Classical Mechanics

1. Oscillatory Motion
   1. Simple Harmonic Motion (SHM)
      1. Characteristics and Definition
         1. Periodicity and Sinusoidal Nature of SHM
         2. Mathematical Representation: x(t) = A cos(ωt + φ)
         3. Definitions and Units of Amplitude, Frequency, and Phase
         4. Angular Frequency and its Relationship to Period
      2. Spring-Mass Systems
         1. Hooke's Law: F = -kx
            1. Proportionality Constant (k) as Spring Constant
            2. Analysis of Restoring Force as Function of Displacement
         2. Energy in SHM
            1. Kinetic Energy and Potential Energy in Springs
            2. Conservation of Mechanical Energy
         3. Deriving Motion Equations
            1. Differential Equations: d²x/dt² + (k/m)x = 0
         4. Real-world Applications
            1. Measurements of Oscillations in Engineering
            2. Vibrational Analysis in Mechanical Systems
      3. Pendulums
         1. Simple Pendulum
            1. Derivation of Period: T = 2π√(l/g)
            2. Assumptions: Small Angle Approximation
            3. Analyzing Forces and Energies
         2. Physical Pendulum
            1. Extension to Compound Bodies
            2. Calculating Periods for Irregular Shapes
         3. Applications and Limitations
            1. Use in Clocks and Timekeeping
            2. Pendulum Designs in Engineering
   2. Damped and Driven Harmonic Oscillators
      1. Damped Oscillations
         1. Types of Damping: Light, Critical, and Overdamping
         2. Damping Equation: mx'' + bx' + kx = 0
         3. Exponential Decay and Effect on Amplitude
      2. Driven Oscillations
         1. Forced Oscillators: Understanding Driving Forces
         2. Resonance Phenomenon in Driven Systems
         3. Steady-State Solution for Forced Oscillations
         4. Phase Shift between Driving Force and Displacement
   3. Resonance
      1. Definition and Concepts
         1. Natural Frequency and its Role in Resonance
         2. Amplitude Response and Frequency Matching
      2. Applications in Engineering and Science
         1. Resonance in Bridges and Buildings
         2. Tuning of Musical Instruments
         3. Electromagnetic Resonance in Circuits
      3. Risks and Mitigation Strategies
         1. Structural Failure Due to Resonance
         2. Designing for Damping and Control