Q: A closed organ pipe of length L and an open organ pipe contain gases of densities ρ1 and ρ2 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 organ pipe is

(a) L/3

(b) 4L/3

(c)$\large \frac{4L}{3}\sqrt{\frac{\rho_1}{\rho_2}}$

(d) $\large \frac{4L}{3}\sqrt{\frac{\rho_2}{\rho_1}}$

Ans: (c)

Sol: Since , both the pipes are vibrating in their first overtone with same frequency.

$\large f_c = f_o $

$\large 3 (\frac{v_c}{4l_c}) = 2 (\frac{v_o}{2l_o})$

$\large l_o = \frac{4}{3} (\frac{v_o}{v_c}) l_c$

As , $\large v \propto \sqrt{\frac{1}{\rho}}$

$\large l_o = \frac{4}{3} \sqrt{\frac{\rho_1}{\rho_2}}L$