Question

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $y(x,t)=\left(0.01m\right)\left[\text{sin}\right(62.8m^{-1}\left)x\right]\text{cos}\left[\right(628s^{-1}\left)t\right]$.
Assuming $\pi =3.14,$ the correct statement(s) is (are)

• A. The number of nodes is $5$
• B. The length of the string is $0.25\text{m}$
• C. The maximum displacement of the mid-point of the string from its equillibrium position is $0.01\text{m}$
• D. The fundamental frequency is $100\text{Hz}$

Answer 1

Answer: (B), (C)

The correct options are
B The length of the string is $0.25\text{m}$
C The maximum displacement of the mid-point of the string from its equillibrium position is $0.01\text{m}$


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