University Physics: Oscillations and Simple Harmonic Motion Study Notes

Fundamental Principles of Oscillations and Units

• Definition of Oscillation: A repetitive motion about an equilibrium position.
• Equilibrium and Motion: An object in equilibrium is not necessarily at rest; however, oscillatory motion occurs around this stable state.
• Characteristics of Oscillation:     * Period ($T$): The time required to complete one full cycle of motion.     * Frequency ($f$): The number of cycles completed per unit of time ($f = \frac{1}{T}$). The standard unit of frequency is the hertz ($Hz$), where $1\,Hz = 1\,s^{-1}$.
• Common Units and Their Relationships (Table 14.1):     * Kilohertz ($kHz$): $10^{3}\,Hz = 1\,kHz$. The corresponding period is $1\,ms$ ($10^{-3}\,s$).     * Megahertz ($MHz$): $10^{6}\,Hz = 1\,MHz$. The corresponding period is $1\,\mu s$ ($10^{-6}\,s$).     * Gigahertz ($GHz$): $10^{9}\,Hz = 1\,GHz$. The corresponding period is $1\,ns$ ($10^{-9}\,s$).

Simple Harmonic Motion (SHM) and Physical Models

• Defining SHM: Simple Harmonic Motion is an oscillation that can be described mathematically by a sinusoidal function (specifically a cosine or sine function).
• The Condition for SHM: SHM occurs whenever the net restoring force acting on an object is linearly proportional to the object’s displacement from its equilibrium position. This is known as a linear restoring force.
• Primary Systems Modeling SHM:     * Mass on a Spring:         * A mass ($m$) is attached to a spring with spring constant ($k$).         * The restoring force is governed by Hooke's Law: $F_{net} = -kx$.         * The period depends on the mass and the stiffness of the spring.         * Biological Example: Vibrations in the ear. Sound waves cause the oscillation of the cochlear membrane. The membrane's thickness acts as the mass, and its rigidity acts as the stiffness ($k$).     * Pendulum:         * A mass (bob) is suspended from a pivot by a light string or rod of length ($L$).         * The restoring force is the tangential component of gravity: $F_{net} = -mg\sin(\theta)$.         * The period depends on the length ($L$) and the free-fall acceleration ($g$).         * Biological Example: The motion of a walking animal's legs (e.g., giraffes) can be modeled as pendulum motion.

Mechanics of Mass-Spring Systems

• Horizontal Motion: The net force is solely the spring force: $F_{x} = -kx$.
• Vertical Motion and Equilibrium:     * For a hanging weight, the equilibrium position is where the block hangs motionless. The spring is stretched by a distance $\Delta L$ due to gravity.     * Newton's First Law at Equilibrium: $(F_{sp})<em>{y} + w</em>{y} = k\Delta L - mg = 0$.     * Equilibrium Stretch Equation: $\Delta L = \frac{mg}{k}$.     * Net Force during Oscillation: When displaced by distance $y$ from equilibrium, the spring stretch is $\Delta L - y$. The net force is:       $F_{net} = k(\Delta L - y) - mg = (k\Delta L - mg) - ky = -ky$.     * The negative sign indicates the force is always directed opposite to the displacement.

Mathematical Description of SHM

• Position ($x(t)$): Described as a cosine function:     $x(t) = A\cos\left(\frac{2\pi t}{T}\right) = A\cos(2\pi ft)$     * Amplitude ($A$): The maximum displacement from equilibrium. The object moves between $x = A$ and $x = -A$.
• Velocity ($v_{x}(t)$): Described as an inverted sine function:     $v_{x}(t) = -v_{max}\sin\left(\frac{2\pi t}{T}\right) = -v_{max}\sin(2\pi ft)$     * Maximum Speed ($v_{max}$): $v_{max} = \frac{2\pi A}{T} = 2\pi fA$.
• Acceleration ($a_{x}(t)$): Described as an inverted cosine function (opposite to position):     $a_{x}(t) = -a_{max}\cos\left(\frac{2\pi t}{T}\right) = -a_{max}\cos(2\pi ft)$     * Maximum Acceleration ($a_{max}$): $a_{max} = (2\pi f)^{2}A = \frac{k}{m}A$.
• Phase Relationships:     * Position and velocity are out of phase: When displacement is maximum ($A$), velocity is zero. When velocity is maximum ($v_{max}$), displacement is zero.     * Velocity and acceleration are out of phase: When velocity is maximum, acceleration is zero.

Connection to Uniform Circular Motion

• Uniform circular motion projected onto one dimension (e.g., the x-axis) results in Simple Harmonic Motion.
• Component Relationships:     * If a particle moves in a circle of radius $A$ with angular velocity $\omega$, its x-position is $x = A\cos(\phi)$.     * The angle at any time $t$ is $\phi = \omega t$.     * Angular Velocity ($\omega$): $\omega = 2\pi f = \frac{2\pi}{T}$.

Energy in Simple Harmonic Motion

• Energy Conservation: In the absence of friction, the total mechanical energy ($E$) is constant.     $E = K + U = \frac{1}{2}mv^{2} + \frac{1}{2}kx^{2} = \text{constant}$
• Energy at Extremes:     * At Maximum Displacement ($x = \pm A$): Velocity is zero. Energy is purely potential: $E = \frac{1}{2}kA^{2}$.     * At Equilibrium ($x = 0$): Potential energy is zero. Energy is purely kinetic: $E = \frac{1}{2}mv_{max}^{2}$.
• Derived Relationships:     * Equating maximum potential and kinetic energies: $\frac{1}{2}mv_{max}^{2} = \frac{1}{2}kA^{2}$.     * This leads to $v_{max} = A\sqrt{\frac{k}{m}}$.
• Frequency Dependence: Frequency ($f$) depends on mass and stiffness ($k$), not on amplitude ($A$).     $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$T = 2\pi\sqrt{\frac{m}{k}}$

The Pendulum and Small-Angle Approximation

• Restoring Force: For a pendulum of length $L$, the net force along the arc is $F_{t} = -mg\sin(\theta)$.
• Small-Angle Approximation: For small angles (\theta < 10^{\circ}), $\sin(\theta) \approx \theta$ (in radians).     * Since the arc length $s = L\theta$, then $\theta = \frac{s}{L}$.     * Substituting this: $F_{t} \approx -mg\left(\frac{s}{L}\right) = -\left(\frac{mg}{L}\right)s$.
• Pendulum Frequency and Period:$f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$$T = 2\pi\sqrt{\frac{L}{g}}$     * Note: The period of a simple pendulum is independent of its mass.
• Physical Pendulum: A pendulum with mass distributed along its length (like a leg).     $f = \frac{1}{2\pi}\sqrt{\frac{mgd}{I}}$     Where $I$ is the moment of inertia and $d$ is the distance from the pivot to the center of gravity.

Damped and Driven Oscillations

• Damped Oscillation: An oscillation where mechanical energy is dissipated (e.g., by air resistance or friction), causing the amplitude to decrease over time.     * Exponential Decay: The maximum displacement decreases according to the function: $x_{max}(t) = Ae^{-t/\tau}$.     * Time Constant (\tau): A measure of the decay rate. After one time constant ($t = \tau$), the amplitude decreases by a factor of $1/e$ (to approximately $37\%$ of its initial value).     * Damping Levels:         * If $\tau \gg T$, the system oscillates many times before stopping.         * If $\tau \ll T$, the system damps quickly.
• Driven Oscillations: A system subjected to a periodic external force with driving frequency $f_{ext}$.
• Resonance: Occurs when the driving frequency ($f_{ext}$) matches the system's natural frequency ($f_{0}$). This results in a large-amplitude response.

Example Problems and QuickChecks

• QuickCheck 14.5: A mass oscillates horizontally. At a point where displacement is positive and moving toward equilibrium, velocity ($v_{x}$) is negative and force ($F_{x}$) is negative.
• QuickCheck 14.6: In a vertical spring, at the lowest point of oscillation, the acceleration ($a_{y}$) is positive (directed upward toward equilibrium).
• QuickCheck 14.9: A block of mass $m$ oscillates with period $T = 2.0\,s$. If a second identical block is added ($2m$), the new period is $T\sqrt{2} \approx 2.8\,s$.
• QuickCheck 14.10: Doubling the amplitude of a mass-spring system does not change the period; it remains $2.0\,s$.
• QuickCheck 14.11: A graph of kinetic energy vs. position shows $K_{max} = 8\,J$ at $x = 0$ and $K = 0$ at $x = 2\,m$.     $E = 8\,J = \frac{1}{2}k(2\,m)^{2} \Rightarrow 8 = 2k \Rightarrow k = 4\,N/m$.
• QuickCheck 14.18: A pendulum on Planet X has $T = 2.0\,s$. If moved to a moon where $g$ is half as large, the period increases: $T \propto \frac{1}{\sqrt{g}}$. Result: $2.0\,s \times \sqrt{2} \approx 2.8\,s$.
• Example 14.2 (Glider on Spring): Glider oscillates between points $12\,cm$ apart ($A = 6\,cm$) at $0.50\,Hz$.
• Example 14.7 (Mass on Hanging Spring): A $25\,g$ mass stretches a spring from $10\,cm$ to $15\,cm$ ($\Delta L = 0.050\,m$).     $k = \frac{mg}{\Delta L} = \frac{0.025\,kg \times 9.8\,m/s^{2}}{0.050\,m} = 4.9\,N/m$.     $f = \frac{1}{2\pi}\sqrt{\frac{4.9}{0.025}} = 2.2\,Hz$.
• Example 14.10 (Clock Pendulum): For a clock with a "tick" every $1.00\,s$, the period is $T = 2.00\,s$.     $L = g\left(\frac{T}{2\pi}\right)^{2} = 9.80\left(\frac{2.00}{2\pi}\right)^{2} = 0.993\,m$.
• Example 14.12 (Decay Time): A grandfather clock with $\tau = 300\,s$. Find time to reach half amplitude.     $\frac{1}{2} = e^{-t/300} \Rightarrow \ln(0.5) = -t/300 \Rightarrow t = 300\ln(2) = 208\,s$.

Questions & Discussion

• Conceptual Example 14.5 (Playground Swing): Starting at the farthest forward point (maximum potential energy, zero kinetic energy), the energy transforms into kinetic energy as you swing toward the bottom (equilibrium), then back to potential energy as you reach the farthest backward point. In one full cycle, this energy transformation happens twice.
• Defining Resonance Frequency: The speaker notes that natural frequency ($f_{0}$) is often referred to as the resonance frequency because that is where the maximum amplitude response occurs when a system is driven by an external force.