The velocity of a particle in simple harmonic motion is given as

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

and magnitude of its acceleration is

$a=\omega^{2} x$

Given $|v|=|a|$

$\therefore \omega \sqrt{A^{2}-x^{2}}-\omega^{2} x$

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

${\omega^{2}=\frac{A^{2}-x^{2}}{x^{2}}=\frac{9-4}{4}=\frac{5}{4}}$

${\omega=\frac{\sqrt{5}}{2}}$

Time period, $T=\frac{2 \pi}{\omega}=2 \pi \cdot \frac{2}{\sqrt{5}}=\frac{4 \pi}{\sqrt{5}} \mathrm{s}$