Question

Exercise:

[ A spring with a spring constant of 1200 N m${}^{-1}$ is mounted on a horizontal table as shown in the 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. ]

In Exercise, let us take the position of mass when the spring is unstretched 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 function for SHM differ from each other in frequency, in amplitude or the initial phase?

Answer 1

Distance travelled by the mass sideways, $a=2.0cm$
Angular frequency of oscillation:

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

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

(a) As time is noted from the mean position, hence using

$x=asin\omega t$ we hav $x=2sin20t$

(b) At maximum stretched position, the body is at the extreme right position, with an intial phase of $\pi /2$ rad. Then,

$x=asin(\omega t+\frac{\pi }{2})=acos\omega t=2cos20t$

(c) At maximum compressed position, the body is at left position, with an intial phase of $3\pi /2rad$. Then,

$x=asin(\omega t+\frac{3\pi }{2})=-acost=-2cos20t$

The functions neither differ in amplitude nor in frequency. They differ in intial phase.