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Three resonant frequencies of a string are 90, 150 and 210 Hz. (a) Find the highest possible fundamental frequency of vibration of this string. (b) Which harmonics of the fundamental are the given frequencies ? (c) Which overtones are these frequencies ? (d) If the length of the string is 80 cm, what would be the speed of a transverse wave on this string ?

Answer

The resoN/Ant frequencies of a string are
\({{f}_{{1}}=}{90}{H}{z}\)
\({{f}_{{2}}=}{150}{H}{z}\)
\({{f}_{{3}}=}{210}{H}{z}\)
a. The highest possible fundamental frequency of the string is
\({f}={30}{H}{z}\)
[because \({{f}_{{1}},}{{f}_{{2}}{\quad\text{and}\quad}}{{f}_{{3}}}\) are integral multiples of \({30}{H}{z}\)]
b. The frequencies are
\({{f}_{{1}}=}{3}{f}\)
\({{f}_{{2}}=}{5}{f}\)
\({{f}_{{3}}=}{7}{f}\)
so, \({{f}_{{1}},}{{f}_{{2}}}\) and \({{f}_{{3}}}\) are 3rd harmonic, \({5}\)th harmonic and \({7}\)th harmonic respectively.
c. The frequecy in the string are \({f},{2}{f},{3}{f},{4}{f},{5}{f},\).........
So, \({3}{f}={2}\)nd overtone and \({3}{r}{d}{h}{a}{r}{m}{o}{n}{i}{c}\)
\({5}{f}={4}\)th overtone and \({5}{t}{h}{h}{a}{r}{m}{o}{n}{i}{c}\) l \({7}{t}{h}={6}{t}{h}\) overteone and 7th harmonic.
d. Length of the string is
\({L}={80}{c}{m}\)
\(\rightarrow{{f}_{{1}}=}{\left(\frac{{3}}{{2}}\right)}{v}\)
(v=velocity of the wave)
\(\rightarrow{90}={\left\lbrace\frac{{3}}{{{2}\times{80}}}\right\rbrace}\times{K}\)
\(={30}\times{2}\times{80}\)
\(={4800}\frac{{{c}{m}}}{{\sec{}}}\)
\(={48}\frac{{m}}{{s}}\)
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