Question

The x-t graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t=\frac{2}{3}s$ is

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

Answer 1

Answer: (B)

Using the general equation of $SHM$
$x=Asin \omega t$
as per graph $A = 1 \,cm, T = 8 \,sec $
$\Rightarrow w=\frac{2 \pi }{T}=\frac{\pi }{4} \\ x=sin\frac{\pi }{4}t$ acceleration $a=\frac{d^{2} x}{d t^{2}}$
${l}\Rightarrow a=\frac{-\pi^2}{16} \sin \frac{\pi}{4} t $
$ \text { at } \mathrm{t}=\frac{2}{3}, a=\frac{-\pi^2}{16} \sin \left(\frac{\pi}{4} \cdot \frac{2}{3}\right)=\frac{-\pi^2}{16} \cdot \sin \left(\frac{\pi}{6}\right) $
$\Rightarrow a=\frac{-\pi^2}{16} \times \frac{1}{2}=\frac{-\pi^2}{32} \mathrm{~cm} / \mathrm{s}^2$