Question

A hollow pipe of length$0.8m$ is closed at one end. At its open end, a $0.5m$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is $50N$ and the speed of sound is $320m{s}^{-1}$, the mass of the string is

• A. $10g$
• B. $20g$
• C. $40g$
• D. $5g$

Answer 1

The correct option is A

$10g$

Step 1. Given data

Length of the hollow pipe, $l_1=0.8m$

Length of the string, $l_2=0.5m$

Tension in the wire, $T=50N$

Speed of the sound, $v=320m{s}^{-1}$

Step 2. Finding the mass of the string

The fundamental frequency of pipe $=$ frequency of the second harmonic of the string

$\begin{array}{rcl}& \Rightarrow & \frac{v}{4l_1}=\frac{2}{2l_2}\sqrt{\frac{T}{\mu }}\\ & \Rightarrow & \frac{320}{4x0.8}=\frac{1}{0.5}\sqrt{\frac{50}{\mu }}\\ & \Rightarrow & \mu =0.02kg/m\\ & & \end{array}$ [Where, $\mu$ is reduced mass]

We know that $m=\mu l_2$ [$m$is mass of the string]

$$\begin{gathered} =0.02x0.5 \\ =10g \end{gathered}$$

Hence, option $\left(A\right)$ is correct.