Explanation:
In linear simple harmonic motion, the velocity of particle is given by
\(v = \omega \sqrt {{A^2} - {x^2}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
where, \(\omega=\) angular frequency
\(A = \) maximum displacement of amplitude and
\(x = \) displacement from mean position
The acceleration of a particle in simple harmonic motion(SHM), is given by
\(a = {\omega ^2}x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
Here, \(x=\dfrac{A}{2}\)
Alos, \(v = a\) (given)
\(\omega \sqrt{\left(A^{2}-x^{2}\right)}=\omega^{2} x \quad\) from eq.(1) and (2), we get
\(\Rightarrow \sqrt{\left(A^{2}-\dfrac{A^{2}}{4}\right)}=\omega \dfrac{A}{2} \Rightarrow \dfrac{\sqrt{3} A}{2}=\omega \times \dfrac{A}{2}\)
\(\Rightarrow \dfrac{2 \pi}{T}=\sqrt{3}\left[\because \omega=\dfrac{2 \pi}{T}\right]\)
\(\Rightarrow T=\dfrac{2 \pi}{\sqrt{3}} s\)