Question 78: Waves

Question

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)

Accepted Answer

First overtone of open pipe $$ = {{{v_2}} \over {{L_2}}}$$
First overtone of closed pipe at one end $$ = {{3v} \over {4L}}$$
As per question,
$${{3V} \over {4L}} = {{{V_2}} \over L}$$
$$ \Rightarrow \sqrt {{B \over {{\rho _1}}}} \,.\,{3 \over {4L}} = \sqrt {{B \over {{\rho _2}}}} \,.\,{1 \over {{L_2}}}$$ ($$\because$$ $$V = \sqrt {{B \over \rho }} $$)
$$ \Rightarrow {L_2} = {{4L} \over 3}\sqrt {{{{\rho _1}} \over {{\rho _2}}}} $$ ..... (i)
According to question, the length of the open pipe is
$${x \over 3}L\sqrt {{{{\rho _1}} \over {{\rho _2}}}} $$ ..... (ii)
Comparing Eqs. (i) and (ii), we get
$$x = 4$$

Waves

Integer / Numeric Answer Type