Question

A particle executes SHM. 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

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

$v=\omega \sqrt{A^2-x^2}$

$\Rightarrow v^2={\omega }^2(A^2-x^2)$

$\Rightarrow v^2={\omega }^2{A}^2-{\omega }^2{x}^2$

$\Rightarrow v^2+{\omega }^2{x}^2={\omega }^2{A}^2$

$\Rightarrow \frac{v^2}{{\omega }^2{A}^2}+\frac{{\omega }^2{x}^2}{{\omega }^2{A}^2}=1$

$\Rightarrow \frac{v^2}{(\omega A{)}^2}+\frac{x^2}{(A{)}^2}=1$

The above equation is similar to the equation of an ellipse $i.e.(\frac{y^2}{a^2}+\frac{x^2}{b^2}=1)$.

So, the graph between velocity and displacement will be an ellipse.