🔧Intro to Mechanics Unit 7 – Oscillations and Simple Harmonic Motion

Key Concepts and Definitions

• Oscillation involves periodic motion or fluctuation between two states or positions
• Period ($T$) represents the time required for one complete oscillation cycle
• Frequency ($f$) measures the number of oscillations per unit time, related to period by $f = \frac{1}{T}$
• Amplitude ($A$) defines the maximum displacement from the equilibrium position
• Angular frequency ($\omega$) describes the rate of change of the oscillation phase angle, expressed as $\omega = 2\pi f$
• Restoring force acts to return the oscillating system to its equilibrium position (Hooke's law for springs, $F = -kx$)
• Phase angle ($\phi$) indicates the position and direction of motion in an oscillation cycle
• Resonance occurs when the frequency of an external force matches the natural frequency of the oscillating system

Types of Oscillations

• Free oscillations occur when a system oscillates without any external force, driven by its own restoring force (pendulum, spring-mass system)
• Damped oscillations involve the gradual decrease in oscillation amplitude due to energy dissipation (friction, air resistance)
  • Overdamped systems return to equilibrium without oscillating
  • Critically damped systems return to equilibrium in the shortest possible time without oscillating
  • Underdamped systems oscillate with decreasing amplitude until reaching equilibrium
• Forced oscillations result from an external periodic force acting on the system (driven pendulum, AC circuits)
• Coupled oscillations involve the transfer of energy between two or more oscillating systems (coupled pendulums, resonating tuning forks)
• Parametric oscillations occur when a system parameter varies periodically, leading to oscillation amplification (parametric pendulum, child on a swing)

Simple Harmonic Motion (SHM)

• SHM is a special case of periodic motion where the restoring force is directly proportional to the displacement from equilibrium
• Characteristics of SHM include sinusoidal displacement, velocity, and acceleration functions
• Examples of SHM systems include mass-spring systems, simple pendulums, and certain electrical circuits (LC circuits)
• The motion in SHM is symmetric about the equilibrium position, with equal time spent on either side
• The velocity is maximum at the equilibrium position and zero at the extremes of the oscillation
• The acceleration is proportional to the displacement but in the opposite direction, always pointing towards the equilibrium position

Mathematical Description of SHM

• The displacement $x(t)$ in SHM is described by the equation $x(t) = A \cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the initial phase angle
• The velocity $v(t)$ is the first derivative of displacement, given by $v(t) = -A\omega \sin(\omega t + \phi)$
• The acceleration $a(t)$ is the second derivative of displacement, expressed as $a(t) = -A\omega^2 \cos(\omega t + \phi)$
• The period of SHM is related to the angular frequency by $T = \frac{2\pi}{\omega}$
• For a mass-spring system, the angular frequency is given by $\omega = \sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass
• For a simple pendulum, the angular frequency is approximated by $\omega \approx \sqrt{\frac{g}{L}}$ for small amplitudes, where $g$ is the acceleration due to gravity and $L$ is the pendulum length

Energy in SHM Systems

• The total energy in an SHM system remains constant and is the sum of kinetic and potential energy
• Kinetic energy $KE = \frac{1}{2}mv^2$ is maximum at the equilibrium position and zero at the extremes
• Potential energy $PE = \frac{1}{2}kx^2$ (for a spring) or $PE = mgh$ (for a pendulum) is maximum at the extremes and zero at the equilibrium position
• Energy is continuously converted between kinetic and potential forms during the oscillation
• The total energy is proportional to the square of the amplitude, $E = \frac{1}{2}kA^2$ (for a spring) or $E = mgA$ (for a pendulum)
• In the absence of damping, the total energy remains constant, and the oscillation continues indefinitely

Damped Oscillations

• Damped oscillations occur when a dissipative force, such as friction or air resistance, acts on the oscillating system
• The amplitude of damped oscillations decreases exponentially over time, described by the equation $A(t) = A_0 e^{-\gamma t}$, where $A_0$ is the initial amplitude and $\gamma$ is the damping coefficient
• The damping coefficient depends on the properties of the system and the surrounding medium (viscosity, friction coefficient)
• Critically damped systems have a specific damping coefficient that results in the fastest return to equilibrium without oscillating
• Overdamped systems have a higher damping coefficient and return to equilibrium slowly without oscillating
• Underdamped systems have a lower damping coefficient and oscillate with decreasing amplitude until reaching equilibrium

Forced Oscillations and Resonance

• Forced oscillations occur when an external periodic force acts on an oscillating system
• The external force can have a frequency different from the natural frequency of the system
• The resulting oscillation frequency matches the frequency of the external force, while the amplitude depends on the relative frequencies and the damping
• Resonance occurs when the frequency of the external force matches the natural frequency of the system
• At resonance, the oscillation amplitude is maximum, as the external force continuously adds energy to the system
• Resonance can be beneficial (musical instruments, radio tuners) or destructive (bridge collapse, building vibrations during earthquakes)
• The sharpness of the resonance peak depends on the damping in the system; higher damping results in a broader peak

Real-World Applications

• Seismic isolation systems use the principles of damped oscillations to protect buildings from earthquake damage
• Shock absorbers in vehicles employ damped oscillations to provide a smooth ride and improve handling
• Resonance is used in musical instruments to produce and amplify sound (strings, air columns)
• Atomic force microscopes (AFMs) utilize the principles of forced oscillations to image surfaces at the nanoscale
• Magnetic resonance imaging (MRI) exploits the resonance of atomic nuclei in a magnetic field to create detailed images of the human body
• Clocks and timekeeping devices rely on the regularity of oscillations in pendulums or quartz crystals
• Electrical power grids must be carefully designed to avoid resonance and ensure stable operation
• Vibration isolation systems, such as those used in sensitive scientific instruments, minimize the impact of external oscillations