Question

The $x-t$ graph of a particle undergoing simple harmonic motion is as shown in the figure.

The acceleration of the particle at $t=\frac{4}{3}s$ is

• A. $\frac{\sqrt{3}}{32}{\pi }^{2}cm{s}^{-2}$
• B. $-\frac{{\pi }^2}{32}cm{s}^{-2}$
• C. $\frac{{\pi }^2}{32}cm{s}^{-2}$
• D. $-\frac{\sqrt{3}}{32}{\pi }^{2}cm{s}^{-2}$

Answer 1

Answer: (D)

From graph,
$A=1cm,T=8s$
$x=Asin\omega t=Asin\frac{2\pi }{T}t$
At $t=\frac{4}{3}s$
$x=1sin\frac{2\pi }{8}\times \frac{4}{3}$
$=sin\frac{\pi }{3}=\frac{\sqrt{3}}{2}cm$
In $SHM$,
Acceleration $=-{\omega }^{2}x$
$=-\frac{4{\pi }^2}{T^2}\times \frac{\sqrt{3}}{2}cm{s}^{-2}\left(\because \omega =\frac{2\pi }{T}\right)$
$=-\frac{4{\pi }^2}{{\left(8\right)}^2}\times \frac{\sqrt{3}}{2}$
$=-\frac{\sqrt{3}{\pi }^2}{32}cm{s}^{-2}$