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
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
The graphs of displacement, velocity and acceleration versus time are as follows: Conclusions:
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