The question asks about the nature of the graph that represents the velocity of a particle as a function of its displacement when the particle executes Simple Harmonic Motion (SHM). Let's analyze this step-by-step:
Understanding Simple Harmonic Motion: In SHM, a particle moves back and forth about an equilibrium position in such a way that its acceleration is proportional to its displacement from that position and directed towards it. This motion can be described by equations involving sinusoidal functions.
Velocity-Displacement Relationship: If we denote displacement by x and velocity by v, and use the angular frequency \omega, the velocity in SHM is given by the formula:
v = \pm \omega \sqrt{A^2 - x^2}
where A is the amplitude of motion.
This indicates that v^2 = \omega^2 (A^2 - x^2).
Equation of an Ellipse: The equation \frac{v^2}{\omega^2} + \frac{x^2}{A^2} = 1 can be rearranged to \left(\frac{x}{A}\right)^2 + \left(\frac{v}{\omega A}\right)^2 = 1, which is the standard form of an ellipse centered at the origin with semi-major and semi-minor axes.
Graphical Representation: The velocity-displacement graph in SHM is elliptical because the derived equation resembles an ellipse, implying that the trajectory of the velocity versus displacement, when plotted, forms an elliptical shape.
Therefore, the correct answer is Elliptical.
As for ruling out other options:
Circular: This would imply a constant radius, which is not the case in SHM.
Sinusoidal: This is typically used to describe the motion versus time in SHM, not velocity versus displacement.
Straight line: A linear relation would indicate a direct proportionality without any corrective force like in SHM, hence not applicable.
