Unit: 3 Harmonic Oscillator | BSc Physics Notes | Kannur University | Semester 3

 

This unit explains the fundamentals of the harmonic oscillator, focusing on simple harmonic motion (SHM), damped, and forced oscillations. Understand the physics behind restoring forces, differential equations of SHM, and energy distribution in oscillatory systems. Learn how damping affects motion and how resonance occurs when a system is driven at its natural frequency. This topic is crucial for Kannur University BSc Physics students, offering deep insights into mechanical vibrations, time period, amplitude, and phase. Applications include pendulums, springs, and even AC circuits. Master the mathematical formulation and physical understanding needed for conceptual clarity and exam success.

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Semester 3 | Kannur University | Notes | BSc Physics

Unit 1:  Non inertial Systems and Fictious Forces: Link

Unit 2:  Central Force Motions: Link

Unit 3:  Harmonic Oscillator: Link

Unit 4:  Waves: Link

Here's a helpful visual illustrating simple harmonic motion, showcasing plots of displacement $x(t)$, velocity $v(t)$, and acceleration $a(t)$ over time. This diagram is especially useful for understanding the phase relationships and amplitude-time behavior of the harmonic oscillator.

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Unit 3: Harmonic Oscillator

BSc Physics (Kannur University – Semester 3)

1. Syllabus Overview

According to Kannur University's Semester 3 syllabus, this unit spans approximately 8 hours and covers both classical and damped/forced harmonic oscillator systems:

• Simple harmonic motion (SHM): definitions, solutions, energy, and average values

• Damped harmonic oscillator: energy aspects, Q-factor, graphical analysis, solutions

• Forced (undamped) oscillator and concepts of resonance
  (Kannur University, stpius.ac.in)

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2. Core Concepts & Definitions

a) Simple Harmonic Motion (SHM)

• Definition: Motion where restoring force is directly proportional to displacement and directed toward equilibrium (e.g., $F = -kx$); motion is sinusoidal.

• Equation of motion:

  $$m\ddot x + kx = 0 \quad \Rightarrow \quad \ddot x + \omega^2 x = 0,\;\; \omega = \sqrt{\frac{k}{m}}$$

  (Note: Common in physics and general textbooks) (Wikipedia)

• General solution:

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

  where $A$ is amplitude and $\phi$ is the phase. Similarly for velocity and acceleration:

  $$v(t) = -A\omega \sin(\omega t + \phi), \quad a(t) = -\omega^2 A \cos(\omega t + \phi)$$

• Time period and frequency:

  $$T = \frac{2\pi}{\omega}, \quad f = \frac{1}{T} = \frac{\omega}{2\pi}$$

b) Energy in SHM

• Kinetic and potential energies:

  $$K = \frac{1}{2} m v^2, \quad U = \frac{1}{2} k x^2$$

• Total mechanical energy:

  $$E = K + U = \frac{1}{2} k A^2 = \text{constant}$$

• Average energies over a cycle: half the total energy resides in kinetic energy and half in potential energy (on average).

c) Examples of SHM

• Mass-spring systems

• Simple and compound pendulums

• Torsional oscillators

• Vibrations of diatomic molecules (modeled as two-body oscillators) (Scribd)

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3. Damped Harmonic Oscillator

a) Differential Equation with Damping

• Includes damping force (typically proportional to velocity):

  $$m\ddot x + b\dot x + kx = 0$$

  where $b$ is the damping coefficient. (Studocu)

b) Types of Damping

• Underdamped ($b^2 < 4mk$): Oscillatory decay

• Critically damped ($b^2 = 4mk$): Returns to equilibrium most quickly without oscillation

• Overdamped ($b^2 > 4mk$): Slow return, no oscillation

c) Energy Dissipation & Q-Factor

• System loses energy over time due to damping, with amplitude decaying exponentially.

• Quality factor (Q) measures how underdamped an oscillator is:

  $$Q \approx \frac{m\omega}{b}$$

  Higher Q implies slower energy loss per cycle.

d) Graphical Analysis

• Amplitude decays over time, energy dissipates, and response depends on damping regime. (Kannur University, Studocu)

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4. Forced (Undamped) Oscillator & Resonance

• Driven by external force of form $F_0 \cos(\omega_{\text{drive}} t)$:

  $$m\ddot x + b\dot x + kx = F_0 \cos(\omega_{\text{drive}} t)$$

• In the undamped limit ($b \to 0$), the system experiences resonance when driving frequency approaches the natural frequency, leading to large amplitude responses. (Kannur University, stpius.ac.in)

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5. Suggested Notes Outline

1. Introduction

   • Definition of SHM

   • Equations of motion, solution forms

2. Energy Considerations

   • Expressions for kinetic, potential, and total energy

   • Time-average values over a cycle

3. Examples & Nomenclature

   • Pendulum, mass-spring, molecular vibrations (two-body system)

4. Damped Oscillator

   • Equation including damping

   • Classify damping regimes

   • Q-factor and visual amplitude decay

5. Forced & Resonance

   • Driven oscillator equation

   • Resonance concept and undamped limits

6. Practice Problems

   • Compute T and $\omega$

   • Analyze energy distribution

   • Solve damping cases, calculate Q, identify resonant response

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Study Tips

• Start with definitions and physical meaning, then derive equations.

• Use sketches and plots (like $x(t), v(t), a(t)$) to visualize motion.

• Solve mechanics problems: natural frequency, amplitude, damping behavior.

• Understand how Q-factor changes energy dissipation and resonance sharpness.

• Apply concepts to physical systems: molecular vibrations, pendulums, electronic analogs.