Question

A body oscillates with $SHM$ according to the equation $x=6\cos (3\pi t+\frac{\pi }{3})cm$; the displacement, velocity and acceleration at time $t=2sec$ in magnitude are:

• A. $1,9\pi ,27{\pi }^2$
• B. $3,9\pi ,27{\pi }^2$
• C. $3,9\pi \sqrt{3},27{\pi }^2$
• D. $3,9{\pi }^2,27{\pi }^2$

Answer 1

Answer: (C)

$3,9\pi \sqrt{3},27{\pi }^2$

Given,

$x=6cos(3\pi t+\frac{\pi }{3})$

Displacement at $t=2s$

$x{|}_{t=2s}=6cos(3\pi \times 2+\frac{\pi }{3})$

$x{|}_{t=2s}=6cos\left(6.33\pi \right)$

$x{|}_{t=2s}=6\times 0.50=3m$

Velocity, $v=\frac{dx}{dt}$

$v=-6\times 3\pi sin(3\pi t+\pi /3)=-18\pi sin(3\pi t+\pi /3)$

$v{|}_{t=2s}=-18\pi sin\left(6.33\pi \right)$

$v{|}_{t=2s}=-18\pi \times \frac{\sqrt{3}}{2}$

Magnitude, $|v_{t=2s}|=9\pi \sqrt{3}m/s$

Now, the acceleration of the body is ,

$a=\frac{dv}{dt}$

$a=-18\times 3{\pi }^{2}cos(3\pi t+\pi /3)$

$a_{t=2s}=-54{\pi }^{2}cos\left(6.33\pi \right)$

$a_{t=2s}=-54{\pi }^2\times 0.5=-27{\pi }^2$

Magnitude, $|a_{t=2s}|=27{\pi }^{2}m/s^2$

The correct option is C.