Oscillations: Simple Harmonic Motion, Energy, Damping, and Resonance

Oscillations and Simple Harmonic Motion (SHM)

Physical Model of a Mass-Spring System

Oscillatory motion occurs when a system moves back and forth about an equilibrium position. The classic example is a block of mass m attached to a spring of stiffness constant k on a frictionless surface. The equilibrium position is where the spring is neither stretched nor compressed (x = 0).

• Displacement (x): Signed distance from equilibrium.

• Amplitude (A): Maximum displacement from equilibrium.

• Restoring Force: Given by Hooke's Law: $F = -kx$

Kinematics of SHM

In SHM, the displacement, velocity, and acceleration vary sinusoidally with time. The general solution for displacement is:

• $x(t) = A \cos(\omega t + \phi)$

• Velocity: $v(t) = -A\omega \sin(\omega t + \phi)$

• Acceleration: $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$

• Period: $T = \frac{2\pi}{\omega}$

• Frequency: $f = \frac{1}{T}$

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

The maxima and minima of these quantities are:

• Maximum displacement: $A$

• Maximum speed: $A\omega$

• Maximum acceleration: $A\omega^2$

Energy in SHM

The total mechanical energy in a simple harmonic oscillator is conserved (if no damping):

• Potential energy: $U(x) = \frac{1}{2}kx^2$

• Kinetic energy: $K = \frac{1}{2}mv^2$

• Total energy: $E = \frac{1}{2}kA^2 = U + K$

At the turning points ($x = \pm A$), all energy is potential; at equilibrium ($x = 0$), all energy is kinetic.

Damped and Driven Oscillations

Damped Harmonic Motion

Real oscillators lose energy due to friction or resistance. The equation of motion for a damped oscillator is:

• $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$

The solution depends on the damping constant b:

• Underdamped: Oscillatory motion with exponentially decreasing amplitude.

• Critically damped: Returns to equilibrium as quickly as possible without oscillating.

• Overdamped: Returns to equilibrium without oscillating, more slowly than critical damping.

Forced Oscillations and Resonance

If an external periodic force acts on the system, the equation becomes:

• $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega t)$

At steady state, the system oscillates at the driving frequency. The amplitude is largest when the driving frequency matches the system's natural frequency—this is resonance.

• Resonant frequency (weak damping): $\omega \approx \omega_0 = \sqrt{\frac{k}{m}}$

• Quality factor: $Q = \frac{m\omega_0}{b}$ (sharpness of resonance)

Additional Topics

Physical Pendulum and Torque

For a rigid body pivoted at a point O, the restoring torque due to gravity is $\tau = -mgd_\perp$, where $d_\perp = h \sin \theta$ is the perpendicular distance from the pivot to the center of mass. The motion can be analyzed similarly to SHM for small angles.

Summary Table: Types of Damped Motion

Case	Behavior
Underdamped	Oscillates with decreasing amplitude
Critically damped	Returns to equilibrium fastest without oscillating
Overdamped	Returns to equilibrium without oscillating, slower than critical

Additional info: The images included above directly illustrate the physical principles, mathematical relationships, and real-world consequences of oscillatory motion, damping, and resonance as discussed in the study notes.