Question

Derive an expression for instantaneous velocity and acceleration of a particle executing simple harmonic motion

Answer 1

According to the simple equation of particle doing simple harmonic motion, it's equation for it's displacement is given by

x = A * sin(wt + ©)

where, x is it's displacement from mean position,
A = amplitude
w = angular frequency of the oscillation
© = initial phase constant

So,
differentiating this x equation with respect to Time we get Velocity

v = A* cos(wt+©) * w ...(due to chain differentiation)

v = Aw cos(wt+©)

Now, again differentiating this with respect to time , we get acceleration

a = Aw * - sin(wt+©) * w ...(again due to chain differentiation)

So,
a = -Aw^2 sin(wt+©)

Now, x = A * sin(wt + ©)

So,
a = - w^2 * x

So, these will be the equations for instantaneous velocity and acceleration of particle undergoing simple harmonic motion.

Answer 2

the expression for velocity and acceleration, it can be inferred that, for aparticle 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 theparticle and is directed towards the mean ...