Question

A particle executes S.H.M., the graph of velocity as a function of displacement is:
a. a circle
b. a parabola
c. an ellipse
d. a helix

Answer 1

Hint: SHM, or Simple Harmonic Motion, is described as a movement wherein the restoring force is directly related to the body's displacement from its mean position. This restorative power is always directed toward the mean position.

Complete answer:
First, we will discuss about the simple harmonic motion. A simple harmonic motion is an oscillatory motion in which the acceleration of each particle is directly proportional to its displacement from its mean position at any given point. It is an example of oscillatory motion. The sine (or) cosine functions, collectively called sinusoids, can be used to describe displacement, velocity, acceleration, and force in this form of oscillatory motion.

As we know that, for a body to perform simple harmonic motion, the relation between displacement and velocity must be,

Now squaring both the sides of the equation
${{v}^{2}}={{\omega }^{2}}({{A}^{2}}-{{x}^{2}})$
${{v}^{2}}={{\omega }^{2}}{{A}^{2}}-{{\omega }^{2}}{{x}^{2}}$
${{v}^{2}}+{{\omega }^{2}}{{x}^{2}}={{\omega }^{2}}{{A}^{2}}$

Now dividing whole equation by${{\omega }^{2}}{{A}^{2}}$
${{v}^{2}}/{{(\omega A)}^{2}}+{{\omega }^{2}}{{x}^{2}}/{{\omega }^{2}}{{A}^{2}}={{\omega }^{2}}{{A}^{2}}/{{w}^{2}}{{A}^{2}}$
${{v}^{2}}/{{\omega }^{2}}{{A}^{2}}+{{x}^{2}}/{{A}^{2}}=1$
Because the above equation is identical to the classic ellipse equation, the graph of velocity and displacement will become an ellipse.

Therefore, the graph of velocity as a function of displacement is an ellipse. The right response to the question is therefore Option (C).

Note: Simple Harmonic Movements (SHM) are all oscillatory and periodic, however not all oscillatory motions are Simple Harmonic Motion. The most important oscillatory movement is simple harmonic motion, which is often referred to as oscillatory motion or the harmonic motion of all oscillatory movements (SHM).