Simple Harmonic Question regarding acceleration

Question

I would just like to clear something up in regards to acceleration in SHM.

When a question asks to find maximum acceleration, why do the solutions require you to use the absolute value of acceleration to find the result?

Thanks

Answer 1

When you have a simple harmonic motion then the particle position is given by

$$x(t) = A \cos(\omega t + \theta_0),$$

where $\omega$ is the angular frequency, $\theta_0$ is the initial angle and $A >0$ is the motion's amplitude.

The amplitude always multiplies the cosine/sine function and represents the maximum value of the physical entity.

Therefore, in order to find the farthest position you need to compute the largest $|x|$, which is $A$. The same idea applies to $a(t) = d^2 x/dt^2$. In this case

$$a(t) = -A\omega^2 \cos(\omega t + \theta_0)$$

and the maximum acceleration is $a_{\text{max}} = A\omega^2$.