The acceleration of a (t) particle undergoing SHM is graphed in Fig. 15-21. (a) Which of the labeled points corresponds to the particle at$\mathbf{-}{\mathbf{x}}_{\mathbf{m}}$? (b) At point 4, is the velocity of the particle positive, negative, or zero? (c) At point5, is the particle at $\mathbf{-}{\mathit{x}}_{\mathbf{m}}$, or at $\mathbf{+}{\mathit{x}}_{\mathbf{m}}$, at 0, between and, or between 0 and $\mathbf{+}{\mathit{x}}_{\mathbf{m}}$?

The graph of acceleration versus time for a particle undergoing SHM is given.

From the given graph, we can determine the position of the particle. We use the direction of acceleration according to the position of the particle. The velocity of the particle is positive, negative or zero can be determined from it.

Formula:

The acceleration of the body in SHM, $a=-{\omega }^{2}x$ (i)

a)

We know that in acceleration, SHM is always negative corresponding equation (i). When the particle moves away from mean position and acceleration is positive, the particle moves towards mean position.

Here particle is at point 2; acceleration is positive that is the particle is moving toward mean position. Also, the acceleration has maximum value at point 2.

Hence, point 2 corresponds to the particle at $-{\mathrm{x}}_m$.

b)

At point 4, the particle is at mean position because acceleration is zero. It came from $-x_m$and is moving towards $+{\mathrm{x}}_m$so the velocity is positive.

c)

At point 5, the particle is moving from mean position towards$+{\mathrm{x}}_m$and having negative acceleration.

Hence the particle is between 0 and + ${\mathrm{x}}_m$.