Question

When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is :

• A. circular
• B. elliptical
• C. parabolic
• D. straight line

Hint:

If the axes are scaled such that $A = A\omega$ (i.e., $\omega = 1$), the graph becomes a circle. Generally, it is an ellipse.

Answer 1

The Correct Option is B

Step 1: For SHM, $x = A \sin(\omega t)$ and $v = A\omega \cos(\omega t)$.
Step 2: From these, $\frac{x}{A} = \sin(\omega t)$ and $\frac{v}{A\omega} = \cos(\omega t)$.
Step 3: Using $\sin^2\theta + \cos^2\theta = 1$, we get $\frac{x^2}{A^2} + \frac{v^2}{(A\omega)^2} = 1$. This is the equation of an ellipse.