In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Question

In the above Q., 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 initial phase?

Answer 1

Here, the maximum displacement = Amplitude (A) = 2 cm

(a) If the time t = 0 at x = 0, the displacement can be expressed as the function of t.

i.e., x(t) = A sin ωt

But, ω = √{k/m} = √{1200/3} = 20 rad *-1

i.e., x(t) = 2 sin 20 t

(b) If the time t = 0 at x = 2 cm (i.e., positive extreme position), the displacement is expressed as the cosine function.

i.e., x(t) = 2 cos 20 t

(c) If the time t = 0 at x = -2 cm (i.e., negative extreme position), the displacement is expressed as the cosine function given by

x(t) = -cos 20 t

The above functions have same values of frequency and amplitude but they differ in phase.