Chapter: Oscillations

• Periodic and Oscillatory Motion

  • Periodic Motion: Motion that repeats at regular intervals of time
    • Example: Revolution of planets around the Sun
  • Oscillatory Motion: To-and-fro motion about a mean position
    • Example: Swinging pendulum, vibrating tuning fork
    • Note: Every oscillatory motion is periodic, but not all periodic motions are oscillatory
  • Time Period ($T$): Time taken for one complete oscillation ($T = 1/f$)
  • Frequency ($f$): Number of oscillations per second
    • SI Unit: Hertz (Hz)

• Simple Harmonic Motion (SHM)

  • Definition: Oscillatory motion where restoring force is directly proportional to displacement
    • Equation: $F = -k x$
  • Differential Equation: $\frac{d^2x}{dt^2} + \omega^2 x = 0$
  • Displacement ($x$): $x(t) = A \cos(\omega t + \phi)$
    • Amplitude ($A$): Maximum displacement from mean position
    • Angular Frequency ($\omega$): $\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T}$
    • Phase ($\omega t + \phi$): Defines the state of motion at time $t$
    • Initial Phase ($\phi$): Phase angle at $t = 0$
  • Velocity ($v$): $v = \pm \omega \sqrt{A^2 - x^2}$
    • Maximum Velocity: $v_{max} = A\omega$ (at mean position, $x = 0$)
    • Minimum Velocity: $v = 0$ (at extreme positions, $x = \pm A$)
  • Acceleration ($a$): $a = -\omega^2 x$
    • Maximum Acceleration: $a_{max} = \omega^2 A$ (at extremes)
    • Minimum Acceleration: $a = 0$ (at mean position)

• Energy in SHM

  • Kinetic Energy ($K$): $K = \frac{1}{2} m \omega^2 (A^2 - x^2)$
    • Maximum at mean position
  • Potential Energy ($U$): $U = \frac{1}{2} m \omega^2 x^2$
    • Maximum at extreme positions
  • Total Mechanical Energy ($E$): $E = K + U = \frac{1}{2} m \omega^2 A^2$
    • Constant at all positions (Conserved)

• Systems Executing SHM

  • Simple Pendulum
    • Restoring torque provided by gravity
    • Time Period: $T = 2\pi \sqrt{\frac{L}{g}}$
    • Independent of the mass of the bob
  • Spring-Mass System
    • Restoring force provided by spring elasticity
    • Time Period: $T = 2\pi \sqrt{\frac{m}{k}}$
    • Series Springs: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$
    • Parallel Springs: $k_{eq} = k_1 + k_2$

• Types of Oscillations

  • Free Oscillations: Oscillates with natural frequency without external force
  • Damped Oscillations
    • Amplitude decreases exponentially over time due to dissipative forces
    • Damping force: $F_d = -b v$
  • Forced (Driven) Oscillations
    • Driven by external periodic force $F(t) = F_0 \cos(\omega_d t)$
  • Resonance
    • Special case of forced oscillation when driving frequency equals natural frequency ($\omega_d = \omega_0$)
    • Results in maximum amplitude of oscillation