The displacement of a particle in simple harmonic motion can be expressed as

PHYSICS · Subjective · AP · Physics 1: Algebra-based · Oscillations · Oscillations

Problem

x(t) = A\cos(\omega t + \phi)

, where

A

and

\phi

are constants to be fixed by initial conditions. (i) Show that if the initial displacement is

x_0

and the initial velocity is

v_0

, then the amplitude is given by

A = \sqrt{x_0^{\,2} + \left(\dfrac{v_0}{\omega}\right)^{\!2}}

. (ii) Prove that the initial phase satisfies

\tan \phi = -\dfrac{v_0}{\omega x_0}

. (iii) Using the results above, write the explicit equation of motion for a particle for which

x_0 = 0.020\,\text{m}

v_0 = 1.2\,\text{m}\,\text{s}^{-1}

and

\omega = 40\,\text{rad}\,\text{s}^{-1}