⚙️AP Physics C: Mechanics (2025) Unit 7 – Oscillations

Key Concepts and Definitions

• Oscillation involves repetitive motion or variation over time around an equilibrium position
• 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 $(A)$ quantifies the maximum displacement from the equilibrium position
• Angular frequency $(ω)$ characterizes the rate of change of the oscillation phase, given by $ω = 2πf$
• Phase $(φ)$ describes the position of an oscillating system at a specific time relative to its cycle
• Simple harmonic motion (SHM) refers to oscillations with a restoring force directly proportional to the displacement from equilibrium
• Hooke's law states that the restoring force $(\vec{F})$ is proportional to the displacement $(\vec{x})$ from equilibrium, expressed as $\vec{F} = -k\vec{x}$, where $k$ is the spring constant

Simple Harmonic Motion Basics

• SHM occurs when a restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction
• Examples of systems exhibiting SHM include mass-spring systems and simple pendulums
• The motion of an object undergoing SHM is sinusoidal, with displacement varying as a function of time
• The restoring force in SHM is conservative, meaning energy is conserved within the system
• The acceleration of an object in SHM is proportional to its displacement and always directed towards the equilibrium position
  • Acceleration reaches its maximum magnitude at the extremes of the oscillation, where displacement is greatest
  • Acceleration is zero at the equilibrium position, where the object momentarily has no net force acting on it
• Velocity in SHM is maximum at the equilibrium position and zero at the extremes of the oscillation
• The period of a mass-spring system is given by $T = 2π\sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant

Mathematical Models of Oscillations

• The displacement of an object undergoing SHM can be modeled using the equation $x(t) = A\cos(ωt + φ)$, where $A$ is the amplitude, $ω$ is the angular frequency, $t$ is time, and $φ$ is the initial phase
• Velocity in SHM is the first derivative of displacement with respect to time, given by $v(t) = -Aω\sin(ωt + φ)$
• Acceleration in SHM is the second derivative of displacement or the first derivative of velocity, expressed as $a(t) = -Aω^2\cos(ωt + φ)$
• The angular frequency $ω$ is related to the period $T$ by $ω = \frac{2π}{T}$
• For a mass-spring system, the angular frequency is given by $ω = \sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass
• The phase angle $φ$ determines the initial position and velocity of the oscillating object
  • If $φ = 0$, the object starts at its maximum positive displacement with zero initial velocity
  • If $φ = \frac{π}{2}$, the object starts at the equilibrium position with maximum positive velocity
• Combining the equations for displacement, velocity, and acceleration reveals their phase relationships in SHM

Energy in Oscillating Systems

• In SHM, energy is continuously converted between potential and kinetic energy
• Total mechanical energy remains constant in the absence of non-conservative forces (friction or damping)
• Potential energy $(U)$ in a mass-spring system is given by $U = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium
  • Potential energy is maximum at the extremes of the oscillation and zero at the equilibrium position
• Kinetic energy $(K)$ in SHM is expressed as $K = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity
  • Kinetic energy is maximum at the equilibrium position and zero at the extremes of the oscillation
• The total mechanical energy $(E)$ of an undamped oscillating system is the sum of its potential and kinetic energies, given by $E = U + K = \frac{1}{2}kA^2$, where $A$ is the amplitude
• The conservation of energy principle applies to SHM, with energy continuously converting between potential and kinetic forms

Damped Oscillations

• Damped oscillations occur when a non-conservative force, such as friction or air resistance, dissipates energy from the system
• The amplitude of a damped oscillation decreases exponentially over time due to the dissipative force
• The equation for displacement in a damped oscillation is $x(t) = Ae^{-bt}\cos(ω_dt + φ)$, where $b$ is the damping coefficient and $ω_d$ is the damped angular frequency
• The damped angular frequency $ω_d$ is lower than the natural angular frequency $ω$ due to the presence of the damping force
• The damping coefficient $b$ determines the rate at which the amplitude decreases over time
  • For underdamped systems $(b < ω)$, the oscillation gradually decays while still exhibiting periodic behavior
  • For critically damped systems $(b = ω)$, the system returns to equilibrium in the shortest possible time without oscillating
  • For overdamped systems $(b > ω)$, the system slowly returns to equilibrium without oscillating
• The quality factor $Q$ quantifies the degree of damping in an oscillating system, given by $Q = \frac{ω}{2b}$
  • Higher $Q$ values indicate less damping and more oscillations before the system settles to equilibrium

Forced Oscillations and Resonance

• Forced oscillations occur when an external periodic force is applied to an oscillating system
• The external force can be modeled as $F(t) = F_0\cos(ω_ft)$, where $F_0$ is the amplitude of the force and $ω_f$ is the angular frequency of the force
• The resulting motion is a combination of the natural frequency of the system and the frequency of the external force
• The steady-state solution for the displacement in a forced oscillation is $x(t) = A\cos(ω_ft - δ)$, where $A$ is the amplitude and $δ$ is the phase difference between the force and the displacement
• The amplitude of the forced oscillation depends on the ratio of the forcing frequency to the natural frequency of the system
• Resonance occurs when the forcing frequency matches the natural frequency of the system $(ω_f = ω)$
  • At resonance, the amplitude of the oscillation is maximum, and the system absorbs the most energy from the external force
  • The phase difference between the force and the displacement is 90° at resonance
• The quality factor $Q$ also determines the sharpness of the resonance peak and the system's sensitivity to the forcing frequency
• Practical applications of forced oscillations and resonance include tuning musical instruments, designing vibration isolators, and creating resonant circuits in electronics

Applications and Real-World Examples

• Mass-spring systems are used in various applications, such as vehicle suspension systems, vibration isolation in machinery, and seismic protection for buildings
• Simple pendulums are employed in timekeeping devices (pendulum clocks) and as a model for understanding more complex oscillating systems
• Torsional pendulums, consisting of a disk suspended by a wire, are used to measure the rotational inertia of objects and the shear modulus of materials
• Resonance is exploited in musical instruments to amplify sound (acoustic guitars, violins) and create distinct tones
• Microelectromechanical systems (MEMS) utilize oscillating structures for sensing and actuation in devices like accelerometers, gyroscopes, and resonators
• Atomic force microscopy (AFM) relies on the oscillation of a cantilever to map surface topography and measure forces at the nanoscale
• Resonance in structures, such as bridges and buildings, can lead to catastrophic failures if not properly accounted for in the design process (Tacoma Narrows Bridge collapse)
• Oscillations in electrical circuits, such as LC circuits and crystal oscillators, form the basis for many electronic applications, including radio and television broadcasting, GPS, and computer clocks

Problem-Solving Strategies

• Identify the type of oscillation (simple harmonic, damped, or forced) and the relevant variables (amplitude, frequency, spring constant, mass, damping coefficient)
• Determine the appropriate equations for the given scenario, such as the displacement function, velocity, acceleration, or energy expressions
• For mass-spring systems, use the equation $T = 2π\sqrt{\frac{m}{k}}$ to find the period or $ω = \sqrt{\frac{k}{m}}$ for the angular frequency
• In simple pendulums, employ the equation $T = 2π\sqrt{\frac{L}{g}}$ to calculate the period, where $L$ is the pendulum length and $g$ is the acceleration due to gravity
• Apply the principle of conservation of energy to relate potential and kinetic energies at different points in the oscillation
• For damped oscillations, use the equation $x(t) = Ae^{-bt}\cos(ω_dt + φ)$ to model the displacement and determine the damping coefficient $b$ and damped angular frequency $ω_d$
• In forced oscillations, identify the natural frequency of the system and the frequency of the external force to predict resonance conditions
• Utilize the steady-state solution $x(t) = A\cos(ω_ft - δ)$ to find the amplitude and phase difference in forced oscillations
• Remember to consider initial conditions when solving for constants in the displacement, velocity, or acceleration functions