8.3 Periodic and Harmonic Motion Modeling

8.3 Periodic and Harmonic Motion Modeling

Periodic phenomena—ranging from circular motion on a rotating Ferris wheel to ocean tides, sound wave vibrations, and mass-spring mechanical oscillations—repeat at regular, deterministic time intervals. Trigonometric functions serve as standard mathematical models for periodic behavior, capturing cyclic motion through precise parameters for amplitude, frequency, period, midline, and phase shift.

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1. Mathematical Architecture of Sinusoidal Models

A sinusoidal model is any function that can be expressed in the general form:

$f(t) = A \sin(B(t - C)) + D \quad \text{or} \quad f(t) = A \cos(B(t - C)) + D$

where $t$ represents time (or another continuous independent variable), and the coefficients represent specific geometric and physical properties of the oscillation:

Parameter Mapping to Physical Characteristics

1. Midline (Vertical Shift $D$): The horizontal line $y = D$ representing the central equilibrium position about which the function oscillates: $D = \frac{\text{Maximum Value} + \text{Minimum Value}}{2}$
2. Amplitude ($|A|$): The maximum displacement of the function above or below the midline: $|A| = \frac{\text{Maximum Value} - \text{Minimum Value}}{2}$ (Note: If $A < 0$, the wave is reflected vertically across the midline).
3. Period ($T$): The duration of time required for the function to execute one complete cycle: $T = \frac{2\pi}{|B|} \implies |B| = \frac{2\pi}{T}$
4. Frequency ($f$): The number of complete cycles executed per unit time: $f = \frac{1}{T} = \frac{|B|}{2\pi}$
5. Phase Shift ($C$): The horizontal shift of the baseline curve. In factored form $B(t - C)$, $C$ represents the time offset of the initial peak (for cosine) or initial rising midline intercept (for sine).

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2. Rotational Motion Modeling: Ferris Wheel Scenarios

Rotational motion around a central axis translates directly into sinusoidal vertical height functions over time.

Worked Example 1: Ferris Wheel Height Dynamics

A giant Ferris wheel has a diameter of $50\text{ meters}$ and completes one full counterclockwise revolution every $80\text{ seconds}$. The bottom of the wheel sits $5\text{ meters}$ above ground level. Passengers board their capsule at the lowest point ($t = 0$).

1. Determine the maximum height, minimum height, midline, and amplitude of a passenger capsule.
2. Determine the period $T$ and frequency coefficient $B$.
3. Construct a cosine function $h(t)$ that models the height in meters of a passenger capsule above ground as a function of time $t$ in seconds.
4. Calculate the height of the capsule $20\text{ seconds}$ after boarding.

Solution:

1. Extrema, Midline, and Amplitude:
   • Minimum height: $\text{Min} = 5\text{ meters}$.
   • Maximum height: $\text{Max} = \text{Min} + \text{Diameter} = 5 + 50 = 55\text{ meters}$.
   • Midline $D$: $D = \frac{55 + 5}{2} = 30\text{ meters}$.
   • Amplitude $A$: $A = \frac{55 - 5}{2} = 25\text{ meters}$.
2. Period and Frequency Coefficient:
   • One full cycle takes $80\text{ seconds}$, so $T = 80\text{ s}$.
   • $B = \frac{2\pi}{T} = \frac{2\pi}{80} = \frac{\pi}{40}\text{ rad/s}$.
3. Model Construction:
   • Since the passenger starts at the lowest point at $t = 0$ ($h(0) = 5$), we use an inverted (negated) cosine model with zero phase shift ($C = 0$): $h(t) = -A \cos(B t) + D = -25 \cos\left(\frac{\pi}{40} t\right) + 30$
4. Height at $t = 20\text{ seconds}$:
   • Substitute $t = 20$: $h(20) = -25 \cos\left(\frac{\pi}{40} \cdot 20\right) + 30 = -25 \cos\left(\frac{\pi}{2}\right) + 30$
   • Since $\cos\left(\frac{\pi}{2}\right) = 0$: $h(20) = -25(0) + 30 = 30\text{ meters}.$
   • At $20\text{ seconds}$ (one-quarter of a full cycle), the capsule is exactly at the midline height of $30\text{ meters}$.

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3. Natural Oscillations: Tides and Mass-Spring Systems

Modeling Ocean Tides

Ocean water levels follow periodic gravitational oscillations driven by the lunar cycle. The time between consecutive high tides is approximately $12.5\text{ hours}$.

Worked Example 2: Oceanic Tidal Model

In a coastal bay, high tide reaches water depth of $16\text{ feet}$ at 4:00 AM ($t = 4\text{ hours}$ after midnight), and low tide reaches a depth of $4\text{ feet}$ at 10:15 AM ($t = 10.25\text{ hours}$).

1. Derive the parameters $A$, $B$, $C$, and $D$ for a cosine model $d(t) = A \cos(B(t - C)) + D$.
2. Calculate the depth of the water in the bay at 1:00 PM ($t = 13\text{ hours}$).

Solution:

1. Deriving Parameters:
   • $\text{Max} = 16\text{ ft}$, $\text{Min} = 4\text{ ft}$.
   • Midline $D = \frac{16 + 4}{2} = 10\text{ feet}$.
   • Amplitude $A = \frac{16 - 4}{2} = 6\text{ feet}$.
   • Time elapsed from high tide to low tide is half the period: $\frac{T}{2} = 10.25 - 4 = 6.25\text{ hours} \implies T = 12.5\text{ hours}$
   • Frequency coefficient $B = \frac{2\pi}{12.5} = \frac{4\pi}{25}$.
   • Since high tide ($\text{Max}$) occurs at $t = 4$, we use an unreflected cosine wave phase-shifted to the right by $C = 4\text{ hours}$: $d(t) = 6 \cos\left(\frac{4\pi}{25}(t - 4)\right) + 10$
2. Evaluating Depth at $t = 13\text{ hours}$ (1:00 PM):
   • Substitute $t = 13$: $d(13) = 6 \cos\left(\frac{4\pi}{25}(13 - 4)\right) + 10 = 6 \cos\left(\frac{36\pi}{25}\right) + 10$
   • Evaluate $\frac{36\pi}{25}\text{ rad} \approx 4.5239\text{ rad}$: $\cos(4.5239) \approx -0.2225$ $d(13) = 6(-0.2225) + 10 = -1.335 + 10 = 8.665\text{ feet}.$

Simple Harmonic Motion (Springs and Pendulums)

An object moving in simple harmonic motion satisfies the differential relationship where acceleration is proportional and opposite to displacement. Its position equation is given by:

$y(t) = A \cos(\omega t) \quad \text{or} \quad y(t) = A \sin(\omega t)$

where $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency (in rad/s) for a mass $m$ attached to a spring with spring constant $k$.

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4. Scannable Summary of Sinusoidal Model Properties

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5. CLEP Exam Traps & Common Errors

• Trap 1: Confusing Unfactored Phase Shift with True Phase Shift
  In the unfactored form $y = A \sin(B t - C) + D$, the horizontal shift is not $C$. You must factor out $B$ as $y = A \sin\left(B\left(t - \frac{C}{B}\right)\right) + D$. The true phase shift is $\frac{C}{B}$.
• Trap 2: Period vs. Frequency Swap
  Remember that period $T$ has units of time per cycle (e.g., seconds per cycle), whereas frequency $f$ has units of cycles per time (e.g., cycles per second or Hz). They are reciprocal quantities: $f = \frac{1}{T}$.
• Trap 3: Selecting Sine vs. Cosine for Initial Boundary Conditions
  • If the motion starts at $t = 0$ at maximum height, use $+ \cos(\omega t)$.
  • If the motion starts at $t = 0$ at minimum height, use $- \cos(\omega t)$.
  • If the motion starts at $t = 0$ at the midline going upward, use $+ \sin(\omega t)$.
  • If the motion starts at $t = 0$ at the midline going downward, use $- \sin(\omega t)$.