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

Answer 1

Answer: (B)

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