Question

A particle is executing SHM given by X = A $sin(\pi t+\varphi )$ . The intial displacement of particle is 1 cm and its initial velocity is $\pi cm/sec$. Find the amplitude of motion and initial phase of the particle?

Answer 1

Given,

Position function, $X=A\sin (\pi t+\varphi )......\left(1\right)$

Velocity function, $v=\frac{dX}{dt}=A\pi \cos (\pi t+\varphi )........\left(2\right)$

Condition, (at $t=0,$ velocity $v=\pi cm{s}^{-1}$ and$x=1cm$ )

From equation (1), $1=A\sin (\pi \times 0+\varphi )......\left(3\right)$

From equation (2), $\pi =A\pi \cos (\pi \times 0+\varphi )........\left(4\right)$

From (3) and (4)

$\tan \varphi =1\Rightarrow \varphi =\pi /4$

Put value in equation (3)

$1=A\sin (\pi \times 0+\pi /4)=A\sin \left(\pi /4\right)$

$A=\sqrt{2}cm$

Hence, amplitude is $\sqrt{2}cm$ and phase is $\frac{\pi }{4}$