shm x, v, a

related notes Link to heading

• m09-2: oscillations
• sin cos 4 quadrants

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.

  Variable	Meaning	Unit
  $x$	displacement	m
  $x_m$	maximum displacement, amplitude	m
  $\omega$	angular frequency	rad/s
  $t$	time	s
  $\theta$	phase	rad
  $\phi$	initial phase, phase angle, phase constant	rad

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.