Question

Prove that the following two equations represent the simple harmonic motion-

X = A cosωt + B sinωt

X = A eiωt

Answer 1

The equation of harmonic oscillations is $x(t)=A'\sin(\omega t+\phi)$ or $x(t)=A'\cos(\omega t+\phi)$ . So,

a) $x(t)=A'\sin(\omega t+\phi)=A'(\sin\omega t\cdot\ \cos\phi+\cos\omega t\cdot \sin\phi)=$

$=A'\sin\omega t\cdot\ \cos\phi+A'\cos\omega t\cdot \sin\phi=B\sin\omega t+A\cos\omega t=$

$=A\cos\omega t+B\sin\omega t$ . Proved

b) For the simple harmonic motion

$\frac{d^2x(t)}{dt^2}+x(t)\omega^2=0$

$x(t)=Ae^{i\omega t}$

$-A\omega^2e^{i\omega t}+Ae^{i\omega t}\omega^2=0$ . Proved