Question

A particle is performing simple harmonic motion.
(i) its velocity-displacement graph is parabolic in nature
(ii) its velocity-time graph is sinusoidal in nature
(iii) its velocity-acceleration graph is elliptical in nature
Correct answer is:

• A. (i), (ii) and (iii)
• B. (ii) and (iii)
• C. (i) and (ii)
• D. (i) and (iii).

Answer 1

The correct option is B (ii) and (iii)

Let the displacement equation of the particle executing $SHM$ is given as $x=Asinwt$ .....(1)

$⟹sinwt=\frac{x}{A}$ ..........(1)

Thus the velocity equation is given by $v=Awcoswt⟹coswt=\frac{v}{Aw}$ ..............(2)

Hence velocity- time graph is a sinusoidal in nature.

And acceleration equation is $a=-A{w}^{2}sinwt⟹sinwt=\frac{a}{-A{w}^2}$ ...............(3)

Now squaring and adding (1) & (2), $⟹1=\frac{x^2}{A^2}+\frac{v^2}{A^2{w}^2}$ which represents an ellipse

Squaring and adding (3) & (2), $⟹1=\frac{a^2}{A^2{w}^4}+\frac{v^2}{A^2{w}^2}$ which also represents an ellipse

Hence options $\left(ii\right)$ and $\left(iii\right)$ are correct.