_{ij}\) and the concept of force per unit area
- Strain tensor \(\epsilon_{ij}\) as a measure of local deformation
- Hooke's law for isotropic materials: \(\sigma_{ij} = \lambda\epsilon_{kk}\delta_{ij} + 2\mu\epsilon_{ij}\)
- Navier-Stokes equation for viscous fluids: \(\rho\left(\frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}\right) = -\nabla p + \mu\nabla^2\vec{v} + \vec{f}\)
- Bernoulli's principle for inviscid and incompressible fluid flow
- Conservation of mass and the continuity equation: \(\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\vec{v}) = 0\)
- Conservation of momentum expressed in continuum mechanics form
- Wave propagation in elastic media including P-waves and S-waves
- Viscosity and the rheology of non-Newtonian fluids
Learning Outcomes:
Elective:
- Interpret the physical meaning of the components of the stress tensor [Familiarity]
- Calculate strain tensor components for simple deformations [Usage]
- Apply Hooke's law to relate stress and strain for isotropic elastic materials [Usage]
- Simplify the Navier-Stokes equation for special cases such as steady or inviscid flow [Usage]
- Apply Bernoulli's principle to calculate pressure variations in fluid flow [Usage]
- Derive the continuity equation from the principle of mass conservation [Usage]
- Analyze different wave propagation modes in elastic solids [Assessment]
4.10.8. CME/Elasticity (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Elastic properties of solids under axial, shear, and volumetric load: stress, strain, and scalar elastic moduli.
Topics:
Core
- Definition of stress (force/area) and strain (\(\Delta L/L\))
- Young's modulus and Hooke's law for longitudinal strain: \(\sigma = E\epsilon\)
- Shear modulus (rigidity) and shear strain
- Bulk modulus (compressibility) and volumetric strain
- Elastic limit, plastic region, and fracture point on the stress-strain diagram
Elective
- Elastic energy stored in a deformed material
Learning Outcomes:
Core:
- Apply the scalar Hooke's law to calculate the deformation of a rod under axial load [Usage]
- Calculate Young's, shear, or bulk modulus from experimental stress-strain data [Usage]
- Identify the elastic region, elastic limit, and fracture point on a stress-strain curve [Familiarity]
Elective:
- Calculate the elastic potential energy stored in a deformed spring or rod [Usage]
4.10.9. CME/Fluids: Hydrostatics, Hydrodynamics, and Viscosity (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Statics and dynamics of incompressible fluids: pressure, buoyancy, the continuity and Bernoulli equations, and an introduction to viscous flow.
Topics:
Core
- Density, pressure, and its variation with depth in a fluid at rest: \(p = p_0 + \rho g h\)
- Pascal's principle and its applications (hydraulic press)
- Archimedes' principle, buoyant force, and flotation
- Pressure measurement with manometers and barometers
- Continuity equation for incompressible flow: \(A_1 v_1 = A_2 v_2\)
- Bernoulli's equation and energy conservation in ideal flow
- Applications: Venturi tube, Torricelli's theorem, Pitot tube
Elective
- Viscosity and Poiseuille's law for laminar flow in pipes
- Reynolds number and the transition to turbulent flow
- Stokes' law for the drag force on a sphere in a viscous fluid
Learning Outcomes:
Core:
- Calculate the pressure at a point in a fluid at rest given depth and density [Usage]
- Apply Archimedes' principle to determine whether an object floats, sinks, or remains in equilibrium [Usage]
- Apply the continuity and Bernoulli equations to solve flow problems in pipes of varying cross-section [Usage]
- Interpret manometer and barometer readings to determine absolute and gauge pressures [Assessment]
Elective:
- Apply Poiseuille's law to calculate the flow rate of a viscous fluid in a cylindrical tube [Usage]
- Estimate the Reynolds number of a flow and predict whether it is laminar or turbulent [Assessment]
4.10.10. CME/Vibrations and Waves (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Simple harmonic motion, damped and forced oscillations, and the propagation, superposition, and resonance of mechanical waves.
Topics:
Core
- Simple harmonic motion (SHM): equation of motion, position, velocity, and acceleration
- Oscillating systems: spring-mass and simple/physical pendulum
- Energy in simple harmonic motion
- Damped and forced oscillations; introduction to resonance
- Transverse and longitudinal waves; propagation speed
- Wave equation and speed of a wave on a taut string: \(v = \sqrt{T/\mu}\)
- Superposition principle and standing waves
- Sound waves, intensity, and sound intensity level (decibels)
Elective
- Doppler effect for sound waves
- Beats and interference of waves of nearby frequencies
Learning Outcomes:
Core:
- Derive the equation of motion and the period of a spring-mass system and a simple pendulum [Usage]
- Calculate the kinetic, potential, and total energy of a simple harmonic oscillator at any instant [Usage]
- Analyze the amplitude response of a forced oscillator near the resonance frequency [Assessment]
- Calculate the propagation speed of a wave on a string from the tension and linear density [Usage]
- Determine the frequencies and wavelengths of the normal modes of a standing wave on a fixed string [Usage]
- Calculate the sound intensity level in decibels from the intensity of a sound wave [Usage]
Elective:
- Apply the Doppler effect formula for a moving source and/or observer [Usage]
4.10.11. CME/Theory of Small Oscillations and Coupled Oscillators (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Small oscillations about equilibrium: normal modes, eigenfrequencies, damping, forcing, and coupled systems, extending single-oscillator SHM to engineering systems with several degrees of freedom.
Topics:
Non Core
- Small oscillation approximation via Taylor expansion about a stable equilibrium
- Coupled oscillators and the formulation of equations of motion in matrix form
- Normal modes and eigenfrequencies derived from the eigenvalue problem
- Damped oscillators: underdamped, critically damped, and overdamped regimes
Learning Outcomes:
NonCore:
- Linearize equations of motion about a stable equilibrium point [Usage]
- Find normal mode frequencies and eigenvectors for systems of Two coupled oscillators [Usage]
4.10.12. CME/Transport Phenomena (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top
Introduction to transport phenomena in continuous media: heat conduction and mass diffusion as gradient-driven flux processes.
Topics:
Core
- Heat conduction and Fourier's law: \(\vec{q} = -k\nabla T\)
- Thermal conductivity of materials and thermal resistance in composite walls
- Molecular diffusion and Fick's first law: \(\vec{J} = -D\nabla C\)
Elective
- Analogy between heat, mass, and momentum transport (Prandtl and Schmidt numbers)
Learning Outcomes:
Core:
- Apply Fourier's law to calculate the heat flux through a plane or composite wall [Usage]
- Apply Fick's first law to calculate the diffusive flux of a species in a concentration gradient [Usage]
Elective:
- Identify mathematical analogies between heat, mass, and momentum transport processes [Familiarity]