Question

A closed organ pipe of length L and an open organ pipe contain gass of densities rho_(1) and rho_(2), respectively. The compressibility of gass are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency . The length of the open orange pipe is (a) (L)/(3) (4l) / (3) (c ) (4l)/(3)sqrt((rho_(1))/(rho_(2))) (d) (4l)/(3)sqrt((rho_(2))/(rho_(1)))

Answer 1

Given:

• 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.

To find:

• Length of open organ pipe

Answer:

• Velocity in closed pipe, v₁ = √(B/ρ₁)

        Velocity in open pipe, v₂ = √(B/ρ₂)

             where B = Bulk modulus [ As compressibility is equal, B is equal.]

• Frequency of first over tone of closed pipe = frequency of first overtone of open pipe

        ⇒ $\frac {3 v_1} {4 L_1} = \frac {v_2}{L_2}$

        ⇒ $\frac{3}{4 L_1}$(√ρ₂) = $\frac{1}{L_2}$ (√ρ₁)

        ⇒ L₂ = $\frac{4}{3} L$ (√ρ₁/√ρ₂) = Length of open organ pipe