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## QuestionPhysicsClass 11

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 ?

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)
$$\rightarrow{90}={\left\lbrace\frac{{3}}{{{2}\times{80}}}\right\rbrace}\times{K}$$
$$={30}\times{2}\times{80}$$
$$={4800}\frac{{{c}{m}}}{{\sec{}}}$$
$$={48}\frac{{m}}{{s}}$$