Question

Two closed pipes of length $33$ m and $33.3$ m are sounded together and they produce $225$ beats in $100$ seconds at $0^{o}C$. if the temperature is increased to ${30}^{o}C$ what will be the number of beats?

• A. $237$
• B. $255$
• C. $227$
• D. $247$

Answer 1

The correct option is A $237$

Frequency produced in closed pipe $\nu =\frac{v}{2L}$
where $L$ is the length of pipe and $v$ is the velocity of sound.
Beat frequency heard at $0^{o}C$, $b_o={\nu }_1-{\nu }_2=\frac{v_o}{2}[\frac{1}{L_1}-\frac{1}{L_2}]$
where $L_1=33m$ and $L_2=33.3m$
Velocity of sound at ${30}^{o}C$, $v_T=\sqrt{\frac{T}{T_o}}{v}_o$
Putting $T_o=273K$, $T=30+273=303K$
We get $\frac{v_T}{v_o}=\sqrt{\frac{303}{273}}=1.05$
Beat frequency heard at ${30}^{o}C$, $b_T=\frac{v_T}{2}[\frac{1}{L_1}-\frac{1}{L_2}]$
$\therefore$ $\frac{b_T}{b_o}=\frac{v_T}{v_o}$
Or $\frac{b_T}{225}=1.05$
$⟹b_T=237$ beats in 100 seconds