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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?
Solution
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 $\left(0, \frac{\pi}{2}, \frac{3 \pi}{2}\right)$
