Oscillations: Simple Harmonic Motion, Damped and Driven Oscillations

Oscillations

Introduction to Oscillations

Oscillations are repetitive variations, typically in time, of some measure about a central value or between two or more different states. Many physical systems exhibit oscillatory behavior, such as pendulums, springs, and electrical circuits.

• Oscillatory motion is motion that repeats itself in a regular cycle.

• Simple Harmonic Motion (SHM) is a special type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

15.1 Simple Harmonic Motion (SHM)

Definition and Characteristics

• Simple Harmonic Motion is defined by the equation: $\frac{d^2x}{dt^2} = -\omega^2 x$ where $x$ is displacement and $\omega$ is the angular frequency.

• The general solution is: $x(t) = A \cos(\omega t + \phi)$ where $A$ is amplitude and $\phi$ is the phase constant.

• Period (T): The time for one complete cycle. $T = \frac{2\pi}{\omega}$

• Frequency (f): Number of cycles per second. $f = \frac{1}{T}$

Kinematics of SHM

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

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

• Maximum speed: $v_{max} = \omega A$

• Maximum acceleration: $a_{max} = \omega^2 A$

Units of Frequency

Quantity	Symbol	SI Unit
Period	T	second (s)
Frequency	f	hertz (Hz)
Angular Frequency	\omega	radian/second (rad/s)

15.2 SHM and Circular Motion

Relationship to Circular Motion

• SHM can be visualized as the projection of uniform circular motion onto one axis.

• If a particle moves in a circle of radius $A$ with angular speed $\omega$, its projection on the x-axis follows: $x(t) = A \cos(\omega t + \phi)$

• The phase constant $\phi$ determines the initial position at $t=0$.

15.3 Energy in SHM

Mechanical Energy

• Total mechanical energy in SHM: $E = \frac{1}{2} k A^2$ where $k$ is the spring constant and $A$ is amplitude.

• Kinetic energy: $K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2)$

• Potential energy: $U = \frac{1}{2} k x^2$

• Energy is conserved in ideal SHM (no friction).

15.4 The Dynamics of SHM

Restoring Force and Equation of Motion

• The restoring force is given by Hooke's Law: $F = -kx$

• Newton's second law gives: $m \frac{d^2x}{dt^2} = -kx$

• Solution leads to SHM equations as above.

15.5 Vertical Oscillations

Oscillations in a Vertical Spring

• Gravity shifts the equilibrium position but does not affect the period.

• Period for a mass $m$ on a spring: $T = 2\pi \sqrt{\frac{m}{k}}$

15.6 The Pendulum

Simple Pendulum

• A mass $m$ suspended from a string of length $L$ exhibits SHM for small angles ($\theta$): $T = 2\pi \sqrt{\frac{L}{g}}$

• Small-angle approximation: $\sin\theta \approx \theta$ (in radians) for $\theta$ small.

Physical Pendulum

• For an extended object swinging about a pivot: $T = 2\pi \sqrt{\frac{I}{mgd}}$ where $I$ is the moment of inertia and $d$ is the distance from pivot to center of mass.

15.7 Damped Oscillations

Damping and Energy Loss

• Damping is the effect of friction or resistance, causing amplitude to decrease over time.

• Equation for damped SHM: $x(t) = A e^{-bt/2m} \cos(\omega' t + \phi)$ where $b$ is the damping constant and $\omega'$ is the damped angular frequency.

• Energy decays exponentially: $E(t) = E_0 e^{-bt/m}$

15.8 Driven Oscillations and Resonance

Forced Oscillations and Resonance

• When a periodic force drives an oscillator, the system can resonate if the driving frequency matches the natural frequency.

• At resonance, amplitude is maximized.

• Equation for steady-state amplitude: $A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}$

15.9 Coupled Oscillations and Normal Modes

Coupled Oscillators

• Systems with two or more oscillators connected can exchange energy, leading to normal modes of oscillation.

• Normal modes are patterns where all parts of the system oscillate at the same frequency.

• Equations for coupled oscillators involve solving simultaneous differential equations.

Summary Table: Key Equations and Concepts

Concept	Equation
SHM Position	$x(t) = A \cos(\omega t + \phi)$
Velocity	$v(t) = -\omega A \sin(\omega t + \phi)$
Acceleration	$a(t) = -\omega^2 x(t)$
Period (mass-spring)	$T = 2\pi \sqrt{\frac{m}{k}}$
Period (pendulum)	$T = 2\pi \sqrt{\frac{L}{g}}$
Total Energy	$E = \frac{1}{2} k A^2$
Damped Oscillator	$x(t) = A e^{-bt/2m} \cos(\omega' t + \phi)$
Resonance Amplitude	$A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}$

Applications and Importance

• Oscillatory systems are fundamental in physics, engineering, and nature (e.g., clocks, musical instruments, molecular vibrations).

• Understanding resonance is crucial for designing stable structures and devices.