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04 Wave Phenomena :ocean: (Simple Harmonic Motion (Mass-Spring…

- - - - \(ma=-kx\)
        
        \(a=-\frac{k}{m}x\)
    - - \(\omega^2=\frac{k}{m}\)
        
        \(\omega=\sqrt{\frac{k}{m}}\)
        
        \(T=\frac{2\pi}{\omega}\)
  - - - \(ma=-mg\sin\theta\)
        
        \(a=-g\sin\frac{x}{L}\)
        
        \(a=-\frac{g}{L}x\)
        
        only proportional if \(\sin\theta=\theta\)
        
        \(\theta=\frac{x}{L}\)
    - - \(\omega^2=\frac{g}{L}\)
        
        \(\omega=\sqrt{\frac{g}{L}}\)
        
        \(T=\frac{2\pi}{\omega}\)
  - - - \(\frac{1}{2}m(-\omega x_0\sin(\omega t))^2\)
        
        \(\frac{1}{2}m\omega^2x_0^2\sin^2(\omega t)\)
    - - Equivalent to
        max KE
        
        \(\frac{1}{2}m\omega^2x_0^2\)
    - - \(\frac{1}{2}m\omega^2x_0^2-\frac{1}{2}m\omega^2x_0^2\sin^2(\omega t)\)
        
        \(\frac{1}{2}m\omega^2x_0^2(1-\sin^2(\omega t))\)
        
        \(\frac{1}{2}m\omega^2x^2\)
  - - - \(\frac{shift}{period}\times 360º\)
    - - Points in same half wavelength IN PHASE
        
        Points in adjacent half wavelengths ANTIPHASE
- - - - \(I=k(E_0\cos\theta)^2\)
        
        \(I=k(E_0)^2\cos^2\theta\)
        
        \(I=I_0\cos^2\theta\)
- - - - 1st harmonic
        \(L=\frac{1}{4}\lambda_1\)
        \(\lambda_1=4L\)
        
        3rd harmonic
        \(L=\frac{3}{4}\lambda_3\)
        \(\lambda_3=\frac{4}{3}L\)
        
        nth harmonic
        \(\lambda_n=\frac{4L}{n};\)
        \(n=1,3,5\dots\)
    - - 1st harmonic
        \(L=\frac{1}{2}\lambda_1\)
        \(\lambda_1=2L\)
        
        2nd harmonic
        \(L=\lambda_2\)
        \(\lambda_2=L\)
        
        nth harmonic
        \(\lambda_n=\frac{2L}{n}\)
        \(n=1,2,3,4,5\dots\)
- - - - \(n_{m}=\frac{c}{c_m}\)
        
        \(n\) is :one: for vaccum
        
        \(n\) is MORE than :one: for optically denser materials
  - - - Refractive Index
        BIG \(n\) :arrow_right: small \(n\)
      - Wave Speed
        Slow :snail: :arrow_right: fast :racehorse:
      - Increase in speed
        
        Decrease in refractive index
        
        Increase in \(\theta\)
    - - Refractive Index
        small \(n\) :arrow_right: BIG \(n\)
      - Wave Speed
        Fast :racehorse: :arrow_right: slow :snail:
      - Decrease in speed
        
        Increase in refractive index
        
        Decrease in \(\theta\)