Chapter 11, Problem 18P

To determine

To Find:The ratio of the amplitudes.

Answer to Problem 18P

The ratio of the amplitudes of the two vibrations is $2.65$ .

Explanation of Solution

Introduction:

Frequency and masses of two vibrations are same. Spring constant of two vibrations are also same.

Spring energy is

  $E=\frac{1}{2}\times k\times A^2$

Here,

  $E$ is spring energy,

  $k$ is spring constant,

  $A$ is amplitude of spring.

Total energy is given by maximum potential energy. So, it is expressed by the following way.

  $\begin{array}{l}{E}_1=\frac{1}{2}\times k\times A_1{}^2\\ E_2=\frac{1}{2}\times k\times A_2{}^2\\ \frac{E_1}{E_2}=\frac{\frac{1}{2}\times k\times A_1{}^2}{\frac{1}{2}\times k\times A_2{}^2}=\frac{A_1{}^2}{A_2{}^2}=7\\ \frac{A_1}{A_2}=\sqrt{7}=2.65\end{array}$

Conclusion:

The ratio of the amplitudes of the two vibrations is $2.65$ .