If $\dpi{100} x,v\; and\; a$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $\dpi{100} T$ , then, which of the following does not change with time? Option 1) $a^{2}T^{2}+4\pi ^{2}v^{2}\;$ Option 2) $\; \; \; aT/x\; \;$ Option 3) $\; aT+2\pi v\;$ Option 4) $\; \; aT/v$

As we learnt in

Simple Harmonic Motion -

It is special case of oscillatory motion of a vibrating particle in which the acceleration at any position is directly propotional to displacement from mean position & it is directed towards mean position.

- wherein

$\vec{a}\; \alpha -\vec{x}$

$x=A\ sin(\omega t+\phi)$

$a=-A\omega^{2}sin(\omega t+ \phi)$

So $\frac{aT}{x}=-\omega^{2}x$

Correct option is 2.

Option 1)

$a^{2}T^{2}+4\pi ^{2}v^{2}\;$

This is an incorrect option.

Option 2)

$\; \; \; aT/x\; \;$

This is the correct option.

Option 3)

$\; aT+2\pi v\;$

This is an incorrect option.

Option 4)

$\; \; aT/v$

This is an incorrect option.

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