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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.
| t | 0 | `"T"/4` | `"T"/2` | `"3T"/4` | T |
| ωt | 0 | `π/2` | π | `"3π"/2` | 2π |
| x | 0 | A | 0 | −A | 0 |
| v | ωA | 0 | −ωA | 0 | ωA |
| a | 0 | −ω2A | 0 | ω2A | 0 |
Using the values in the table we can plot graphs of displacement, velocity, and acceleration with time
Graphs of displacement, velocity, and acceleration with time for a particle in SHM starting from the mean position
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.