Simple Harmonic Motion

• Periodic Motion - Any motion that repeats itself after a finite segment of time, period, T
• Oscillation - One complete cycle of the periodic motion
• Frequency of periodic motion (f) - The number of cycles per until time (1 second), measured in Hz
  • $f=\frac{1}{T}$
• Simple harmonic motion (SHM) - Any periodic motion that can be described mathematically with a sin or cos function
  • $x(t)=Acos(\omega t+\varphi )$
  • A - Amplitude, represents the maximum displacement of the object from it’s equilibrium position
  • $\omega$ - Angular Frequency, Related to linear frequency, f, and is measured in [rad/s]
    • $\omega =2\pi f=\frac{2\pi }{T}$
  • $\varphi$ - Phase constant, angle measured in [rad] that represents the original angular displacement of the object at time $t_0=0$
• Velocity of SHM
  • $v(t)=\frac{dx(t)}{dt}=\frac{d}{dt}[Acos(\omega t+\varphi )]$
  • $v(t)=-\omega Asin(\omega t+\varphi )$
• Acceleration of SHM
  • $a(t)=\frac{dv(t)}{dt}=\frac{d}{dt}[-\omega Asin(\omega t+\varphi )]$
  • $a(t)=-{\omega }^{2}Acos(\omega t+\varphi )=-{\omega }^{2}x(t)$