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Key Concepts in Mechanics and SHM: Newton's Laws of Motion are defined, including the laws of inertia, force proportionality to acceleration, and equal and opposite action/reaction. Simple Harmonic Motion (SHM) is a periodic motion where acceleration (or force) is directly proportional to displacement and directed towards a fixed point. Total Energy of SHM is $E = \frac{1}{2} m \omega^2 a^2$. Damped Harmonic Motion occurs when a body's amplitude continuously decays due to dissipative forces like friction. The differential equation for damped motion is $m\frac{d^2x}{dt^2} = -kx - L\frac{dx}{dt}$. Motion is classified as overdamped (dead beat) if $K > \omega$, critically damped if $K = \omega$, and damped oscillatory if $K < \omega$. Relaxation Time ($\tau$) is the time taken for the amplitude to fall to $1/e$ times its initial amplitude. $\tau$ is also defined as the time for energy to fall to $1/e$ of its initial value, where $\tau = \frac{1}{2K}$. Forced Vibration is produced when a body vibrates under an applied strong periodic force. Velocity Resonance occurs at the natural frequency, $\omega$. Amplitude Resonance occurs at a resonant frequency $q_{res} = \sqrt{\omega^2 - 2K^2}$. Quality Factor (Q) is $2\pi$ times the ratio of average energy stored to the energy dissipated per cycle. For weak damping, $Q \approx \frac{\omega}{2K}$. Key Concepts in Vector Calculus: A vector field $\vec{A}$ is solenoidal if $\nabla \cdot \vec{A} = 0$. A vector field $\vec{A}$ is irrotational if $\nabla \times \vec{A} = 0$. Stokes' Theorem relates the surface integral of the curl of a vector field to the line integral around the boundary of the surface. Gauss's Divergence Theorem relates the volume integral of the divergence of a vector field to the surface integral of the vector field. For a conservative field, the line integral around a closed path is zero ($\oint \vec{E} \cdot d\vec{l} = 0$)
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