Question

Although California is known for earthquakes, it has large regions dotted with precariously balanced rocks that would be easily toppled by even a mild earthquake. The rocks have stood this way for thousands of years, suggesting that major earthquakes have not occurred in those regions during that time. If an earthquake were to put such a rock into sinusoidal oscillation (parallel to the ground) with a frequency of$\mathbf{2}\mathbf{.}\mathbf{2}\mathbf{Hz}$, an oscillation amplitude of$\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{cm}$would cause the rock to topple. What would be the magnitude of the maximum acceleration of the oscillation, in terms of g?

Short answer

The magnitude of maximum acceleration in terms of g is$0.19\text{g}$

Step-by-step solution

Given

1. Frequency of oscillation of the rock$f=2.2\text{Hz}.$
2. Maximum displacement of the rock$x_m=1.0\text{cm}=0.01\text{m}$

Understanding the concept

A precariously balanced rock oscillates harmonically parallel to the ground when earthquakes occur. Hence, we can applytheequations of SHM to studythemotion of the rock.

The angular frequency of oscillation is given as-

$\omega =2\pi \text{f}$

The maximum acceleration is given as-

${\text{a}}_{max}={\omega }^2{\text{x}}_m$

Calculate the magnitude of the maximum acceleration of the oscillation, in terms of

The frequency of oscillation$\left(f\right)$ of diaphragm is related to angular frequency (ω) by the relation,

$\omega =2\pi \text{f}$

Putting the values, we get

$\begin{array}{rcl}\omega & =& 2\pi \text{f}\\ & =& 2\times 3.14\times 2.2\text{Hz}\\ & =& 13.8\text{rad/s}\\ & & \end{array}$

For SHM, the magnitude of maximum acceleration is given by${\text{a}}_{max}={\omega }^2{\text{x}}_m$

Putting the values,

$$\begin{gathered} \begin{array}{rcl}{\text{a}}_{max}& =& {\omega }^2{\text{x}}_m{ \\ }^{\mathit{\text{=}}{\left(13.8\raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{$s$}\right.\right)}^{\mathit{2}}}\times \left(0.01\text{m}\right)\\ & =& 1.90{\text{m/s}}^{\text{2}}\end{array} \end{gathered}$$

In terms of g, the acceleration is,

$$\begin{gathered} {\text{a}}_{max}=\frac{1.90{\displaystyle \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.}}{9.8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.}g \\ =0.19\text{g} \end{gathered}$$

The maximum acceleration of the oscillations is $0.19\text{g}$.