A closed organ pipe of length L and an open organ pipe contain gases of densities $\rho$1 and $\rho$2 respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is ${x \over 3}L\sqrt {{{{\rho _1}} \over {{\rho _2}}}}$ where x is ___________. (Round off to the Nearest Integer)
Answer (integer)
4
Solution
<p>First overtone of open pipe $= {{{v_2}} \over {{L_2}}}$</p>
<p>First overtone of closed pipe at one end $= {{3v} \over {4L}}$</p>
<p>As per question,</p>
<p>${{3V} \over {4L}} = {{{V_2}} \over L}$</p>
<p>$$ \Rightarrow \sqrt {{B \over {{\rho _1}}}} \,.\,{3 \over {4L}} = \sqrt {{B \over {{\rho _2}}}} \,.\,{1 \over {{L_2}}}$$ ($\because$ $V = \sqrt {{B \over \rho }}$)</p>
<p>$\Rightarrow {L_2} = {{4L} \over 3}\sqrt {{{{\rho _1}} \over {{\rho _2}}}}$ ..... (i)</p>
<p>According to question, the length of the open pipe is</p>
<p>${x \over 3}L\sqrt {{{{\rho _1}} \over {{\rho _2}}}}$ ..... (ii)</p>
<p>Comparing Eqs. (i) and (ii), we get</p>
<p>$x = 4$</p>
About this question
Subject: Physics · Chapter: Waves · Topic: Wave Motion and Types
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