The equation x= A sin `omega` t gives the relation between the time t and the corresponding displacement x of a moving particle where A and `omega` are constant. Prove that the acceleration of the particle is proportional to its displacement and is directed opposite to it.

Explanation

X = A sin `omega` t
`:.` Velocity, `v = (dx)/(dt) = omega "A cos" omega`t
and, acceleration a = `(dv)/(dt) = omega A (-omega "sin" omega`t)
`= -omega^(2) "A sin" omega t = -omega^(2)`x
So, `a prop -` x
This means that (i) a is proportional to x (ii) the direction of a is opposite to that of x, as indicated by the negative sign.