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