In this article, we shall study graphical representation of S.H.M. i.e. variation in displacement, velocity, and acceleration with time for a body performing S.H.M. starting from a) the mean position and b) from the extreme position.
Graphical Representation of Linear S.H.M. of a Particle Starting from Mean Position:
The general equation for the displacement of a particle performing linear S.H.M. at any instant ‘t’ is given by
x = a sin (ωt + α )
Where a = amplitude of S.H.M., ω = angular speed of S.H.M.,
α = Initial phase of S.H.M.
As particle is starting from mean position, α = 0
x = a sin ωt …….. (1)
Velocity of particle performing S.H.M.can be obtained by differentiating above expression
v = dx/dt = a cos ωt . ω = ωa cos ωt
v = ωa cos ωt …….. (2)
Acceleration of particle performing S.H.M. can be obtained by differentiating above expression
f = dv/dt = ωa (-sin ωt) ω
f = dv/dt = – ω²a sin ωt …….. (3)
From equation (1) and (3) we have
f = dv/dt = – ω²x …….. (4)
Using equations (1), (2) and (4) and knowing ω = 2π/T we prepare following table
| Time (t) | Phase Φ = ωt = (2π/T)t | Displacement (x) | Velocity (v) | Acceleration (f) |
| 0 | 0 | 0 | aω | 0 |
| T/4 | π/2 | a | 0 | – aω² |
| T/2 | π | 0 | – aω | 0 |
| 3T/4 | 3π/2 | – a | 0 | a ω² |
| T | 2π | 0 | aω | 0 |
The graphs of displacement, velocity and acceleration versus time are as follows:
Graphical Representation of Linear S.H.M. of a Particle Starting from Extreme Position:
The general equation for displacement of a particle performing linear S.H.M. at any instant ‘t’ is given by
x = a sin (ωt + α )
Where a = amplitude of S.H.M., ω = angular speed of S.H.M., α = Initial phase of S.H.M.
As particle is starting from mean position, α = π/2
x = a sin (ωt + π/2 )
x = a cos ωt …….. (1)
Velocity of particle performing S.H.M.can be obtained by differentiating above expression
v = dx/dt = a (- sin ωt) . ω = – ωa sin ωt
v = – ωa sin ωt …….. (2)
Acceleration of particle performing S.H.M. can be obtained by differentiating above expression
f = dv/dt = – ωa (cos ωt) ω
f = dv/dt = – ω²a cos ωt …….. (3)
From equation (1) and (3) we have
f = dv/dt = – ω²x …….. (4)
Using equations (1), (2) and (4) and knowing ω = 2π/T we prepare following table
| Time (t) | Phase Φ = ωt = (2π/T)t | Displacement (x) | Velocity (v) | Acceleration (f) |
| 0 | 0 | a | 0 | – aω² |
| T/4 | π/2 | 0 | aω | 0 |
| T/2 | π | – a | 0 | aω² |
| 3T/4 | 3π/2 | 0 | – aω | 0 |
| T | 2π | a | o | – aω² |
The graphs of displacement, velocity and acceleration versus time are as follows:
Conclusions:
- Graphs are drawn for displacement, velocity and acceleration against time and curves are obtained as shown. As the curves have the shape same as the sine curve, the curves are called as harmonic curves.
- From the graph, we can conclude that the displacement, velocity, and acceleration are the periodic functions of time.
- From the graph, we can see that velocity is 90° (π/2 radians) out of phase with displacement, whereas acceleration is 180° (π radians) out of phase with displacement. Similarly, acceleration is 90° (π/2 radians) out of phase with velocity.
- The velocity leads the displacement by a phase difference of π/2 radians.
- The acceleration lags behind displacement by a phase of π radians.
- The displacement and acceleration are maximum at the extreme position while velocity is minimum at the same position. Similarly, the displacement and acceleration are minimum at the mean position while velocity is maximum at the same position.
- All curves repeat after a phase of 2π radians.
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Science > Physics > Oscillations: Simple Harmonic Motion > Graphical Representation of S.H.M.
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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.