What is the phase difference between the displacement and velocity displacement and acceleration of a particle performing SHM starting from extreme position?

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Consider a particle performing S.H.M., with amplitude A and period T = 2π/ω starting from the mean position towards the positive extreme position where w is the angular frequency. Its displacement from the mean position (x), velocity (v), and acceleration (a) at any instant are

x = A sin  ωt = A sin `((2π)/"T""t")` .........`(∴ ω = (2π)/"T")`

v = `"dv"/"dt"` = ωA cos ωt = ωA cos `((2π)/"T""t")`

a = − ω2A sin  ωt = − ω2A sin `((2π)/"T""t")`

as the initial phase x = 0.

Using these expressions, the values of x, v, and a at the end of every quarter of a period, starting from t = 0, are tabulated below.

Using the values in the table we can plot graphs of displacement, velocity, and acceleration with time

Conclusions:

• Displacement, velocity and acceleration of S.H.M. are periodic functions of time.
• Displacement time curve and acceleration time curves are sine curves and velocity time curve is a cosine curve.
• There is a phase difference of `π/2` radian between displacement and velocity.
• There is a phase difference of `π/2` radian between velocity and acceleration.
• There is a phase difference of π radian between displacement and acceleration.