PX155 - D4 - initial conditions

in $A \cos (ω t) + B \sin (ω t)$
- $A^{'} \cos (ω t + ϕ)$
- 2 constants to find which specify the problem, ie: particular harmonic oscillation, usually from initial conditions
eg: a mass $m = 2 k g$ on a spring with $k = 8 N m^{- 1}$ . at $t = 0 s$ , $x = 2 m$ , and $\dot{x} = 6 m s^{- 1}$
- to find: $x (t)$
  $x = A \cos (ω t) + B \sin (ω t)$
  $\dot{x} = - ω A \sin (ω t) + B ω \cos (ω t)$
  - at $t = 0$ :
    $x = A + = 2 m$
    $\dot{x} = B ω = 6 m s^{- 1}$
    $ω = \sqrt{\frac{k}{m}} = \sqrt{\frac{8}{2}} = 2 s^{- 1}$
    so, $B = 3 m$
    $∴ x (t) = (2 \cos 2 t + 3 \sin 2 t) m$