Question

Match List-I with List-II on the basis of two simple harmonic signals of the same frequency and various phase differences interacting with each other:

LIST-I (Lissajous Figure)		LIST-II (Phase Difference)
A.	Right handed elliptically polarized vibrations	I.	Phase difference = \( \frac{\pi}{4} \)
B.	Left handed elliptically polarized vibrations	II.	Phase difference = \( \frac{3\pi}{4} \)
C.	Circularly polarized vibrations	III.	No phase difference
D.	Linearly polarized vibrations	IV.	Phase difference = \( \frac{\pi}{2} \)

Choose the correct answer from the options given below:

• A. A - I, B - II, C - III, D - IV
• B. A - I, B - III, C - II, D - IV
• C. A - I, B - II, C - IV, D - III
• D. A - III, B - IV, C - I, D - II

Hint:

For Lissajous figures with two SHMs of the same frequency: - Phase diff \(0\) or \(\pi\) \(\rightarrow\) Straight Line - Phase diff \(\pi/2\) or \(3\pi/2\) (and equal amplitudes) \(\rightarrow\) Circle - Any other phase diff \(\rightarrow\) Ellipse

Answer 1

The Correct Option is C

Step 1: Analyze the superposition of two perpendicular simple harmonic motions (SHMs) with identical frequencies. The resultant motion's form is contingent upon the phase difference \(\delta\).

• D. Linearly Polarized Vibrations: The resulting motion is rectilinear. This occurs when the phase difference is \(\delta = 0\) or \(\pi\). This corresponds to III (No phase difference).
• C. Circularly Polarized Vibrations: The resulting motion is circular. This occurs when amplitudes are equal and the phase difference is \(\delta = \pi/2\) or \(3\pi/2\). This corresponds to IV.
• A. & B. Elliptically Polarized Vibrations: For any other phase difference, the resulting motion is elliptical.
  • For phase differences between \(0\) and \(\pi\), such as \(\pi/4\) and \(3\pi/4\), the polarization is elliptical. By convention, phase differences in this range determine right or left-handedness based on the tracing direction. \(\delta = \pi/4\) results in a right-handed ellipse (A matches I), and \(\delta = 3\pi/4\) results in another ellipse, defined here as left-handed (B matches II).

Step 2: Formulate the correct matching sequence based on the analysis.

• A \(\rightarrow\) I
• B \(\rightarrow\) II
• C \(\rightarrow\) IV
• D \(\rightarrow\) III

This sequence is A - I, B - II, C - IV, D - III, corresponding to option (3).