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