Question

Which of the following combinations of Lissajous figure will be like infinite ( $\infty$ )
(A) $x = a\sin \omega t$ $y = b\sin \omega t$
(B) $x = a\sin 2\omega t{\text{ }}y = b\sin \omega t$
(C) $x = a\sin \omega t{\text{ }}y = b\sin 2\omega t$
(D) $x = a\sin 2\omega t{\text{ }}y = b\sin 2\omega t$

Answer 1

Hint : For a lissajous figure which looks like infinity, the phase angle of the horizontal input must be half the phase angle of the vertical. The phase angles are the quantities after the sine.

Formula used: In this solution we will be using the following formula;
$x = A\sin (at + d)$ and $y = B\sin bt$ where $A$ can be said to be the amplitude for x axis, and $B$ is amplitude for y axis.
The value $at + d$ and $bt$ are phase angles, and $d$ is the angle in which a line drawn along the length of the figure makes with the x axis, or the angle of rotation.

Complete step by step answer
The lissajous figure (also known as lissajous curve or Bowditch curve) is the curve of a system traced out by parametric equations. The significance in physics is that they describe complex harmonic motions (as opposed to simple harmonic motions).
Generally, the lissajous figure can be gotten from these two equations, which are
$x = A\sin (at + d)$ and $y = B\sin bt$ where $A$ can be said to be the amplitude for x axis, and $B$ is amplitude for y axis. The value $at + d$ and $bt$ are phase angles, and $d$ is the angle in which a line drawn along the length of the figure makes with the x axis, or the angle of rotation. When $d$ is zero, the x and y components are said to be in phase
Using this, for a lissajous figure to look like infinity, the shift $d$ must be equal to zero.
Also the ratio of $a$ to $b$ must be equal to 1:2.
Hence, by observation of the options, the option which will give this ratio is option C.
Thus, the correct option is C.

Note
For clarity, the $a$ in option A is $\omega$ and $b$ is also $\omega$ , hence the ratio is 1:1
The $a$ in option B is $2\omega$ and $b$ is $\omega$ , hence the ratio is 2:1
Also, $a$ in option D is $2\omega$ and $b$ is also $2\omega$ , hence the ratio is 2:2 which is 1:1
But in option C, $a = \omega$ and $b = 2\omega$ hence the ratio is 1:2 as wanted.