Question

The displacement of a simple harmonic oscillator is represented by x=a sin wt. If the displacement x and velocity v are plotted in rectangular axes, then prove that-
(i) the locus of (x,v ) points will be ellipse and
(ii) this ellipse represents a path of constant energy

Answer 1

Question 13297

a) $x = a\sin \omega t$ , so velocity $\nu = \dot{x} = a\omega \cos \omega t$ . One knows, that the parametric equation of the ellipse in coordinates $(x,y)$ is $x = A\sin \omega t,y = B\cos \omega t$ . Comparing it with equations for $x,\nu$ , in $(x,\nu)$ they will represent an ellipse with $A = a,B = a\omega$

b) Let the energy $E = \frac{m\dot{x}^2}{2} + \frac{k x^2}{2}$ be constant. One might then rewrite the last equation as $\frac{m\dot{x}^2}{2E} + \frac{k x^2}{2E} = 1$ , or $\frac{\dot{x}^2}{2E} + \frac{x^2}{2E} = 1$ . The equation of ellipse is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ . So, constant energy represents the ellipse with $a = \sqrt{2\frac{E}{k}}$ , $b = \sqrt{2\frac{E}{m}}$ , which are constant when $E = const$ .