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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.

Knowledge Area (KA)	CS Core	KA Core
3.12.1 Rotational Motion and Angular Momentum	Elective
3.12.2 Systems of Particles and Linear Momentum	Elective
3.12.3 Kinematics and Dynamics of Particles	Elective
3.12.4 Work, Energy, and Conservation Laws	Elective
3.12.5 Rigid Body Dynamics	Elective
3.12.6 Oscillations and Waves	Elective
3.12.7 Lagrangian and Hamiltonian Mechanics	Elective

3.12.1. CME/Rotational Motion and Angular Momentum ↑ 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:

1. Relate linear and angular kinematic quantities for circular motion [Usage]
2. Calculate torque about a given axis for a force applied at a specific point [Usage]
3. Determine the moment of inertia for simple geometries such as a rod, hoop, disk, or sphere [Usage]
4. Apply conservation of angular momentum to analyze changes in rotational speed when the moment of inertia changes [Assessment]
5. Compute rotational kinetic energy for a rotating rigid body [Usage]
6. Analyze the motion of objects rolling without slipping down an incline [Assessment]
7. Predict the direction of precession for a spinning top or gyroscope [Assessment]

Elective:

1. 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 ↑ 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:

1. Locate the center of mass for a discrete set of particles in One, Two, and Three dimensions [Usage]
2. Determine the total linear momentum of a system from the motion of its center of mass [Usage]
3. Apply conservation of linear momentum to solve collision problems in One and Two dimensions [Usage]
4. Classify collisions as elastic, inelastic, or completely inelastic based on kinetic energy conservation [Familiarity]
5. Calculate the coefficient of restitution from pre-collision and post-collision velocities [Usage]
6. Solve the rocket equation to determine velocity as a function of ejected mass [Usage]
7. Analyze how internal forces affect individual particles but not the total momentum of a system [Assessment]

Elective:

1. 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 ↑ 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:

1. Identify all forces acting on a particle in a given mechanical scenario and draw a correct free-body diagram [Familiarity]
2. Implement a numerical integration algorithm such as the Runge-Kutta method to compute the trajectory of a particle under variable forces [Usage]
3. 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 ↑ 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:

1. Explain the distinction between conservative and non-conservative forces and the conditions under which mechanical energy is conserved [Familiarity]
2. Apply energy conservation methods to compute speeds and positions in one-dimensional motion problems common in physics engine implementations [Usage]
3. Analyze elastic and inelastic collision events, computing post-collision velocities and quantifying energy dissipation [Assessment]

3.12.5. CME/Rigid Body Dynamics ↑ 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:

1. Explain how the moment of inertia governs the rotational response of a rigid body to applied torques [Familiarity]
2. Compute the angular velocity and orientation of a rigid body given an applied torque history and initial conditions [Usage]
3. 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 ↑ 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:

1. Describe the physical behavior of underdamped, critically damped, and overdamped oscillators and relate each regime to applications in control and audio systems [Familiarity]
2. Solve the damped harmonic oscillator equation analytically and numerically for given initial conditions and forcing functions [Usage]
3. 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 ↑ 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:

1. Explain the variational principle underlying Lagrangian mechanics and contrast it with the Newtonian force-based approach [Familiarity]
2. Construct the Lagrangian for a mechanical system with holonomic constraints and derive its equations of motion via the Euler-Lagrange equations [Usage]
3. Evaluate the long-run energy conservation of symplectic versus non-symplectic integrators through numerical experiments on a Hamiltonian system [Assessment]