Chapter 11, Problem 76GP

a.

To determine

To Find: The frequency of the vibration.

Answer to Problem 76GP

The frequency of the vibration is $1.455\text{Hz}$ .

Explanation of Solution

Given Information:

Mass of wooden raft, $\mathrm{m}=220\text{kg}$ ,

Factor of compression, $\mathrm{x}=4.0\text{cm}$ ,

Weight of man, $\mathrm{M}=75\text{kg}$ ,

Acceleration due to gravity $\mathrm{g}=9.8{\text{m/s}}^{\text{2}}$ .

Formula used:

According to Hooke's law:

  $\mathrm{k}=\frac{\mathrm{f}}{\mathrm{x}}$

And the frequency of vibration is

  $\mathrm{k}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$

Here,

Calculation:

From Hooke's law$\mathrm{k}=\frac{\mathrm{f}}{\mathrm{x}}$ = $\frac{\mathrm{m}\mathrm{g}}{\mathrm{x}}$

  $\mathrm{k}=\frac{(75\text{kg})(9.8{\text{m/s}}^{\text{2}})}{4.0\times {10}^{-2}\text{m}}$

  $\mathrm{k}=1.8375\times {10}^4\text{N/m}$

Then the frequency is

  $\begin{array}{l}\mathrm{f}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{1.8375\times {10}^4\text{N/m}}{220\text{kg}}}\\ \mathrm{f}=1.455\text{Hz}\end{array}$

Conclusion:

The frequency of the vibration is $1.455\text{Hz}$ .

b.

To determine

To Find: The total energy of vibration.

Answer to Problem 76GP

The total energy of vibration is $14.7\text{J}$ .

Explanation of Solution

Given Information:

Factor of compression, $\mathrm{x}=4.0\text{cm}$

The spring constant, $\mathrm{k}=1.8375\times {10}^4\text{N/m}$

Formula used:

The total energy of vibration, ${\mathrm{E}}_{\text{total}}=\frac{1}{2}\mathrm{k}{\mathrm{x}}^2$

Here, k is the spring constant.

Calculation:

The total energy of vibration

  ${\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\frac{1}{2}\mathrm{k}\times {\mathrm{x}}^2$

Substitute the values in the above expression.

  $\begin{array}{c}{\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\frac{1}{2}\times 1.8375\times {10}^4\times {4.0}^2\times {10}^{-4}\\ =14.7\text{J}\end{array}$

Conclusion:

The total energy of vibration is $14.7\text{J}$ .