The string of length $2 \mathrm{~m}$ is fixed at both ends. If the string vibrates in its fourth normal mode…
The string of length $2 \mathrm{~m}$ is fixed at both ends. If the string vibrates in its fourth normal mode with a frequency of $500 \mathrm{~Hz}$, then the waves would travel on it with a velocity of
$125 \mathrm{~m} / \mathrm{s}$
$250 \mathrm{~m} / \mathrm{s}$
$500 \mathrm{~m} / \mathrm{s}$
$1000 \mathrm{~m} / \mathrm{s}$
Solution
In general, $n$th mode of a string fixed at ends has frequency,
$v=\frac{n v}{2 l}$
where, $n=1,2,3, \ldots$
where, $v$ is the velocity of wave and $l$ is the length of string.
In fourth normal mode, $n=4$
$v=\frac{4 v}{2 l}$
Given, $v=500 \mathrm{~Hz}, l=2 \mathrm{~m}$
Hence, $500=\frac{4 v}{2 \times 2}$
or $\quad 0=\frac{500 \times 4}{4}=500 \mathrm{~m} / \mathrm{s}$