A spring having with a spring constant $1200\; N m ^{-1}$ is mounted on a hortzontal table as shown in Figure A mass of $3 \;kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \;cm$ and released 

let us take the position of mass when the spring is unstreched as $x=0,$ and the direction from left to right as the positive direction of $x$ -axis. Give $x$ as a function of time $t$ for the oscillating mass if at the moment we start the stopwatch $(t=0),$ the mass is

$(a)$ at the mean position,

$(b)$ at the maximum stretched position, and

$(c)$ at the maximum compressed position. In what way do these functions for $SHM$ differ from each other, in frequency, in amplitude or the inittal phase?

The functions have the same frequency and amplitude, but different initial phases

Distance travelled by the mass sideways, $A=2.0 \,cm$

Force constant of the spring, $k=1200\, N m ^{-1}$

Mass, $m=3 \,kg$

Angular frequency of oscillation:

$\omega=\sqrt{\frac{k}{m}}$

$=\sqrt{\frac{1200}{3}}=\sqrt{400}=20 \,rad s ^{-1}$

When the mass is at the mean position, initial phase is $0 .$

Displacement, $x=A \sin \omega t$

$=2 \sin 20 t$

At the maximum stretched position, the mass is toward the extreme right. Hence, the

initial phase is $\frac{\pi}{2}$

Displacement, $x=A \sin \left(\omega t+\frac{\pi}{2}\right)$

$=2 \sin \left(20 t+\frac{\pi}{2}\right)$

$=2 \cos 20 t$

At the maximum compressed position, the mass is toward the extreme left. Hence, the initial phase is $\frac{3 \pi}{2}$

$x=A \sin \left(\omega t+\frac{3 \pi}{2}\right)$

Displacement,

$=2 \sin \left(20 t+\frac{3 \pi}{2}\right)=-2 \cos 20 t$

The functions have the same frequency $\left(\frac{20}{2 \pi} Hz \right)$ and amplitude $(2 \,cm ),$ but different initial phases $\left(0, \frac{\pi}{2}, \frac{3 \pi}{2}\right)$