In this article, we will learn about the mathematical equation of velocity and acceleration of a simple harmonic motion. Let us consider a particle P moving in a circle with a tangential velocity of v and the angular speed can be given as,
ω, as v = ω A, where ω is the angular velocity and A is the radius of the circle.
The projection of its tangential velocity on the x-axis is given by,
v(t) = –ωA sin (ωt + φ )
From the figure, we can see that the velocity component of P is directed towards the negative X direction, hence the projection is negative. Here, the above equation gives the instantaneous velocity of the particle Q, which is the projection of particle P. We can say that the particle Q is executing simple harmonic motion.
For a particle executing uniform circular motion, the radial acceleration is directed towards the centre. For the particle P moving with a tangential acceleration of magnitude ‘at’, we can write the radial acceleration as, ar = ω2A.
The projection of radial acceleration on the x-axis is given by,
ar(t) = ω2A cos (ωt + φ) = –ω2x (t),
which can be assumed to be the acceleration of a particle Q, which is the projection of particle P. Here, we can say that, particle Q is executing simple harmonic motion and the above equation gives the magnitude of the acceleration of this particle executing simple harmonic motion. From the expression for velocity and acceleration, it can be inferred that, for a particle executing simple harmonic motion, the acceleration can be given as the differentiation of velocity with respect to time and that acceleration is proportional to the displacement of the particle and is directed towards the mean position. From the expressions, we can see that the phase of velocity is shifted by a phase angle π/2 with respect to the displacement and that of acceleration is shifted by a phase angle of π. Hence, when the magnitude of displacement is the least, that of the velocity is the highest and when displacement has a positive value, the acceleration has a negative value.
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