Solved Problem on Simple Harmonic Motiom (SHM)
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The figures represent the composition of two SHM of frequencies f1 and f2 on the orthogonal axes x and y. If the frequency of motion, in the x direction is equal to 300 Hz for all figures, what is the frequency in the y direction in each case?
Problem data:
- Frequency of motion in the x direction: f1 = f2 = 300 Hz.
To calculate the frequency in the y direction in each of the graphs, we draw two secant lines to the curves, one parallel to the x-axis and the other to the y-axis (the lines must not coincide with the coordinate axes). Using the following expression
\[
\begin{gather}
\bbox[#99CCFF,10px]
{\frac{f_{y}}{f_{x}}=\frac{n_{x}}{n_{y}}} \tag{I}
\end{gather}
\]
where
- fx and fy are the frequencies in the x and y directions, respectively;
- nx and ny are the number of intersections of the secant lines with the Lissajours curves.
In Figure (1), we have nx = 2 and ny = 2.
Substituting these values and the given frequency fx, in expression (I), we calculate fy
Substituting these values and the given frequency fx, in expression (I), we calculate fy
\[
\begin{gather}
\frac{f_{y}}{300}=\frac{2}{2}
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{f_{y}=300\;\text{Hz}}
\end{gather}
\]
Figure 1
In Figure (2), we have nx = 3 and ny = 2.
For the calculation of fy
For the calculation of fy
\[
\begin{gather}
\frac{f_{y}}{300}=\frac{3}{2}\\[5pt]
f_{y}=\frac{3}{2}\times 300\\[5pt]
f_{y}=\frac{900}{2}
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{f_{y}=450\;\text{Hz}}
\end{gather}
\]
Figure 2
In Figure (3), we have nx = 4 and ny = 3.
For the calculation of fy
For the calculation of fy
\[
\begin{gather}
\frac{f_{y}}{300}=\frac{4}{2}\\[5pt]
f_{y}=2\times 300
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{f_{y}=600\;\text{Hz}}
\end{gather}
\]
Figure 3
In Figure (4), we have nx = 2 and ny = 4.
For the calculation of fy
For the calculation of fy
\[
\begin{gather}
\frac{f_{y}}{300}=\frac{2}{4}\\[5pt]
f_{y}=\frac{2}{4}\times 300\\[5pt]
f_{y}=\frac{600}{4}
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{f_{y}=150\;\text{Hz}}
\end{gather}
\]
Figure 4
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .