A particle executes S.H.M., the graph of velocity as a function of displacement is:c
Question:

A particle executes S.H.M., the graph of velocity as a function of displacement is:

1. (1) a circle

2. (2) a parabola

3. (3) an ellipse

4. (4) a helix

Correct Option: , 3

Solution:

(3)

For a body performing SHM, relation between velocity and displacement $v=\omega \sqrt{A^{2}-x^{2}}$ now, square both side $v^{2}=w^{2}\left(A^{2}-x^{2}\right)$

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

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

divide whole equation by $\omega^{2} \mathrm{~A}^{2} \frac{\mathrm{v}^{2}}{\omega^{2} \mathrm{~A}^{2}}+\frac{\omega^{2} \mathrm{X}^{2}}{\omega^{2} \mathrm{~A}^{2}}=\frac{\omega^{2} \mathrm{X}^{2}}{\omega^{2} \mathrm{~A}^{2}}$

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

above equation is similar as standard equation of ellipes, so graph between velocity and displacement will be ellipes.