Question

Two simple harmonic motions, as shown below, are at right angles. They are combined to form Lissajous figures.
$x(t) = A \, \sin \, (at + \delta )$
$y(t) = B \, \sin \, ( bt)$
Identify the correct match below.

• A. Parameters $- A \neq B , a = b ; \delta=0$
  Curve - Parabola
• B. Parameters - $A=B, a=b ; \delta=\pi / 2$
  Curve - Line
• C. Parameters - $A \neq B, a=b ; \delta=\pi / 2$
  Curve - Ellipse
• D. Parameters $-A=B, a=2 b ; \delta=\pi / 2$
  Curve - Circle

Answer 1

Answer: (C)

The given simple harmonic motions to form Lissajous figures are $x(t)=A \sin (a t+\delta)$ and $y(t)=B \sin (b t)$.
For parabola, conditions should be$\,\,\,\,\,A=B$ or $A \neq B, a=2 b, \delta=\pi / 2$
For line, conditions should be$\,\,\,\,\,A=B, a=b, \delta=\pi$
For cir ould be$\,\,\,\,\,A=B, a=b ; \delta=\pi / 2$
For ellipse, condition should be$\,\,\,\,\,A \neq B, a=b ; \delta=\pi / 2$
Therefore, we obtain an ellipse.