Oscillations & Simple Harmonic Motion Full Study Guide

PHY1031F Vibrations & Waves: Course Overview - Course Details: * Course Title: PHY1031F Vibrations & Waves. * Unit Duration: 18 Lectures. * Lecturer: Dr. S.M. Wheaton. * Location: Room LT3A, R.W. James Building. - Recommended Resource Material: * Chapter 13: OpenStax. * Chapter 14: Knight, Jones, and Field. ## Fundamentals of Equilibrium and Free-Body Diagrams - Equilibrium Definition: An object is in equilibrium when the net force (

$F<em>{net}$) and net torque ( $\tau</em>{net}$) acting upon it are zero. - ConcepTest: Equilibrium States: * Fact: An object in equilibrium need not be at rest. * Fact: An object at rest must be in equilibrium. - Six-Step Procedure for Constructing Free-Body Diagrams (FBDs): 1. Draw a Picture: Create a visual representation of the physical situation. 2. Define the System: Circle the specific system of interest on the picture. 3. Identify Forces: Identify all significant forces acting on the system. Identify these by asking, "Who or what interacts with the system?" Draw these as labeled arrows. 4. Coordinate Representation: Redraw the system as a single dot and representing the forces as arrows. The length of the arrow should represent the magnitude of the force. 5. Establish Axes: Draw convenient coordinate axes, usually centered on the dot representing the system. 6. Net Force Representation: Draw the net force vector alongside the specific physical force vectors. - Equilibrium in FBDs: When drawing FBDs for oscillating systems, the origin of the coordinate system is typically placed at the equilibrium position (the position where the system would be at rest). ## Mathematical Description of Simple Harmonic Motion (SHM) - Sinusoidal Nature: By definition, the graphs representing Simple Harmonic Motion are sinusoidal. This fixes the general shape, though specific details (sine, cosine, or phase-shifted sine) depend on initial conditions. - The Special Case (Release from Maximum Displacement): Consider a horizontal mass-spring system where the mass is released from position
$x = A$ at time
$t = 0$. * Position $x(t), y(t)$: Represented as a cosine function starting at maximum amplitude.
$y(t) = A ext{cos} rac{2 ext{ ext{π}}t}{T}$ * Velocity $v<em>x(t), v</em>y(t)$: The derivative of position; velocity is zero when displacement is at a maximum or minimum.
$v<em>y(t) = - rac{(2 ext{π}f) A ext{sin} rac{2 ext{π}t}{T}}{T}$ * Acceleration $a</em>x(t), a<em>y(t)$: Proportional to the negative of displacement. Acceleration is max in the negative direction when displacement is max in the positive direction. $a</em>y(t) = - rac{(2 ext{π}f)^2 A ext{cos} rac{2 ext{π}t}{T}}{T}$ - Key Kinematic Relationships in SHM: * When displacement is maximum
$A$ or minimum
$-A$, velocity is zero. * When displacement is zero (passing equilibrium), velocity is at its maximum
$v<em>{max}$. * When displacement is at its positive maximum, acceleration is at its negative maximum. * Required Argument Format: All arguments in these trigonometric formulas must be in radians. ## Completely General Equations for SHM - Limitations of Specialized Equations: The standard sine/cosine equations require a specific starting point $t=0$. If these conditions are not met, general equations are required. - General Kinematic Set: * Position: $x(t) = A ext{cos}( rac{2 ext{π}t}{T} + ext{φ})$ * Velocity: $v</em>x(t) = -(2 ext{π}f) A ext{sin}( rac{2 ext{π}t}{T} + ext{φ})$ * Acceleration:
$a<em>x(t) = -(2 ext{π}f)^2 A ext{cos}( rac{2 ext{π}t}{T} + ext{φ})$ - The Phase Constant ( φ): * This is a constant determined by the specific initial conditions of the system. * The special case (release from $x=A$) is recovered when $ext{φ} = 0$. * Example Case 1: Timing begins as the body passes $x=0$ traveling in the positive x-direction. This requires a shift of $ext{φ} = - rac{ ext{π}}{2}$. * Example Case 2: Timing begins as the body passes $x=0$ traveling in the negative x-direction. This requires a shift of $ext{φ} = + rac{ ext{π}}{2}$. - Trigonometric Co-ratios for Shifts: * $ext{sin}( ext{θ}) = ext{cos}( ext{θ} - rac{ ext{π}}{2})$ * $ext{cos}( ext{θ}) = ext{sin}( ext{θ} + rac{ ext{π}}{2})$ ## Dynamics of SHM: Forces and Hooke’s Law - Defining Property of SHM: Simple Harmonic Motion occurs whenever the net restoring force is directly proportional to the object's displacement from its equilibrium position. Not all periodic motion qualifies as SHM. - Hooke’s Law: Specifically for springs, the restoring force ( $F</em>s$) is directly proportional to the amount the spring is stretched or compressed. * Formula:
$F<em>s = -kx$ * $k$ = spring constant (magnitude of spring stiffness). * $x$ = displacement from the equilibrium position ( $x=0$ at equilibrium). * Negative sign indicates the force is a restoring force, always directed opposite to the displacement. - Newton’s Second Law Application: For a horizontal mass-spring system (ignoring friction): * $(F</em>{net})<em>x = ma</em>x$ *
$-kx = ma<em>x$ * $a</em>x = - rac{k}{m}x$ - Non-Constant Acceleration: In SHM, acceleration depends on position. Therefore, acceleration is not constant, and standard constant-acceleration "equations of motion" cannot be applied. ## Vertical Mass-Spring Systems and Pendulums - Vertical Mass-Spring System: * The equilibrium position is where the upward spring force balances the downward gravitational force:
$k ext{Δ}L = mg$. * If moved from this point, the net force causes oscillation around the new equilibrium point
$y=0$. - The Simple Pendulum: * Restoring force:
$(F<em>{net})</em>t = -mg ext{sin}( ext{θ})$. * For SHM, the force must be proportional to displacement, not the sine of the angle. * Small Angle Approximation: If the angle
$ext{θ}$ is small,
$ext{sin}( ext{θ}) ≈ ext{θ}$. * Restoring force for small angles:
$F ≈ - rac{mg}{L}x$, where
$x = L ext{θ}$. - Frequency and Period Formulas: * Mass-Spring System:
$T = 2 ext{π} ext{√{ rac{m}{k}}}$ and
$f = rac{1}{2 ext{π}} ext{√{ rac{k}{m}}}$ * Simple Pendulum (Small Angles):
$T = 2 ext{π} ext{√{ rac{L}{g}}}$ and
$f = rac{1}{2 ext{π}} ext{√{ rac{g}{L}}}$ * Critical Note: The frequency of a pendulum depends only on its length (
$L$) and gravity (
$g$), not on the mass attached. ## Conservation of Energy in Oscillating Systems - Isolated Systems: For an isolated system with no external forces (like friction), total mechanical energy (
$E$) is conserved. *
$E = K + U = ext{constant}$ *
$K<em>i + U</em>i = K<em>f + U</em>f$ - Energy Components: * Kinetic Energy ( $K$):
$K = rac{1}{2}mv^2$. This is maximum at the equilibrium position. * Potential Energy ( $U$): For a spring,
$U = rac{1}{2}kx^2$. This is zero at the equilibrium position. * Total Energy at Amplitude: At
$x = A$, velocity is zero, so total energy
$E = rac{1}{2}kA^2$. - Energy Transformation: During oscillation, energy transforms from potential to kinetic and back. Inertia carries the mass past the equilibrium position, compressing the spring and turning kinetic energy back into potential energy. - Speed Calculation at Any Position: * Using
$E = K + U$:
$rac{1}{2}mv^2 + rac{1}{2}kx^2 = rac{1}{2}kA^2$ *
$v = ± ext{√{ rac{k}{m}(A^2 - x^2)}}$ ## Damping and Real-World Systems - Reality vs. Ideality: Only ideal systems oscillate indefinitely. Real systems experience friction/damping, reducing total mechanical energy over time. - Air Resistance: In pendulums, energy is lost to air resistance (drag). Drag is greatest when the pendulum travels fastest. - Amplitude Decay: The "envelope" of the oscillation follows a dying exponential curve. * The rate of drop is determined by the time constant (
$ext{τ}$) of the decay. - Mathematical Constants and Logs for Decay Calculations: *
$e ≈ 2.7182$ * Solving for exponents requires natural logs (
$ext{ln}$). ## Driven Oscillators and Resonance - Natural Frequency ( $f<em>0$): Every system has a specific frequency at which it oscillates naturally when disturbed. - Driven Oscillators: Systems subject to an external force moving at frequency
$f</em>{ext}$. The system will oscillate at this external frequency regardless of its natural frequency. - Resonance: Occurs when the driving frequency (
$f<em>{ext}$) matches the natural frequency ( $f</em>0$). The resulting oscillation amplitude becomes very large. - Resonance Example: In a set of pendulums on a flexible beam, if one (Pendulum A) is set in motion, others will vibrate. The pendulum with a length (and thus
$f<em>0$) matching Pendulum A will oscillate with the greatest amplitude. - PhET Simulation Data Examples: * System 1: $M</em>1 = 2.5330 ext{ kg}$,
$k<em>1 = 100 ext{ N/m}$, $f</em>1 = 1.0 ext{ Hz}$. * System 2:
$M<em>2 = 1.6887 ext{ kg}$, $k</em>2 = 150 ext{ N/m}$,
$f<em>2 = 1.5 ext{ Hz}$. * System 3: $M</em>3 = 1.2665 ext{ kg}$,
$k<em>3 = 200 ext{ N/m}$, $f</em>3 = 2.0 ext{ Hz}$. ## Worked Examples and Quantitative Data - Example: Standard SHM Parameters: * Given: Body oscillates between two points
$10 ext{ cm}$ apart (implies amplitude
$A = 5 ext{ cm}$), frequency
$f = 0.25 ext{ Hz}$. Released from
$x = +5 ext{ cm}$ at
$t = 0$. * Period ( $T$):
$1/f = 4.0 ext{ s}$. * Position at $t = 1.5 ext{ s}$:
$x = -3.54 ext{ cm}$. * Distance traveled in first $1.5 ext{ s}$:
$8.54 ext{ cm}$ (
$5 ext{ cm}$ to equilibrium +
$3.54 ext{ cm}$ beyond). * Velocity at $1.5 ext{ s}$:
$-5.55 ext{ cm/s}$. * Maximum Speed ( $v<em>{max}$):
$v</em>{max} = (2 ext{π}f)A = 7.85 ext{ cm/s}$. * Maximum Acceleration ( $a<em>{max}$):
$a</em>{max} = (2 ext{π}f)^2 A = 12.3 ext{ cm/s}^2$. - Example: Horizontal 500g Block: *
$500 ext{ g}$ block, stretched by
$10 ext{ cm}$, released. Speed at equilibrium =
$1.0 ext{ m/s}$. * Period ( $T$):
$0.63 ext{ s}$. * Speed at compression of $5.0 ext{ cm}$:
$0.87 ext{ m/s}$. - Example: Vertical 100g Block: * Mass =
$100 ext{ g}$, stretches spring
$16 ext{ cm}$ at rest. Pulled down additional
$26 ext{ cm}$. * Question: Speed when
$13 ext{ cm}$ above equilibrium? * Answer:
$1.76 ext{ m/s}$. - Example: Damping Decay: * Mass on spring with amplitude
$20 ext{ cm}$ and time constant
$τ = 50 ext{ s}$. * Time to reach half amplitude ( $10 ext{ cm}$):
$35 ext{ s}$. * Time to reach half energy:
$17 ext{ s}$. ## Questions & Discussion - Question: If the mass of a toy car on a spring is doubled, what happens to the period? - Answer: The period will increase (since
$T ∝ ext{√{m}}$). - Question: Two pendula have the same length but different masses. How do periods compare? - Answer: The period is the same for both cases (mass does not affect pendulum period). - Question: For SHM, which pair of vector quantities can never point in the same direction? - Answer: Position (displacement from equilibrium) and acceleration (always directed toward equilibrium). - Question: If the amplitude of SHM is doubled, which quantity remains unchanged? - Answer: The period (period is independent of amplitude in SHM). - Question: What is the total distance traveled by a mass with amplitude
$A$ during one full period
$T$? - Answer:
$4A$ (travels from
$A$ to
$0$,
$0$ to
$-A$,
$-A$ back to
$0$, and
$0$ back to
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