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

The x-t graph of a particle undergoing simple harmonic motion is as shown in the figure. The acceleration of the particle at t = 43\frac{4}{3}s is

A

332π2cms−2\frac{\sqrt{3}}{32}\pi^{2}\text{cm}s^{- 2}

B

−π232cms−2- \frac{\pi^{2}}{32}\text{cm}s^{- 2}

C

π232cms−2\frac{\pi^{2}}{32}\text{cm}s^{- 2}

D

332π2cms−2\frac{\sqrt{3}}{32}\pi^{2}\text{cm}s^{- 2}

Answer

332π2cms−2\frac{\sqrt{3}}{32}\pi^{2}\text{cm}s^{- 2}

Explanation

Solution

From graph,

A = 1 cm, T = 8 s

x=Asin⁡ωT=Asin⁡2πTtx = A\sin\omega T = A\sin\frac{2\pi}{T}t

At t=43s,t = \frac{4}{3}s,

x=1sin⁡2π8×43=sin⁡π3=32cmx = 1\sin\frac{2\pi}{8} \times \frac{4}{3} = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}cm

In SHM,

Acceleration = −ω2x- \omega^{2}x

=−4π2T2×32cms−2(∵ω=2πT)= - \frac{4\pi^{2}}{T^{2}} \times \frac{\sqrt{3}}{2}cms^{- 2}(\because\omega = \frac{2\pi}{T})

=4π2(8)2×32=−3πr232cms−2= \frac{4\pi^{2}}{(8)^{2}} \times \frac{\sqrt{3}}{2} = - \frac{\sqrt{3}\pi r^{2}}{32}cms^{- 2}