7.3 Applications

Differential equations model real-world systems like springs and circuits. They help us understand how things move and change over time. By tweaking variables, we can see how different factors affect the system's behavior.

These equations are powerful tools for engineers and scientists. They let us predict how systems will react to different inputs and conditions. Understanding these models is key to designing everything from car suspensions to electronic devices.

Modeling Physical Systems with Differential Equations

Second-order equations for harmonic motion

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• Model simple harmonic motion (SHM) using the second-order differential equation $\frac{d^2x}{dt^2} + \omega^2x = 0$ where $x$ represents displacement from equilibrium and $\omega$ is the angular frequency calculated by $\omega = \sqrt{\frac{k}{m}}$ ($k$ is the spring constant and $m$ is the mass)
• Incorporate a damping term proportional to velocity in the equation for damped harmonic motion $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega^2x = 0$ where $\gamma$ is the damping coefficient
  • Determine the system's behavior based on the value of $\gamma$ relative to $\omega$:
    • Underdamped ($\gamma < \omega$) results in oscillatory motion with decreasing amplitude (pendulum with friction)
    • Critically damped ($\gamma = \omega$) returns to equilibrium in the shortest time without oscillating (car suspension system)
    • Overdamped ($\gamma > \omega$) slowly returns to equilibrium without oscillating (door closer)
• Include an external driving force $F(t)$ in the equation for forced harmonic motion $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega^2x = F(t)$ where the system oscillates at the frequency of the driving force with an amplitude dependent on the driving force's frequency and the system's natural frequency (AC circuits, vibrating machines)

Applications to RLC circuits

• Model the charge $q$ in an RLC series circuit (consisting of a resistor, inductor, and capacitor) using the second-order differential equation $L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t)$ where $E(t)$ is the electromotive force (voltage) as a function of time
• Calculate the current $I$ in the circuit using the relationship $I = \frac{dq}{dt}$
• Analyze the system's behavior based on the relative values of R, L, and C:
  • Underdamped ($R < 2\sqrt{\frac{L}{C}}$) leads to oscillatory current and charge
  • Critically damped ($R = 2\sqrt{\frac{L}{C}}$) results in current and charge returning to equilibrium without oscillating
  • Overdamped ($R > 2\sqrt{\frac{L}{C}}$) causes current and charge to slowly return to equilibrium without oscillating

Oscillation Characteristics

• Amplitude: The maximum displacement from equilibrium in an oscillating system
• Frequency: The number of oscillations per unit time, related to the angular frequency $\omega$ by $f = \frac{\omega}{2\pi}$
• Equilibrium: The position or state where the net force on the system is zero
• Velocity: The rate of change of displacement, given by $v = \frac{dx}{dt}$
• Acceleration: The rate of change of velocity, given by $a = \frac{d^2x}{dt^2}$

Interpreting and Comparing Solutions

Physical interpretation of solutions

• Interpret the solution $x(t)$ for simple harmonic motion as the displacement from equilibrium over time with the amplitude of oscillation determined by initial conditions (displacement and velocity at $t = 0$) and the period of oscillation calculated by $T = \frac{2\pi}{\omega}$
• Understand the solution $x(t)$ for damped harmonic motion as the displacement from equilibrium over time with the amplitude decreasing exponentially for underdamped systems at a decay rate determined by the damping coefficient $\gamma$
• Interpret the solution $q(t)$ for RLC circuits as the charge in the circuit over time with the current $I(t)$ being the derivative of the charge with respect to time and the amplitude and decay rate of oscillations (if any) determined by the values of R, L, and C

Comparison of harmonic motion types

• Contrast undamped harmonic motion, which exhibits periodic oscillations with constant amplitude (simple pendulum) at a frequency determined by the system's parameters, with damped and forced harmonic motion
• Compare damped harmonic motion, which may exhibit oscillations with exponentially decaying amplitude (underdamped case) or no oscillations (critically damped or overdamped cases) depending on the damping coefficient relative to the natural frequency, to undamped and forced harmonic motion
• Distinguish forced harmonic motion, which exhibits oscillations at the frequency of the external driving force with an amplitude dependent on the driving force's frequency relative to the system's natural frequency and can lead to resonance when the frequencies match (tuning forks), from undamped and damped harmonic motion