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3.12. Classical Mechanics (CME)
Classical Mechanics provides the foundational principles governing the motion of particles and rigid bodies. This computing-focused subset emphasizes kinematics, energy methods, rigid-body rotation, oscillatory phenomena, and variational mechanics relevant to robotics, game engines, physics simulations, and physics-informed machine learning.
3.12.1. CME/Rotational Motion and Angular Momentum (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Rotational kinematics and dynamics, torque, moment of inertia, rotational energy, and conservation of angular momentum for particles and systems.
Topics:
Core
- Rotational kinematics including angular displacement, velocity, and acceleration (\(\theta, \omega, \alpha\))
- Torque definition: \(\vec{\tau} = \vec{r} \times \vec{F}\) and its magnitude calculation
- Angular momentum of a particle: \(\vec{L} = \vec{r} \times \vec{p}\)
- Rotational form of Newton's second law: \(\vec{\tau}_{net} = \frac{d\vec{L}}{dt}\)
- 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 for isolated systems
- Gyroscopic motion and the phenomenon of precession
- Rolling without slipping dynamics: \(v_{CM} = R\omega\)
Elective
- Coriolis effect as a manifestation of rotational dynamics in non-inertial frames
Learning Outcomes:
Core:
- Relate linear and angular kinematic quantities for circular motion [Usage]
- Calculate torque about a given axis for a force applied at a specific point [Usage]
- Determine the moment of inertia for simple geometries such as a rod, hoop, disk, or sphere [Usage]
- Apply conservation of angular momentum to analyze changes in rotational speed when the moment of inertia changes [Assessment]
- Compute rotational kinetic energy for a rotating rigid body [Usage]
- Analyze the motion of objects rolling without slipping down an incline [Assessment]
- Predict the direction of precession for a spinning top or gyroscope [Assessment]
Elective:
- Explain the Coriolis effect in terms of rotating reference frames and its effect on projectiles [Familiarity]
3.12.2. CME/Systems of Particles and Linear Momentum (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Dynamics of multi-particle systems: center of mass motion, linear momentum conservation, internal and external forces, and variable mass systems.
Topics:
Core
- Definition of center of mass: \(\vec{R}_{CM} = \frac{1}{M}\sum m_i \vec{r}_i\)
- Linear momentum of a particle and system: \(\vec{p} = m\vec{v}\) and \(\vec{P} = M\vec{V}_{CM}\)
- Newton's second law for systems: \(\vec{F}_{ext} = \frac{d\vec{P}}{dt}\)
- Conservation of linear momentum for isolated systems
- Distinction between internal and external forces in a system
- Two-body collisions including elastic, inelastic, and completely inelastic cases
- Coefficient of restitution and its physical meaning in collisions
- Systems with variable mass such as the rocket equation: \(M\frac{dV}{dt} = v_{ex}\frac{dM}{dt}\)
- Many-particle systems and their center of mass dynamics
Elective
- Continuous mass distributions and their center of mass calculation using integration
Learning Outcomes:
Core:
- Locate the center of mass for a discrete set of particles in One, Two, and Three dimensions [Usage]
- Determine the total linear momentum of a system from the motion of its center of mass [Usage]
- Apply conservation of linear momentum to solve collision problems in One and Two dimensions [Usage]
- Classify collisions as elastic, inelastic, or completely inelastic based on kinetic energy conservation [Familiarity]
- Calculate the coefficient of restitution from pre-collision and post-collision velocities [Usage]
- Solve the rocket equation to determine velocity as a function of ejected mass [Usage]
- Analyze how internal forces affect individual particles but not the total momentum of a system [Assessment]
Elective:
- Integrate to find the center of mass of symmetric continuous bodies such as rods, disks, and spheres [Usage]
3.12.3. CME/Kinematics and Dynamics of Particles (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Mathematical description of particle motion and the forces governing it, with emphasis on phase space analysis and numerical integration methods used in robotics and simulation.
Topics:
Core
- Vector description of position, velocity, and acceleration; Newton's three laws and inertial reference frames
- Formulation and analytical solution of equations of motion for constant and variable forces
- Phase space diagrams (position vs. velocity) for visualizing mechanical state, stability, and periodicity
- Numerical integration of equations of motion using Euler, Verlet, and Runge-Kutta methods
- Constrained motion and holonomic constraints relevant to robotic linkages and articulated bodies
Learning Outcomes:
Core:
- Identify all forces acting on a particle in a given mechanical scenario and draw a correct free-body diagram [Familiarity]
- Implement a numerical integration algorithm such as the Runge-Kutta method to compute the trajectory of a particle under variable forces [Usage]
- Construct and interpret phase space diagrams to characterize the stability and periodicity of a simulated mechanical system [Assessment]
3.12.4. CME/Work, Energy, and Conservation Laws (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Energy methods and conservation principles underpinning physics engines, collision resolution, and energy-aware optimization in computing systems.
Topics:
Core
- Work-energy theorem relating net work done to change in kinetic energy
- Potential energy functions and the force-gradient relation for conservative force fields
- Conservation of total mechanical energy and conditions for its applicability
- Energy diagrams for identifying equilibrium points, turning points, and stability
- Elastic and inelastic collision models: momentum and energy exchange in physics engine implementations
Learning Outcomes:
Core:
- Explain the distinction between conservative and non-conservative forces and the conditions under which mechanical energy is conserved [Familiarity]
- Apply energy conservation methods to compute speeds and positions in one-dimensional motion problems common in physics engine implementations [Usage]
- Analyze elastic and inelastic collision events, computing post-collision velocities and quantifying energy dissipation [Assessment]
3.12.5. CME/Rigid Body Dynamics (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Rotational dynamics of rigid bodies including torque, moment of inertia, and angular momentum, directly applicable to game engine physics and robot kinematics.
Topics:
Core
- Torque definition and moment of inertia for standard geometries: rod, disk, sphere, and box
- Angular momentum vector and its conservation for isolated rotating systems
- Rotational equations of motion and coupling between translational and rotational degrees of freedom
- Euler angles and quaternion representations for three-dimensional orientation tracking
- Rolling without slipping condition and constraint forces in rigid body systems
Learning Outcomes:
Core:
- Explain how the moment of inertia governs the rotational response of a rigid body to applied torques [Familiarity]
- Compute the angular velocity and orientation of a rigid body given an applied torque history and initial conditions [Usage]
- Design a rigid body simulation for a game engine scenario, selecting appropriate numerical integrators and orientation representations [Assessment]
3.12.6. CME/Oscillations and Waves (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Simple harmonic motion, damping, resonance, and wave propagation, providing the mechanical foundations for signal processing, audio synthesis, and vibration analysis in computing applications.
Topics:
Core
- Simple harmonic oscillator: frequency, period, amplitude, phase, and energy
- Damped oscillators: underdamped, critically damped, and overdamped regimes and their time-domain signatures
- Forced oscillations, resonance frequency, amplitude response, and the quality factor Q
- Coupled oscillators and normal mode decomposition for multi-degree-of-freedom systems
- Mechanical wave propagation: wave equation, dispersion relation, and group versus phase velocity
Learning Outcomes:
Core:
- Describe the physical behavior of underdamped, critically damped, and overdamped oscillators and relate each regime to applications in control and audio systems [Familiarity]
- Solve the damped harmonic oscillator equation analytically and numerically for given initial conditions and forcing functions [Usage]
- Analyze the amplitude-frequency response of a driven oscillator and quantify the effect of damping on resonance bandwidth and peak gain [Assessment]
3.12.7. CME/Lagrangian and Hamiltonian Mechanics (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top
Variational formulations of mechanics underpinning symplectic integrators, differentiable physics simulations, and physics-informed machine learning.
Topics:
Core
- Hamilton's principle of stationary action and its role in deriving equations of motion
- Euler-Lagrange equations of motion in generalized coordinates for constrained mechanical systems
- Hamiltonian formulation: canonical coordinates, phase flow, and energy interpretation
- Symplectic integrators (Verlet, leapfrog) and their energy-conserving properties for long-run simulations
- Noether's theorem connecting continuous symmetries to conserved quantities
Learning Outcomes:
Core:
- Explain the variational principle underlying Lagrangian mechanics and contrast it with the Newtonian force-based approach [Familiarity]
- Construct the Lagrangian for a mechanical system with holonomic constraints and derive its equations of motion via the Euler-Lagrange equations [Usage]
- Evaluate the long-run energy conservation of symplectic versus non-symplectic integrators through numerical experiments on a Hamiltonian system [Assessment]
3.12.8. CME/Elasticity (CS Core: 1 hr, KA Core: 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]
3.12.9. CME/Fluids: Hydrostatics, Hydrodynamics, and Viscosity (CS Core: 1 hr, KA Core: 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]
3.12.10. CME/Vibrations and Waves (CS Core: 1 hr, KA Core: 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]
3.12.11. CME/Transport Phenomena (CS Core: 1 hr, KA Core: 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]