Lissajous figures are produced by superposition of 2 SHM's in mutually perpendicular directions.
If both SHM have same frequency, the lissajous figure is simple. It may be a circle, ellipse or a straight line depending on the phase difference between two SHMs. Interesting case occurs when one of the SHM is at double frequency of another. Suppose a body executes SHM vertically with frequency $f_{o}$ , but horizontally with a frequency 2 $f_{o}$ and is initially at A, mean position of vertical as well as horizontal SHM. It can be seen to trace out a figure of 8 in space as shown. Since horizontal SHM has half the time period, it executes two horizontal oscillations by the time it completes a vertical oscillation.

Question

Lissajous figures are produced by superposition of 2 SHM's in mutually perpendicular directions.

If both SHM have same frequency, the lissajous figure is simple. It may be a circle, ellipse or a straight line depending on the phase difference between two SHMs. Interesting case occurs when one of the SHM is at double frequency of another. Suppose a body executes SHM vertically with frequency $f_{o}$ , but horizontally with a frequency 2 $f_{o}$ and is initially at A, mean position of vertical as well as horizontal SHM. It can be seen to trace out a figure of 8 in space as shown. Since horizontal SHM has half the time period, it executes two horizontal oscillations by the time it completes a vertical oscillation.

G1 Physics · Accepted Answer

Sub 1: the shown curve is a circle. x=A sinωt and y=A cosωt satisfy x²+y²=A² and reproduce the shown orientation, so A is accepted. Sub 2: the shown Lissajous pattern has three horizontal lobes; the lobe count gives f(vertical)/f(horizontal)=3. Correct answer: A.