A particle is subjected simultaneously to two SHM's, one...

A particle is subjected simultaneously to two SHM's, one along the $x$ -axis and the other along the $y$ -axis. The two vibrations are in phase and have unequal amplitudes. The particle will execute
(a) straight line motion
(b) circular motion
(c) elliptic motion
(d) parabolic motion

Key Concepts

Amplitude Ratio

The amplitude ratio, which is the comparison between the maximum displacements of the two oscillations, plays a key role in shaping the resultant trajectory. When two in-phase SHMs have unequal amplitudes, the difference in magnitude leads to a stretched or compressed path, commonly resulting in an elliptical shape rather than a straight line or circle.

Parametric Equations and Lissajous Figures

When combining two perpendicular SHMs, the motion can be described by parametric equations, resulting in figures known as Lissajous figures. These figures illustrate the complex trajectories that arise from the interplay of amplitude and phase differences, with elliptical paths being a typical outcome when the oscillations are in phase but have unequal amplitudes.

Simple Harmonic Motion

Simple harmonic motion (SHM) refers to a type of periodic oscillation where the restoring force is directly proportional to the displacement. When SHM is applied in two perpendicular directions, the overall motion can be described by combining two independent oscillations, leading to various possible trajectories depending on their amplitudes and phase differences.

Phase Relationship

The phase relationship between two oscillations is crucial in determining the resulting path when they are combined. In-phase vibrations mean that the oscillations reach their maximum and minimum values simultaneously; this synchrony significantly influences the shape of the resulting motion when superimposed.

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