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Chapter 13: Oscillations

13.1 Introduction

• Daily examples of motions include rectilinear motion and projectile motion, which are non-repetitive.

• Uniform circular motion and the orbital motion of planets are periodic as they repeat after certain intervals.

• Oscillatory motion is characterized by repetitive to-and-fro movements about a mean position.

• Examples of oscillatory motion:

  • Rocking in a cradle

  • Swinging on a swing

  • Pendulum of a wall clock

  • Piston in a steam engine

• Importance in physics:

  • Fundamental concepts such as period, frequency, amplitude, and phase arise in oscillatory motion.

  • Musical sounds produced by vibrating strings in instruments.

  • Vibrations of membranes in drums and telecommunication devices.

  • AC power supply shows voltage oscillating above and below mean value.

13.2 Periodic and Oscillatory Motions

• Definition of periodic motion: Motion that repeats itself at regular intervals.

• Examples and graphing:

  1. Insect climbing a ramp

  2. Child climbing and descending steps

  3. Bouncing ball between palm and ground

• Each occurrence derives a height versus time graph demonstrating periodic motion with a defined period (T).

• Equilibrium position: No net external forces act, and if displaced, restoring force arises, leading to oscillations.

• Not all periodic motions are oscillatory; e.g., circular motion.

• Vibrations vs. oscillations:

  • Low frequency = oscillation

  • High frequency = vibration

13.2.1 Period and Frequency

• Period (T): Smallest time interval after which motion repeats (SI unit: second).

• Frequency (v): Number of repetitions per unit time, given by the relation: ν = 1/T.

• SI unit of frequency is hertz (Hz), defined as: 1 Hz = 1 oscillation per second.

13.3 Simple Harmonic Motion (SHM)

• Motion is termed simple harmonic when displacement (x) varies as:

  • x(t) = A cos(ωt + φ)

  • Definitions:

    • A: Amplitude (maximum displacement)

    • ω: Angular frequency

    • φ: Phase constant

• Characteristics of SHM:

  • The motion is periodic and sinusoidal.

  • Maximum speed occurs at the mean position & zero speed at extreme displacements.

  • Mathematical representation:

    • SHM is exhibited as projections of uniform circular motion, demonstrating the interconnectedness of circular and oscillatory motion.

• Graphical Depiction:

  • Trained understanding with plots indicating displacement, velocity, and acceleration.

13.4 SHM and Uniform Circular Motion

• Projection of a particle moving in circular motion on a diameter resembles SHM.

• Illustrative examples link circular motion to oscillatory behavior in simple models.

13.5 Velocity and Acceleration in SHM

• Formulations for velocity and acceleration based on SHM characteristics:

  • Velocity: v(t) = -ωA sin(ωt + φ)

  • Acceleration: a(t) = -ω²A cos(ωt + φ)

• Insights into how acceleration is proportional to displacement and directed towards equilibrium.

13.6 Force Law for SHM

• Applying Newton’s second law:

  • F(t) = ma = -mω² x(t)

• Understanding restoring force related to displacement, leading to SHM definitions.

• Any oscillation under such motion falls under linear harmonic oscillators.

13.7 Energy in SHM

• Kinetic energy (K) and potential energy (U) expressions of a particle in SHM:

  • K = 1/2 mv²

  • U = 1/2 kx²

• Total energy (E) remains constant in an ideal system:

  • E = K + U = 1/2 kA².

13.8 The Simple Pendulum

• Observation of periodic motion in a pendulum setup through small angle displacement, leading to SHM properties.

• Derivation of time period (T) for a simple pendulum:

  • T = 2π √(L/g)

  • L: length of the pendulum, g: acceleration due to gravity.

Summary

1. Periodic motion: repeats at intervals.

2. Frequency and period are intrinsically linked.

3. SHM defined by sinusoidal displacement functions.

4. Circular motion relates to SHM via projections.

5. Energy conservation laws govern kinetic and potential energy changes in SHM.

6. Understanding simple pendulum dynamics reaffirms the principles of oscillation in physical systems.

Points to Ponder

• Periodicity does not imply SHM except under specific force laws.

• The amplitude does not affect the period in SHM.

Exercises

• A variety of exercises can be used to solidify concepts learned about periodic and oscillatory motions, SHM, and related systems.