Chapter 12, Problem 78GP

To determine

To Calculate: The resonant frequency that can be expected for the overpass after the modification.

To identify: Whether the modifications will do good or not.

Answer to Problem 78GP

The resonant frequency that overpass can bear is $8\text{Hz}$ and the modifications in the overpass is good to protect it from earthquake.

Explanation of Solution

Given:

The frequency of the earthquake occurred $\mathrm{f}=4\text{Hz}$ .

Formula Used:

Write the expression for the frequency of the wave.

  $\mathrm{f}=\frac{\mathrm{v}}{\mathrm{\lambda }}$

It can be written as:

  $\mathrm{v}=\mathrm{f}\times \mathrm{\lambda }$

Here, $\mathrm{v}$ is the velocity of the wave

  $\mathrm{\lambda }$ is the wavelength of the wave.

Calculation:

Write the expression for the wavelength for the one full loop when there is no support at the middle.

  $\begin{array}{l}\frac{{\mathrm{\lambda }}_1}{2}=\mathrm{L}\\ {\mathrm{\lambda }}_1=2\mathrm{L}\end{array}$

Write the expression for the wavelength for the one full loop when there is a support at the middle of the bridge.

  $\begin{array}{l}2\frac{{\mathrm{\lambda }}_2}{2}=\mathrm{L}\\ {\mathrm{\lambda }}_2=\mathrm{L}\end{array}$

Write the expression for the velocity before modification.

  $\mathrm{v}=\mathrm{f}\times \mathrm{\lambda }$

Here, $\mathrm{v}=343\text{m/s}$

Substitute the values in above expression.

  $\begin{array}{c}343=4\times 2\times \mathrm{L}\\ \mathrm{L}=42.9\text{m}\end{array}$

After modification,

Calculate the resonant frequency that can be withstand by the overpass.

  $\mathrm{v}={\mathrm{f}}_{\mathrm{b}}\times {\mathrm{\lambda }}_2$

Velocity will be the same.

  $\begin{array}{c}343={\mathrm{f}}_{\mathrm{b}}\times \mathrm{L}\\ {\mathrm{f}}_{\mathrm{b}}=\frac{343}{42.9}\\ =8\text{Hz}\end{array}$

The earthquake frequency ranges from $5\text{Hz}$ or $6\text{Hz}$ which is lesser than the frequency overpass can bear that is $8\text{Hz}$ .

Thus, the modification done in the overpass is good.

Conclusion:

The resonant frequency that overpass can bear is $8\text{Hz}$ and the modifications in the overpass is good to protect it from earthquake.