Part I: Oscillations | Chapter 1

The Harmonic Oscillator

The most fundamental oscillatory system in all of physics

1.1 Why the Harmonic Oscillator Matters

The harmonic oscillator is arguably the single most important model in physics. Any system near a stable equilibrium point can be approximated as a harmonic oscillator for small displacements. This universality makes it appear throughout mechanics, electromagnetism, quantum mechanics, and even general relativity.

Consider a general potential energy $V(x)$ with a minimum at $x = x_0$. Expanding in a Taylor series:

$$V(x) = V(x_0) + \underbrace{V'(x_0)}_{=0}(x - x_0) + \frac{1}{2}V''(x_0)(x - x_0)^2 + \cdots$$

Since $V'(x_0) = 0$ at the minimum, the leading-order behavior is quadratic:$V(x) \approx \frac{1}{2}k(x - x_0)^2$ where $k = V''(x_0)$. This is exactly the harmonic oscillator potential.

Physical examples: A mass on a spring, a pendulum for small angles, an LC circuit, molecular vibrations, phonons in a crystal lattice, and the quantum harmonic oscillator that underpins quantum field theory.

1.2 Equation of Motion and General Solution

Figure. A mass m on a spring with constant k, displaced by x from equilibrium. The restoring force F = -kx drives simple harmonic motion with angular frequency ω₀ = √(k/m).

Derivation 1: Newton's Second Law for SHM

For a mass $m$ attached to an ideal spring with spring constant $k$, the restoring force is $F = -kx$. Applying Newton's second law:

$$m\ddot{x} = -kx$$$$\ddot{x} + \omega_0^2 x = 0 \quad \text{where } \omega_0 = \sqrt{\frac{k}{m}}$$

This is a second-order linear ODE with constant coefficients. We try $x = e^{i\alpha t}$:

$$-\alpha^2 e^{i\alpha t} + \omega_0^2 e^{i\alpha t} = 0 \implies \alpha = \pm \omega_0$$

The general solution is a linear combination of the two independent solutions:

General Solution of SHM

$$x(t) = A\cos(\omega_0 t + \phi)$$

where $A$ is the amplitude and $\phi$ is the initial phase, both determined by initial conditions. Equivalently:$x(t) = C_1\cos(\omega_0 t) + C_2\sin(\omega_0 t)$.

Key Parameters

Angular frequency

$$\omega_0 = \sqrt{k/m}$$

rad/s

Period

$$T = \frac{2\pi}{\omega_0} = 2\pi\sqrt{m/k}$$

seconds

Frequency

$$f = \frac{1}{T} = \frac{\omega_0}{2\pi}$$

Hertz (Hz)

1.3 Energy in Simple Harmonic Motion

Derivation 2: Energy Conservation

The kinetic and potential energies of a spring-mass system are:

$$T = \frac{1}{2}m\dot{x}^2 = \frac{1}{2}m\omega_0^2 A^2 \sin^2(\omega_0 t + \phi)$$$$V = \frac{1}{2}kx^2 = \frac{1}{2}k A^2 \cos^2(\omega_0 t + \phi)$$

Using $k = m\omega_0^2$, the total energy is:

Total Energy

$$E = T + V = \frac{1}{2}kA^2 = \frac{1}{2}m\omega_0^2 A^2 = \text{const}$$

Energy oscillates between kinetic and potential forms, but the total is conserved. The time-averaged kinetic and potential energies are each equal to $E/2$.

This equipartition of energy between kinetic and potential forms is a hallmark of the harmonic oscillator and persists even in the quantum mechanical treatment, where the ground-state energy $E_0 = \frac{1}{2}\hbar\omega_0$ is split equally between kinetic and potential contributions.

1.4 The Simple Pendulum

Derivation 3: Pendulum as a Harmonic Oscillator

A mass $m$ suspended from a string of length $\ell$ swings under gravity. The torque about the pivot is $\tau = -mg\ell\sin\theta$. Using $\tau = I\ddot{\theta}$with $I = m\ell^2$:

$$m\ell^2 \ddot{\theta} = -mg\ell\sin\theta$$$$\ddot{\theta} + \frac{g}{\ell}\sin\theta = 0$$

For small angles, $\sin\theta \approx \theta$, so the equation becomes:

Small-Angle Pendulum

$$\ddot{\theta} + \frac{g}{\ell}\theta = 0 \quad \Longrightarrow \quad \omega_0 = \sqrt{\frac{g}{\ell}}, \quad T = 2\pi\sqrt{\frac{\ell}{g}}$$

The period is independent of mass and amplitude (for small oscillations). This isochronous property was discovered by Galileo around 1602 and later used by Huygens to build the first pendulum clock in 1656.

Beyond Small Angles

For larger amplitudes $\theta_0$, the exact period involves an elliptic integral:

$$T = 4\sqrt{\frac{\ell}{g}} \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - \sin^2(\theta_0/2)\sin^2\phi}} \approx T_0\left(1 + \frac{\theta_0^2}{16} + \cdots\right)$$

1.5 The LC Circuit

Derivation 4: Electrical Oscillations

An inductor $L$ and capacitor $C$ connected in a loop form an electrical oscillator. Using Kirchhoff's voltage law:

$$L\frac{dI}{dt} + \frac{Q}{C} = 0$$$$L\ddot{Q} + \frac{Q}{C} = 0 \quad \Longrightarrow \quad \omega_0 = \frac{1}{\sqrt{LC}}$$

The analogy between mechanical and electrical oscillators is exact:

Mechanical	Electrical	Role
Mass $m$	Inductance $L$	Inertia
Spring constant $k$	$1/C$	Restoring force
Displacement $x$	Charge $Q$	Generalized coordinate
Velocity $\dot{x}$	Current $I$	Generalized velocity
Damping $b$	Resistance $R$	Dissipation

Energy in the LC Circuit

$$E = \frac{1}{2}LI^2 + \frac{Q^2}{2C} = \frac{Q_0^2}{2C} = \text{const}$$

Energy oscillates between magnetic (inductor) and electric (capacitor) forms, analogous to kinetic and potential energy in the mass-spring system.

1.6 Initial Conditions and Phase Space

The general solution $x(t) = A\cos(\omega_0 t + \phi)$ contains two constants determined by the initial conditions $x(0) = x_0$ and$\dot{x}(0) = v_0$:

$$A = \sqrt{x_0^2 + \left(\frac{v_0}{\omega_0}\right)^2}$$$$\phi = -\arctan\!\left(\frac{v_0}{\omega_0 x_0}\right)$$

Alternatively, writing $x(t) = C_1\cos(\omega_0 t) + C_2\sin(\omega_0 t)$, we identify $C_1 = x_0$ and $C_2 = v_0/\omega_0$ directly.

Phase Space Representation

Plotting $x$ vs $p = m\dot{x}$ (or equivalently$x$ vs $\dot{x}/\omega_0$) creates a phase portrait. For the SHO, trajectories are ellipses centered at the origin:

$$\frac{x^2}{A^2} + \frac{\dot{x}^2}{(A\omega_0)^2} = 1$$

Each ellipse corresponds to a constant energy $E = \frac{1}{2}m\omega_0^2 A^2$. The motion is always clockwise (for $x$ vs $p$ axes), reflecting the fact that positive momentum means increasing position. The origin is a stable equilibrium (an elliptic fixed point).

Phase space is a powerful tool because it applies to any dynamical system, not just the SHO. For a nonlinear oscillator, the phase portrait reveals qualitative behavior: stable and unstable equilibria, limit cycles, and separatrices. The SHO phase portrait is the linearization of any oscillatory system near a stable equilibrium.

Liouville's theorem: In Hamiltonian mechanics, the phase-space area enclosed by any trajectory is conserved in time. For the SHO, each ellipse has area $2\pi E/\omega_0$. In quantum mechanics, this area is quantized in units of $h$, leading to $E_n = (n + \frac{1}{2})\hbar\omega_0$.

1.6b Superposition and Lissajous Figures

When a particle undergoes SHM simultaneously in two perpendicular directions with possibly different frequencies and phases:

$$x(t) = A_x\cos(\omega_x t + \phi_x)$$$$y(t) = A_y\cos(\omega_y t + \phi_y)$$

The resulting trajectory in the $xy$-plane is a Lissajous figure. When $\omega_x/\omega_y$ is a rational number $p/q$, the path closes after $q$ periods in $x$ and $p$periods in $y$. Key cases:

$\omega_x = \omega_y$

Ellipse (becomes line for $\delta = 0, \pi$; circle for$\delta = \pm\pi/2$ with equal amplitudes). This is precisely the polarization ellipse of Chapter 7.

$\omega_x/\omega_y = 2$

Figure-eight or bowtie pattern, depending on the phase difference. Used historically to determine frequency ratios in acoustics.

Irrational ratio

The trajectory never closes and eventually fills a rectangle — this is the ergodic motion of a 2D oscillator with incommensurate frequencies.

Lissajous figures were first studied by Nathaniel Bowditch (1815) and later popularized by Jules Antoine Lissajous (1857), who displayed them using tuning forks and mirrors. Today they appear on oscilloscopes for comparing signal frequencies and phases.

1.6c Preview: The Quantum Harmonic Oscillator

In quantum mechanics, the harmonic oscillator becomes even more important. The Hamiltonian is:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega_0^2\hat{x}^2$$

The energy eigenvalues are quantized:

Quantum Energy Levels

$$E_n = \hbar\omega_0\!\left(n + \frac{1}{2}\right) \quad n = 0, 1, 2, \ldots$$

The ground state energy $E_0 = \frac{1}{2}\hbar\omega_0$ is the zero-point energy, a purely quantum effect with no classical analog. Equal spacing $\Delta E = \hbar\omega_0$ between levels means the quantum oscillator emits/absorbs photons of a single frequency, matching the classical result.

The quantum harmonic oscillator is exactly solvable using either the power series method (Hermite polynomials) or the elegant algebraic method of raising and lowering operators ($\hat{a}^\dagger$ and $\hat{a}$). This algebraic approach generalizes to quantum field theory, where each mode of the electromagnetic field is a quantum harmonic oscillator — the "photon" is a quantum of excitation.

Connection to fields: A free quantum field is an infinite collection of harmonic oscillators, one for each momentum mode. The vacuum energy is the sum of all zero-point energies, leading to the Casimir effect and the cosmological constant problem.

1.6 Complex Notation

Derivation 5: Phasor Representation

A powerful technique is to represent oscillations using complex exponentials. We write:

$$\tilde{x}(t) = \tilde{A}e^{i\omega_0 t} \quad \text{where } \tilde{A} = Ae^{i\phi}$$$$x(t) = \text{Re}[\tilde{x}(t)] = A\cos(\omega_0 t + \phi)$$

The complex amplitude $\tilde{A}$ encodes both the amplitude and phase. This notation simplifies algebraic manipulations enormously: differentiation becomes multiplication by $i\omega_0$, and adding oscillations of the same frequency becomes adding complex numbers (phasor addition).

For the velocity and acceleration:

$$\tilde{v} = i\omega_0 \tilde{x} \quad \Longrightarrow \quad v(t) = -A\omega_0\sin(\omega_0 t + \phi)$$$$\tilde{a} = -\omega_0^2 \tilde{x} \quad \Longrightarrow \quad a(t) = -A\omega_0^2\cos(\omega_0 t + \phi)$$

Why complex notation? When we add driven forces and damping in the next chapter, the complex exponential approach turns a second-order ODE into a simple algebraic equation. This is the standard approach in electrical engineering (AC circuit analysis) and throughout wave physics.

1.7 Applications

Quartz Crystal Oscillators

The piezoelectric effect in quartz crystals creates a mechanical resonator with$Q \sim 10^5$. The resonant frequency is extremely stable, making quartz oscillators the basis of modern timekeeping in watches, computers, and telecommunications.

MEMS Accelerometers

Micro-electromechanical systems (MEMS) in smartphones use tiny spring-mass systems etched in silicon. Measuring the displacement of the mass reveals acceleration, enabling screen rotation, step counting, and navigation.

Molecular Vibrations

Atoms in molecules vibrate about their equilibrium bond lengths. These vibrations are well-modeled as harmonic oscillators and are probed by infrared spectroscopy. Each vibrational mode has a characteristic frequency that identifies functional groups.

Gravitational Wave Detectors

LIGO uses suspended mirrors as test masses in a giant interferometer. Understanding the harmonic oscillator response of these suspensions to gravitational waves was essential for the 2015 detection of gravitational waves from merging black holes.

1.8 Historical Context

Galileo Galilei (1602): Discovered the isochronous property of the pendulum by timing the swings of a chandelier in the Pisa Cathedral using his pulse as a clock. He realized that the period of oscillation was independent of amplitude for small swings.

Robert Hooke (1660): Published his law of elasticity, $F = -kx$, initially as an anagram "ceiiinosssttuv" (Latin for "ut tensio, sic vis" — as the extension, so the force). This linear restoring force is the defining feature of the harmonic oscillator.

Christiaan Huygens (1656): Built the first pendulum clock, achieving an accuracy of about 15 seconds per day — a dramatic improvement over earlier mechanical clocks. His work on the cycloidal pendulum showed deep understanding of oscillatory mechanics.

Leonhard Euler (1739): Provided the complete mathematical framework for solving linear differential equations with constant coefficients, including the complex exponential solutions that we use today.

Lord Rayleigh (1877): In "The Theory of Sound," Rayleigh systematized the study of vibrations and oscillations, establishing many results we still use, including the Rayleigh quotient for estimating natural frequencies.

1.9 Python Simulation

This simulation visualizes simple harmonic motion, energy exchange, the pendulum period as a function of amplitude, and the phase-space trajectory.

Simple Harmonic Oscillator: SHM, Energy, Pendulum, and Phase Space

Python

script.py125 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Simple Harmonic Oscillator: Comprehensive Visualization
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Simple Harmonic Oscillator', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: SHM displacement, velocity, acceleration ---
ax1 = axes[0, 0]
omega0 = 2 * np.pi  # 1 Hz
A = 1.0
phi = 0.0
t = np.linspace(0, 3, 1000)

x = A * np.cos(omega0 * t + phi)
v = -A * omega0 * np.sin(omega0 * t + phi)
a = -A * omega0**2 * np.cos(omega0 * t + phi)

ax1.plot(t, x, color='#22d3ee', linewidth=2, label='x(t)')
ax1.plot(t, v / omega0, color='#f97316', linewidth=2, label='v(t) / omega', alpha=0.8)
ax1.plot(t, a / omega0**2, color='#a855f7', linewidth=2, label='a(t) / omega^2', alpha=0.8)
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Normalized amplitude')
ax1.set_title('Displacement, Velocity, Acceleration')
ax1.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax1.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)

# --- Panel 2: Energy exchange ---
ax2 = axes[0, 1]
k = 1.0
m = 1.0 / omega0**2 * k
KE = 0.5 * m * (A * omega0 * np.sin(omega0 * t + phi))**2
PE = 0.5 * k * (A * np.cos(omega0 * t + phi))**2
E_total = KE + PE

ax2.fill_between(t, 0, KE, color='#f97316', alpha=0.4, label='Kinetic energy')
ax2.fill_between(t, KE, KE + PE, color='#22d3ee', alpha=0.4, label='Potential energy')
ax2.plot(t, E_total, color='white', linewidth=2, label='Total energy', linestyle='--')
ax2.set_xlabel('Time (s)')
ax2.set_ylabel('Energy (J)')
ax2.set_title('Energy Exchange in SHM')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Pendulum period vs amplitude ---
ax3 = axes[1, 0]
from scipy import integrate

theta0_values = np.linspace(0.01, np.pi * 0.95, 200)
T_exact = np.zeros_like(theta0_values)
g_over_l = 1.0
T_shm = 2 * np.pi / np.sqrt(g_over_l)

for i, th0 in enumerate(theta0_values):
    def integrand(phi_var):
        k_param = np.sin(th0 / 2)
        val = 1 - k_param**2 * np.sin(phi_var)**2
        if val <= 0:
            return 0.0
        return 1.0 / np.sqrt(val)
    result, _ = integrate.quad(integrand, 0, np.pi / 2)
    T_exact[i] = 4.0 / np.sqrt(g_over_l) * result

T_approx = T_shm * (1 + theta0_values**2 / 16)

ax3.plot(np.degrees(theta0_values), T_exact / T_shm, color='#22d3ee', linewidth=2, label='Exact (elliptic integral)')
ax3.plot(np.degrees(theta0_values), T_approx / T_shm, color='#f97316', linewidth=2, linestyle='--', label='Approx: 1 + theta^2/16')
ax3.axhline(y=1, color='white', linewidth=0.5, alpha=0.3, linestyle=':')
ax3.set_xlabel('Amplitude (degrees)')
ax3.set_ylabel('T / T_SHM')
ax3.set_title('Pendulum Period vs Amplitude')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Phase space trajectory ---
ax4 = axes[1, 1]
# Multiple amplitudes
for A_val in [0.3, 0.6, 1.0, 1.5]:
    x_ps = A_val * np.cos(omega0 * t + phi)
    v_ps = -A_val * omega0 * np.sin(omega0 * t + phi)
    ax4.plot(x_ps, v_ps / omega0, linewidth=1.5, alpha=0.8)

ax4.set_xlabel('x')
ax4.set_ylabel('v / omega')
ax4.set_title('Phase Space (Ellipses for Different Amplitudes)')
ax4.set_aspect('equal')
ax4.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)
ax4.axvline(x=0, color='white', linewidth=0.3, alpha=0.3)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("SIMPLE HARMONIC OSCILLATOR: NUMERICAL RESULTS")
print("=" * 60)
print("\n--- SHM Parameters ---")
print("  omega_0 = " + str(round(omega0, 4)) + " rad/s")
print("  Period T = " + str(round(2 * np.pi / omega0, 4)) + " s")
print("  Frequency f = " + str(round(omega0 / (2 * np.pi), 4)) + " Hz")
print("  Amplitude A = " + str(A))
print("  Total energy E = " + str(round(0.5 * k * A**2, 4)) + " J")
print("\n--- Pendulum Period Corrections ---")
for angle in [5, 15, 30, 60, 90, 120, 150]:
    idx = np.argmin(np.abs(np.degrees(theta0_values) - angle))
    ratio = T_exact[idx] / T_shm
    print("  theta_0 = " + str(angle) + " deg: T/T_SHM = " + str(round(ratio, 4)))
print("\n--- Phase Space ---")
print("  Orbits are ellipses: x^2/A^2 + v^2/(A*omega)^2 = 1")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

EOM: $\ddot{x} + \omega_0^2 x = 0$
Solution: $x = A\cos(\omega_0 t + \phi)$
Energy: $E = \frac{1}{2}kA^2$
Pendulum: $T = 2\pi\sqrt{\ell/g}$
LC circuit: $\omega_0 = 1/\sqrt{LC}$

Key Insights

Any stable equilibrium is approximately harmonic
Energy is conserved: $\langle T \rangle = \langle V \rangle = E/2$
Period is amplitude-independent (isochrony)
Mechanical-electrical analogy is exact
Complex notation simplifies calculations

Practice Problems

Problem 1: Period of a Compound PendulumA uniform rod of length $L = 1.2$ m and mass $M = 2$ kg is pivoted at a point $d = 0.3$ m from its center of mass. Find the period of small oscillations.

Solution:

1. For a compound (physical) pendulum, the period of small oscillations is:

$$T = 2\pi\sqrt{\frac{I_{\text{pivot}}}{Mgd}}$$

2. The moment of inertia of a uniform rod about its center is $I_{cm} = \frac{1}{12}ML^2$. By the parallel axis theorem:

$$I_{\text{pivot}} = I_{cm} + Md^2 = \frac{1}{12}(2)(1.2)^2 + 2(0.3)^2 = 0.24 + 0.18 = 0.42\;\text{kg·m}^2$$

3. The restoring torque about the pivot is $\tau = -Mgd\sin\theta \approx -Mgd\theta$ for small angles.

4. Substituting into the period formula:

$$T = 2\pi\sqrt{\frac{0.42}{2 \times 9.81 \times 0.3}} = 2\pi\sqrt{\frac{0.42}{5.886}}$$

5. Evaluating:

$$\boxed{T = 2\pi\sqrt{0.0713} = 2\pi \times 0.267 = 1.68\;\text{s}}$$

The equivalent simple pendulum length is $\ell_{eq} = I_{\text{pivot}}/(Md) = 0.42/0.6 = 0.70$ m. This is longer than $d$ because the distributed mass increases the effective inertia.

Problem 2: Energy of a Spring-Mass SystemA 0.5 kg mass on a spring ($k = 200$ N/m) oscillates with amplitude 4 cm. Calculate the total energy, maximum velocity, and the position where kinetic and potential energies are equal.

Solution:

1. The total energy of a simple harmonic oscillator equals the maximum potential energy:

$$E = \frac{1}{2}kA^2 = \frac{1}{2}(200)(0.04)^2 = 0.16\;\text{J}$$

2. Maximum velocity occurs at the equilibrium position ($x = 0$) where all energy is kinetic:

$$\frac{1}{2}mv_{\max}^2 = E \implies v_{\max} = \sqrt{\frac{2E}{m}} = \sqrt{\frac{2(0.16)}{0.5}} = \sqrt{0.64}$$

$$\boxed{v_{\max} = 0.80\;\text{m/s}}$$

3. Alternatively, $v_{\max} = \omega A = \sqrt{k/m}\,A = \sqrt{400}\times 0.04 = 20 \times 0.04 = 0.80$ m/s. Confirmed.

4. When KE = PE, each equals $E/2$. Since $PE = \frac{1}{2}kx^2 = E/2$:

$$\frac{1}{2}kx^2 = \frac{1}{2}\left(\frac{1}{2}kA^2\right) \implies x^2 = \frac{A^2}{2}$$

5. Therefore:

$$\boxed{x = \pm\frac{A}{\sqrt{2}} = \pm\frac{4}{\sqrt{2}} \approx \pm 2.83\;\text{cm}}$$

This is the position at phase angle $\pi/4$ (45°) from equilibrium, which is roughly 70.7% of the amplitude.

Problem 3: LC Circuit Resonance FrequencyAn LC circuit has $L = 50$ mH and $C = 20$ $\mu$F. Find the resonance frequency, the period of oscillation, and the peak current if the capacitor is initially charged to 12 V.

Solution:

1. The angular resonance frequency of an LC circuit is:

$$\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{(50 \times 10^{-3})(20 \times 10^{-6})}} = \frac{1}{\sqrt{10^{-6}}} = 1000\;\text{rad/s}$$

2. The frequency and period:

$$\boxed{f_0 = \frac{\omega_0}{2\pi} = \frac{1000}{2\pi} = 159.2\;\text{Hz}, \quad T = \frac{1}{f_0} = 6.28\;\text{ms}}$$

3. The initial energy stored in the capacitor is:

$$E = \frac{1}{2}CV_0^2 = \frac{1}{2}(20 \times 10^{-6})(12)^2 = 1.44 \times 10^{-3}\;\text{J}$$

4. At peak current, all energy is in the inductor ($\frac{1}{2}LI_{\max}^2 = E$):

$$I_{\max} = \sqrt{\frac{2E}{L}} = \sqrt{\frac{2(1.44 \times 10^{-3})}{50 \times 10^{-3}}} = \sqrt{0.0576}$$

5. The peak current is:

$$\boxed{I_{\max} = 0.24\;\text{A} = 240\;\text{mA}}$$

Equivalently, $I_{\max} = V_0\sqrt{C/L} = 12\sqrt{(20\times10^{-6})/(50\times10^{-3})} = 12 \times 0.02 = 0.24$ A. The LC circuit is the electrical analog of the spring-mass system with $L \leftrightarrow m$ and $1/C \leftrightarrow k$.

Problem 4: Quality Factor from Amplitude DecayA damped oscillator with natural frequency $f_0 = 500$ Hz has its amplitude decrease to $e^{-1}$ of its initial value after 200 oscillations. Calculate the damping coefficient $\gamma$ and the quality factor $Q$.

Solution:

1. The amplitude of a damped oscillator decays as $A(t) = A_0\,e^{-\gamma t/2}$. The amplitude drops to $e^{-1}$ when:

$$\frac{\gamma t}{2} = 1 \implies t = \frac{2}{\gamma}$$

2. This occurs after 200 oscillations, so $t = 200T = 200/f_0 = 200/500 = 0.4$ s:

$$\frac{2}{\gamma} = 0.4 \implies \boxed{\gamma = 5.0\;\text{s}^{-1}}$$

3. The quality factor is defined as:

$$Q = \frac{\omega_0}{\gamma} = \frac{2\pi f_0}{\gamma} = \frac{2\pi \times 500}{5.0} = \frac{3141.6}{5.0}$$

4. Therefore:

$$\boxed{Q = 628}$$

5. Physical interpretation: $Q = 2\pi \times (\text{energy stored})/(\text{energy lost per cycle})$. The system loses a fraction $\Delta E/E = 2\pi/Q \approx 0.01$ of its energy per cycle. After $Q/\pi \approx 200$ cycles, the energy drops by $e^{-2}$ (amplitude by $e^{-1}$), consistent with the given data. This high Q indicates a lightly damped resonator.

Problem 5: Coupled Oscillators — Normal ModesTwo identical masses $m$ are connected by springs: each mass is attached to a wall by a spring of constant $k$, and they are coupled to each other by a spring of constant $k_c$. Find the normal mode frequencies. Evaluate for $m = 0.1$ kg, $k = 40$ N/m, $k_c = 10$ N/m.

Solution:

1. The equations of motion for displacements $x_1$ and $x_2$ from equilibrium are:

$$m\ddot{x}_1 = -kx_1 - k_c(x_1 - x_2), \quad m\ddot{x}_2 = -kx_2 - k_c(x_2 - x_1)$$

2. Introduce symmetric and antisymmetric coordinates: $q_+ = x_1 + x_2$, $q_- = x_1 - x_2$:

$$m\ddot{q}_+ = -kq_+, \quad m\ddot{q}_- = -(k + 2k_c)q_-$$

3. The normal mode frequencies are:

$$\omega_+ = \sqrt{\frac{k}{m}}, \quad \omega_- = \sqrt{\frac{k + 2k_c}{m}}$$

4. Substituting numerical values:

$$\omega_+ = \sqrt{\frac{40}{0.1}} = 20\;\text{rad/s} \quad (f_+ = 3.18\;\text{Hz})$$

$$\omega_- = \sqrt{\frac{40 + 20}{0.1}} = \sqrt{600} = 24.5\;\text{rad/s} \quad (f_- = 3.90\;\text{Hz})$$

5. Results:

$$\boxed{\omega_+ = 20.0\;\text{rad/s}\;(\text{in-phase}), \quad \omega_- = 24.5\;\text{rad/s}\;(\text{out-of-phase})}$$

In the symmetric mode ($\omega_+$), both masses move together and the coupling spring is never stretched. In the antisymmetric mode ($\omega_-$), the masses move oppositely, and the coupling spring contributes additional restoring force $2k_c$, raising the frequency. The beat frequency for energy exchange is $\Delta\omega = 4.5$ rad/s.

← Part I Overview Next: Damped Oscillators →

Mechanical	Electrical	Role
Mass \(m\)	Inductance \(L\)	Inertia
Spring constant \(k\)	\(1/C\)	Restoring force
Displacement \(x\)	Charge \(Q\)	Generalized coordinate
Velocity \(\dot{x}\)	Current \(I\)	Generalized velocity
Damping \(b\)	Resistance \(R\)	Dissipation

The Harmonic Oscillator

1.1 Why the Harmonic Oscillator Matters

1.2 Equation of Motion and General Solution

Derivation 1: Newton's Second Law for SHM

General Solution of SHM

Key Parameters

Angular frequency

Period

Frequency

1.3 Energy in Simple Harmonic Motion

Derivation 2: Energy Conservation

Total Energy

1.4 The Simple Pendulum

Derivation 3: Pendulum as a Harmonic Oscillator

Small-Angle Pendulum

Beyond Small Angles

1.5 The LC Circuit

Derivation 4: Electrical Oscillations

Energy in the LC Circuit

1.6 Initial Conditions and Phase Space

Phase Space Representation

1.6b Superposition and Lissajous Figures

\(\omega_x = \omega_y\)

\(\omega_x/\omega_y = 2\)

Irrational ratio

1.6c Preview: The Quantum Harmonic Oscillator

Quantum Energy Levels

1.6 Complex Notation

Derivation 5: Phasor Representation

1.7 Applications

Quartz Crystal Oscillators

MEMS Accelerometers

Molecular Vibrations

Gravitational Wave Detectors

1.8 Historical Context

1.9 Python Simulation

Simple Harmonic Oscillator: SHM, Energy, Pendulum, and Phase Space

Summary

Core Equations

Key Insights

Practice Problems

Solution:

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Solution: