Study Guide: Periodic Motion and Damped Oscillations

Periodic Motion

Simple Harmonic Motion (SHM)

Simple harmonic motion describes the oscillatory motion of a system where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

• Restoring Force: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

• Equation of Motion: $m\ddot{x} + kx = 0$

• Solution: $x(t) = A\cos(\omega t + \phi)$, where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is phase constant.

• Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$

• Period: $T = \frac{2\pi}{\omega}$

• Frequency: $f = \frac{1}{T}$

• Example: A mass $m$ attached to a spring with constant $k$ oscillates with period $T = 2\pi\sqrt{\frac{m}{k}}$.

Small Angle Approximation for Pendulums

For a pendulum of length $l$ and mass $m$, the restoring force is $F = -mg\sin\theta$. For small angles ($\theta \ll 1$ rad), $\sin\theta \approx \theta$.

• Equation of Motion: $m l \ddot{\theta} + mg\theta = 0$

• Angular Frequency: $\omega = \sqrt{\frac{g}{l}}$

• Interpretation: The frequency increases as the length decreases or gravity increases.

Damped Oscillations

Types of Damping

Damping refers to the effect of dissipative forces (like friction or air resistance) on oscillatory motion.

• Underdamped: Oscillations decrease in amplitude over time but persist ($\gamma^2 < 4mk$).

• Critically Damped: System returns to equilibrium as quickly as possible without oscillating ($\gamma^2 = 4mk$).

• Overdamped: System returns to equilibrium without oscillating, but slower than the critically damped case ($\gamma^2 > 4mk$).

Equation of Motion for Damped Oscillator

• General Form: $m\ddot{x} + b\dot{x} + kx = 0$, where $b$ is the damping coefficient.

• Quality Factor (Q): $Q = \frac{\omega_0}{2\beta}$, where $\omega_0 = \sqrt{\frac{k}{m}}$ and $\beta = \frac{b}{2m}$.

• Angular Frequency of Damped Oscillation: $\omega' = \sqrt{\omega_0^2 - \beta^2}$

• Time Constant: $\tau = \frac{1}{\beta}$

• Example: For $m = 0.5$ kg, $k = 200$ N/m, $b = 1$ kg/s, $\omega_0 = 20$ rad/s, $\beta = 1$ s$^{-1}$, $Q = 10$.

Driven Oscillations and Resonance

Forced Oscillations

When an external periodic force acts on a damped oscillator, the system exhibits forced oscillations.

• Equation of Motion: $m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t)$

• Steady-State Amplitude: $A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\beta\omega)^2}}$

• Resonance: Maximum amplitude occurs when the driving frequency $\omega$ is close to the natural frequency $\omega_0$.

• Example: For $F_0 = 10$ N, $m = 1$ kg, $k = 100$ N/m, $b = 2$ kg/s, plot $A(\omega)$ versus $\omega$ to observe resonance peak.

Conceptual Questions and True/False

• Increasing mass in a simple pendulum: Does not affect the frequency ($\omega = \sqrt{g/l}$).

• Damping and frequency: In an underdamped system, the oscillation frequency decreases as damping increases.

• Long-term behavior: In damped systems, the amplitude decays to zero over time.

• Driven oscillators: After a long time, the amplitude and phase are determined by the driving force, not the initial conditions.

Summary Table: Damped Oscillator Types

Type	Condition	Behavior
Underdamped	$\gamma^2 < 4mk$	Oscillatory decay
Critically damped	$\gamma^2 = 4mk$	No oscillation, fastest return to equilibrium
Overdamped	$\gamma^2 > 4mk$	No oscillation, slow return to equilibrium

Key Formulas

• $\omega_0 = \sqrt{\frac{k}{m}}$

• $\omega' = \sqrt{\omega_0^2 - \beta^2}$

• $Q = \frac{\omega_0}{2\beta}$

• $A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\beta\omega)^2}}$

• $T = 2\pi\sqrt{\frac{m}{k}}$

Additional info: This guide covers the main concepts, equations, and qualitative understanding of periodic motion, damped oscillations, and resonance, as relevant to introductory college physics.