simple harmonic motion Link to heading

  • It is periodic motion that is a sinusoidal function of time $$\tag{1} \begin{array}{rcl} x(t) & = & x_m \sin \theta \newline & = & x_m \sin (\omega t + \phi). \end{array} $$
  • It is solution of an ODE $$\tag{2} \frac{d^2x}{dt^2} + \omega^2 x = 0. $$

variables Link to heading

  • Each of variables meaning in Eqn (1) are as follow.
    VariableMeaningUnit
    $x$displacementm
    $x_m$maximum displacement, amplitudem
    $\omega$angular frequencyrad/s
    $t$times
    $\theta$phaserad
    $\phi$initial phase, phase angle, phase constantrad

phase constant Link to heading

  • It shifts the sine or cosine function.
  • For certain $\phi_s$ and $\phi_c$ following relation, $$\tag{3} \cos (\omega t + \phi_c) = \sin (\omega t + \phi_s), $$ will hold.
  • Use trigonometric identities to relate $\phi_s$ and $\phi_c$ as in sin cos 4 quadrants.