Key Points on Simple Harmonic Motion (SHM)

  • Periodic Motion: Repeats at regular intervals (e.g., guitar string, swing).

  • Period (T) and Frequency (f):

    • Period (T): Time for one oscillation (in seconds).
    • Frequency (f): Number of oscillations per time unit (in Hertz, Hz).
    • Relationship: T=1fT = \frac{1}{f} (f = 1\text{ Hz} = \frac{1}{s}

  • Characteristics of SHM:

    • Acceleration (a) in SHM is proportional to displacement (x).
    • Follows Hooke’s Law: F_s = -kx
    • k :springconstant;: spring constant; x : displacement.
    • Amplitude (A): Maximum displacement from equilibrium.
    • Period and frequency independent of amplitude; only mass (m) and spring constant (k) affect T and f.

  • Differential Equation of Motion:

    • F = ma leadstoleads to ma = -kx oror m \frac{d^2x}{dt^2} = -kx

  • SHM Graphical Representation:

    • x(t) = A \cos(\omega t + \phi)
    • Where:
      • A = amplitude
      • \omega = angular frequency
      • \phi = phase constant.

  • Equations of Motion:

    • Position: x(t) = A \cos(\omega t + \phi)
    • Velocity: v(t) = -A\omega \sin(\omega t + \phi)
    • Acceleration: a(t) = -A\omega^2 \cos(\omega t + \phi)

  • Energy in SHM:

    • Total energy: E = \frac{1}{2} kA^2
    • Kinetic Energy: K = \frac{1}{2} mv^2
    • Potential Energy: U = \frac{1}{2} kx^2

  • Simple Pendulum:

    • Period: T = 2\pi \sqrt{\frac{L}{g}} $$ (depends on length of string L and gravitational acceleration g).

  • Damping:

    • Decreases amplitude and frequency over time.
    • Three types:
    • Underdamped: oscillates with decaying amplitude.
    • Critically damped: returns to equilibrium without oscillating.
    • Overdamped: returns slowly without oscillating.