Harmonic oscillator

Simple Harmonic Motion

• A displacement function $x(t)$ is said to describe simple harmonic motion iff it satisfies the differential equation $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0$
• $x(t)=A\cos (\omega t+\varphi )$ is the displacement from the equilibrium position
  • $\varphi$ is the phase angle (in $\mathsf{rad}$)
  • $A$ is the amplitude (the maximum displacement from the equilibrium)
  • $\omega =\sqrt{\frac{k}{m}}$ is the angular frequency (in $rad/s$)
• $F=-kx$ is the restoring force (Hooke’s Law)
• $k$ is the spring constant (related to the stiffness of the spring) (in $\mathsf{N}\cdot {\mathsf{m}}^{-1}$)
• $T=\frac{2\pi }{\omega }$ is the period (in $\mathsf{s}$)
• $v=\omega \sqrt{A^2-x^2}$ is the velocity as a function of position
• $v_{\text{max}}=\omega A$ is the maximum velocity
• $m$ is the mass of the oscillating body (in $\mathsf{kg}$)
• $E=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2$ is the total mechanical energy (in $\mathsf{J}$)
  • $\frac{1}{2}m{v}^2$ is the kinetic energy (in $\mathsf{J}$) (it’s total energy in the moment of equilibrium, $x=0$)
  • $\frac{1}{2}k{x}^2$ is the elastic potential energy (in $\mathsf{J}$) (it’s total energy in the monent of turning point, $v=0$)

Mass-Spring System

• $f_0=\frac{1}{2\pi }\sqrt{\frac{k}{m}}=\frac{1}{T}$ is the natural frequency of the system
• $v_{\text{max}}=A\sqrt{\frac{k}{m}}$
• $v_{\text{max}}=\pm A\sqrt{\frac{k}{m}}$ is the maximum velocity

Pendulum

• The oscillating body is the pendulum bob
• $T=2\pi \sqrt{\frac{L}{g}}$
• $L$ is the length of the pendulum
• $g$ is the acceleration due to gravity
• $\frac{mg}{L}$

Damped Harmonic Motion

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