Question

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation,
$y(x,t)=(0.01m)[sin(628m^{-1})x]cos[(628s^{-1})t].$
Assuming $\pi =3.14,$ the correct statement(s) is (are)

• A. the number of nodes is 5
• B. the length of the string is 0.25 m
• C. the maximum displacement of the mid-point of the string from its equilibrium position is 0.01 m
• D. the fundamental frequency is 100 Hz

Answer 1

Answer: (B), (C)

Number of nodes $=6$
From the given equation, we can see that
$<br/>\begin{array}{c}<br/>k=\frac{2\pi }{\lambda }=62.8m^{-1}\\ <br/>\therefore \lambda =\frac{2\pi }{62.8}m=0.1m\\ <br/>l=\frac{5\lambda }{2}=0.25m<br/>\end{array}<br/>$
The mid-point of the string is $P$, an antinode
$\therefore$ maximum displacement $=0.01m$
$<br/>\begin{array}{l}<br/>\omega =2\pi f=628s^{-1}\\ <br/>\therefore f=\frac{628}{2\pi }=100Hz<br/>\end{array}<br/>$
But this is fifth harmonic frequency.
$\therefore$ Fundamental frequency $f_0=\frac{f}{5}=20Hz$