⚙️AP Physics C: Mechanics Unit 6 – Oscillations in AP Physics C: Mechanics

Key Concepts

• Oscillations involve periodic motion back and forth around an equilibrium position
• Simple harmonic motion (SHM) is a specific type of oscillation where the restoring force is directly proportional to the displacement from equilibrium
• Springs that obey Hooke's law ($F=-kx$) exhibit SHM when stretched or compressed
  • $k$ represents the spring constant, a measure of the spring's stiffness
  • $x$ represents the displacement from the equilibrium position
• Pendulums, both simple and physical, can exhibit SHM for small angles of displacement
• Energy in oscillating systems alternates between kinetic energy (KE) and potential energy (PE) with the total energy remaining constant in the absence of damping
• Damped oscillations occur when energy is dissipated from the system, causing the amplitude of oscillation to decrease over time
  • Examples of damping include air resistance and friction
• Forced oscillations occur when an external periodic force is applied to an oscillating system
  • Resonance occurs when the frequency of the external force matches the natural frequency of the system, leading to large amplitude oscillations

Simple Harmonic Motion

• SHM is characterized by a restoring force that is directly proportional to the displacement from equilibrium and always directed towards the equilibrium position
• The acceleration of an object undergoing SHM is also proportional to the displacement and directed towards the equilibrium position
• The period ($T$) of SHM is the time taken for one complete oscillation and is independent of the amplitude
  • $T=2\pi\sqrt{\frac{m}{k}}$ for a mass-spring system, where $m$ is the mass and $k$ is the spring constant
  • $T=2\pi\sqrt{\frac{L}{g}}$ for a simple pendulum, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity
• The frequency ($f$) of SHM is the number of oscillations per unit time and is related to the period by $f=\frac{1}{T}$
• The angular frequency ($\omega$) of SHM is related to the frequency by $\omega=2\pi f$ and is measured in radians per second
• Displacement ($x$), velocity ($v$), and acceleration ($a$) in SHM can be described using sinusoidal functions
  • $x(t)=A\cos(\omega t+\phi)$, where $A$ is the amplitude and $\phi$ is the initial phase angle
  • $v(t)=-A\omega\sin(\omega t+\phi)$
  • $a(t)=-A\omega^2\cos(\omega t+\phi)$

Springs and Hooke's Law

• Hooke's law states that the restoring force exerted by a spring is directly proportional to the displacement from its equilibrium position
  • Mathematically, $F=-kx$, where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement
• The negative sign in Hooke's law indicates that the restoring force acts in the opposite direction of the displacement
• The spring constant ($k$) is a measure of the stiffness of the spring and is determined by the material and geometry of the spring
  • A higher spring constant results in a stiffer spring and a greater restoring force for a given displacement
• The potential energy stored in a stretched or compressed spring that obeys Hooke's law is given by $PE=\frac{1}{2}kx^2$
• When a mass is attached to a spring that obeys Hooke's law, the system can exhibit SHM with a period of $T=2\pi\sqrt{\frac{m}{k}}$
• Springs in series and parallel can be combined to create equivalent spring constants
  • For springs in series, the equivalent spring constant is given by $\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}+...$
  • For springs in parallel, the equivalent spring constant is given by $k_{eq}=k_1+k_2+...$

Pendulums

• A simple pendulum consists of a mass (bob) suspended by a lightweight, inextensible string from a fixed point
  • For small angles of displacement, a simple pendulum exhibits SHM with a period of $T=2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity
• A physical pendulum is an extended object that oscillates about a fixed axis (not necessarily its center of mass)
  • The period of a physical pendulum is given by $T=2\pi\sqrt{\frac{I}{mgd}}$, where $I$ is the moment of inertia about the axis of rotation, $m$ is the mass of the pendulum, $g$ is the acceleration due to gravity, and $d$ is the distance between the axis of rotation and the center of mass
• The restoring force for a pendulum is the component of the gravitational force tangent to the arc of motion
  • For small angles, this restoring force is approximately proportional to the displacement, allowing the pendulum to exhibit SHM
• The potential energy of a pendulum is given by $PE=mgh$, where $h$ is the height of the bob relative to its lowest position
• The kinetic energy of a pendulum is given by $KE=\frac{1}{2}mv^2$, where $v$ is the velocity of the bob
• Pendulums have numerous applications, such as in clocks, seismometers, and for studying the Earth's gravitational field

Energy in Oscillating Systems

• In an ideal oscillating system (no damping), the total energy remains constant and is continuously converted between kinetic energy (KE) and potential energy (PE)
  • At the equilibrium position, the energy is entirely kinetic
  • At the maximum displacement, the energy is entirely potential
• For a mass-spring system, the potential energy is stored in the spring as elastic potential energy, given by $PE=\frac{1}{2}kx^2$
• For a pendulum, the potential energy is gravitational potential energy, given by $PE=mgh$, where $h$ is the height of the bob relative to its lowest position
• The kinetic energy in an oscillating system is given by $KE=\frac{1}{2}mv^2$, where $m$ is the mass of the oscillating object and $v$ is its velocity
• The total energy in an oscillating system is the sum of the kinetic and potential energies, $E_{total}=KE+PE$
• In the absence of damping, the total energy remains constant, and the system will oscillate indefinitely
  • $E_{total}=\frac{1}{2}kA^2$ for a mass-spring system, where $A$ is the amplitude of oscillation
  • $E_{total}=mgA$ for a pendulum, where $A$ is the maximum height reached by the bob relative to its lowest position
• Power in an oscillating system is the rate at which energy is transferred, given by $P=\frac{dE}{dt}$

Damped Oscillations

• Damped oscillations occur when energy is dissipated from an oscillating system, causing the amplitude of oscillation to decrease over time
• Damping can be caused by various factors, such as air resistance, friction, or electrical resistance
• The damping force is often proportional to the velocity of the oscillating object, $F_d=-bv$, where $b$ is the damping coefficient
• The presence of damping modifies the equation of motion for an oscillating system, resulting in the damped harmonic oscillator equation, $m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0$
• The behavior of a damped oscillator depends on the relative values of the damping coefficient ($b$), mass ($m$), and spring constant ($k$)
  • Underdamped: The system oscillates with gradually decreasing amplitude (most common)
  • Critically damped: The system returns to equilibrium in the shortest possible time without oscillating
  • Overdamped: The system slowly returns to equilibrium without oscillating
• The quality factor ($Q$) is a measure of the damping in an oscillating system, given by $Q=2\pi\frac{E_{stored}}{E_{dissipated per cycle}}$
  • A higher quality factor indicates less damping and a slower decay of oscillations
• Damped oscillations have important applications in shock absorbers, musical instruments, and electrical circuits

Forced Oscillations and Resonance

• Forced oscillations occur when an external periodic force is applied to an oscillating system
• The external force can be described by $F_{ext}=F_0\cos(\omega t)$, where $F_0$ is the amplitude of the force and $\omega$ is the angular frequency of the force
• The equation of motion for a forced harmonic oscillator is $m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=F_0\cos(\omega t)$
• The steady-state solution for a forced harmonic oscillator is given by $x(t)=A\cos(\omega t-\phi)$, where $A$ is the amplitude of oscillation and $\phi$ is the phase difference between the external force and the oscillation
• The amplitude of a forced oscillator depends on the frequency of the external force and is given by $A=\frac{F_0}{\sqrt{(k-m\omega^2)^2+(b\omega)^2}}$
• Resonance occurs when the frequency of the external force matches the natural frequency of the oscillating system
  • At resonance, the amplitude of oscillation is maximized, and the system oscillates with a phase difference of $\phi=\frac{\pi}{2}$ relative to the external force
• The natural frequency of a mass-spring system is given by $\omega_0=\sqrt{\frac{k}{m}}$, while for a pendulum, it is given by $\omega_0=\sqrt{\frac{g}{L}}$
• Resonance can be beneficial or destructive depending on the context
  • Beneficial examples include musical instruments, microwave ovens, and radio tuners
  • Destructive examples include the collapse of the Tacoma Narrows Bridge and the resonance of crystal wine glasses

Real-World Applications

• Oscillations and simple harmonic motion have numerous real-world applications across various fields, including physics, engineering, and biology
• In mechanical engineering, understanding oscillations is crucial for designing systems such as car suspensions, bridges, and buildings to minimize unwanted vibrations and ensure stability
  • Tuned mass dampers, which are essentially pendulums, are used in tall buildings to counteract wind-induced oscillations
• In electrical engineering, oscillations are fundamental to the design and operation of circuits, particularly in the context of alternating current (AC) and radio frequency (RF) systems
  • LC circuits, consisting of an inductor and a capacitor, exhibit electrical oscillations analogous to mass-spring systems
• In acoustics and music, oscillations are central to the production and perception of sound
  • Musical instruments, such as guitars and violins, rely on the oscillations of strings to produce different pitches
  • The human vocal cords oscillate to produce speech and singing
• In biology, oscillations are observed in various physiological processes and biological rhythms
  • The cardiac cycle, which involves the contraction and relaxation of the heart muscle, is an example of a biological oscillation
  • Circadian rhythms, which govern sleep-wake cycles and other physiological processes, are driven by oscillations in gene expression and hormone levels
• In astronomy and astrophysics, oscillations are studied in the context of celestial bodies and gravitational systems
  • The orbits of planets and moons around their parent bodies can be approximated as simple harmonic motion for small eccentricities
  • Stellar oscillations, known as asteroseismology, provide insights into the internal structure and properties of stars
• Understanding oscillations is essential for developing technologies related to energy harvesting, such as piezoelectric and electromagnetic generators that convert mechanical vibrations into electrical energy