4.4 Applications in Classical Mechanics and Electrical Circuits

Second-order ODEs are the backbone of classical mechanics, describing how objects move. They factor in mass, damping, spring constants, and external forces to paint a picture of motion in the physical world.

From simple harmonic motion to pendulums, these equations help us understand oscillations. We can analyze damped and forced systems, uncovering critical concepts like resonance that shape our understanding of mechanical and electrical systems.

Classical Mechanics

Second-order ODEs in mechanics

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Top images from around the web for Second-order ODEs in mechanics

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  Simple Harmonic Motion: A Special Periodic Motion | Physics View original

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  The Simple Pendulum · Physics View original

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• Describe the motion of objects using second-order ODEs incorporates mass $m$, damping coefficient $b$, spring constant $k$, and external force $F(t)$
• Model simple harmonic motion of springs using Hooke's law $F = -kx$ leads to the ODE $m\frac{d^2x}{dt^2} + kx = 0$ with solution $x(t) = A\cos(\omega t + \phi)$ where $A$ represents amplitude, $\omega = \sqrt{\frac{k}{m}}$ angular frequency, and $\phi$ phase shift
• Analyze pendulum motion using the ODE $\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0$ where $\theta$ denotes angular displacement, $g$ gravitational acceleration, and $L$ pendulum length
  • Simplify the pendulum ODE for small angles using the approximation $\sin\theta \approx \theta$ results in $\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$ with solution $\theta(t) = \theta_0\cos(\omega t + \phi)$ where $\theta_0$ represents initial angular displacement and $\omega = \sqrt{\frac{g}{L}}$ angular frequency

Analysis of oscillating systems

• Investigate damped oscillators described by the ODE $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$ exhibits critical damping when $b = 2\sqrt{mk}$, overdamping when $b > 2\sqrt{mk}$, and underdamping when $b < 2\sqrt{mk}$ with solution $x(t) = e^{-\frac{b}{2m}t}(A\cos(\omega t) + B\sin(\omega t))$ where $\omega = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}$ represents angular frequency
• Examine forced oscillators governed by the ODE $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega_0 t)$ where $F_0$ denotes amplitude and $\omega_0$ angular frequency of the driving force
  • Determine the steady-state solution $x(t) = A\cos(\omega_0 t - \phi)$ with amplitude $A = \frac{F_0}{\sqrt{(k - m\omega_0^2)^2 + (b\omega_0)^2}}$ and phase shift $\phi = \arctan\left(\frac{b\omega_0}{k - m\omega_0^2}\right)$
  • Identify resonance when the driving frequency $\omega_0$ closely matches the natural frequency $\sqrt{\frac{k}{m}}$ (mass-spring system, pendulum)

Electrical Circuits

ODEs for electrical circuits

• Model RC circuits using the ODE $\frac{dV}{dt} + \frac{1}{RC}V = \frac{V_s(t)}{RC}$ where $V$ represents voltage across the capacitor, $R$ resistance, $C$ capacitance, and $V_s(t)$ source voltage
  • Solve for a discharging RC circuit ($V_s(t) = 0$) obtaining $V(t) = V_0e^{-\frac{t}{RC}}$ where $V_0$ denotes initial voltage
  • Solve for a charging RC circuit ($V_s(t) = V_0$) obtaining $V(t) = V_0(1 - e^{-\frac{t}{RC}})$
• Analyze RLC circuits described by the ODE $L\frac{d^2I}{dt^2} + R\frac{dI}{dt} + \frac{1}{C}I = V_s(t)$ where $I$ represents current, $L$ inductance, $R$ resistance, $C$ capacitance, and $V_s(t)$ source voltage
  • Classify the solution based on the values of $L$, $R$, and $C$ as overdamped when $R^2 > \frac{4L}{C}$, critically damped when $R^2 = \frac{4L}{C}$, and underdamped when $R^2 < \frac{4L}{C}$

Physical interpretation of ODE solutions

• Relate oscillation frequency $f = \frac{\omega}{2\pi}$ and period $T = \frac{2\pi}{\omega}$ to the angular frequency $\omega$ obtained from ODE solutions
• Understand energy dissipation in damped systems where damping reduces oscillation amplitude over time (mechanical systems) and resistance dissipates energy as heat (electrical systems)
• Recognize resonance in forced systems characterized by large amplitudes when the driving frequency matches the natural frequency (mechanical systems) or maximum power transfer when the driving frequency matches the resonant frequency (electrical systems)
• Distinguish between transient behavior, the initial response of a system to changes in input or initial conditions, and steady-state behavior, the long-term behavior after transients have died out
• Apply ODE solutions to real-world scenarios such as vibration isolation in machines and structures, tuning of musical instruments (guitars, violins), design of filters and oscillators in electrical circuits (radios, televisions), and control systems in robotics and automation (drones, self-driving cars)