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

The correct option is C

an ellipse

Explanation for the correct option(s):

C: An eclipse

For a body performing SHM, the relation between velocity and displacement

$v=\omega \sqrt{(A^2–x^2)}$

Now, square both the sides

$$\begin{gathered} v^2={\omega }^2(A^2–x^2) \\ {v}^2={\omega }^2{A}^2–{\omega }^2{x}^2 \\ {v}^2+{\omega }^2{x}^2={\omega }^2{A}^2 \end{gathered}$$

Divide the whole equation by ${\omega }^2{A}^2$

$$\begin{gathered} v^2/{\omega }^2{A}^2+{\omega }^2{x}^2/{\omega }^2{A}^2={\omega }^2{A}^2/{\omega }^2{A}^2 \\ {v}^2/{\left(\omega A\right)}^2+x^2/{\left(A\right)}^2=1 \end{gathered}$$

The above equation is similar to the standard equation of an ellipse, so the graph between velocity and displacement will be an ellipse.

Hence, option C is correct.