Question

A string of length $2\,m$ is fixed at both ends. If this string vibrates in its fourth normal mode with a frequency of $500 \,Hz$, then the waves would travel on it with a velocity of:

• A. 125 m/s
• B. 250 m/s
• C. 500 m/s
• D. 1000 m/s

Answer 1

Answer: (C)

A normal mode of an oscillating system is a motion in which all particles of the system move sinusoidally with the same frequency. In general, $ pth $ mode of a string fixed at ends has frequency.
$ n=\frac{pv}{2l} $ $ p=1,2,3.. $
where $ v $ is velocity of wave and I is length of string. In fourth normal mode, $ p=4 $
$ \therefore \, $ $ n=\frac{4v}{2l} $
Given, $ n=500\,Hz,l=2\,m $
Hence, $ 500\,=\frac{4v}{2\times 2} $
or $ v=\frac{500\times 4}{4}=500\,m/s $