Review of Simple Harmonic Motion Concepts

Key Concepts of Simple Harmonic Motion (SHM)

  • Basic Definitions:

    • Simple Harmonic Motion (SHM) describes a restoring force proportional to the displacement.

    • Model a mass attached to a spring or pendulum swinging back and forth.

  • Equations Used in SHM:

    • Displacement equation:
      x(t) = A ext{cos}(ωt + φ)
      Where:

    • A = amplitude (maximum displacement)

    • ω = angular frequency (in radians)

    • φ = phase constant (initial phase)

    • Velocity equation:
      v(t) = -Aω ext{sin}(ωt + φ)

    • Acceleration equation:
      a(t) = -Aω^2 ext{cos}(ωt + φ)

  • Key Parameters:

    • Amplitude (A): Maximum extent of displacement from the equilibrium position.

    • Angular Frequency (ω): Defined as:
      ω = 2πf
      Where f = frequency (Hz), measured in cycles per second.

    • Frequency (f): Number of cycles per second. Given by:
      f = rac{1}{T}
      Where T = period of motion.

    • Period (T): Time taken for one complete cycle of motion:
      T = rac{2π}{ω} = rac{1}{f}

Cases of Motion in SHM

Case 1

  • At time t = 0 , the particle is at maximum displacement:

    • x(t) = A ext{cos}(0) = A

    • Velocity at start:
      v(t=0) = -Aω ext{sin}(0 + φ) = 0

    • Outcome: Particle starts from maximum displacement.

Case 2

  • At time t where the value is not maximum:

    • x(t) = A ext{cos}(ωt + φ)

    • The particle's position changes over time.

Case 3 and 4

  • Generalize further motion cases to find solutions based on different initial conditions.

Velocity and Acceleration

  • Velocity and Acceleration are derived from displacement:

  • Analytical relationships define these equations and help in analyzing state changes over time.

    • The acceleration is directly related to displacement, confirming the nature of SHM, where a(t) = -ω^2 x(t) .

Graphical Interpretation

  • Often visualized using sine and cosine waves representing motion over time.

  • The smooth curves exhibit the cyclic nature of motion, indicating energy transfers during oscillations.

Important Notes on Units

  • Angular frequency ω : measured in radians per second (rad/s).

  • Frequency f : measured in Hertz (Hz).

  • Period T : measured in seconds (s).

Practical Implications

  • Applies in physics (pendulums, springs), engineering (vibrating systems), and everyday occurrences (sound waves).

  • Third law implications: the motion is predictable under structural constraints and outside forces.

Example Scenarios

  1. Pendulum: Exhibits periodic motion affected by gravity, modeled with these equations.

  2. Spring Systems: Understand how different masses affect angular frequency and period of oscillation.