A pipe of length , closed at one end is kept in a chamber of gas of density . A second pipe open at both ends is placed in a second chamber of gas of density . The compressibility of both the gases is equal. Calculate the length of the second pipe if freq

Question

A pipe of length $ℓ_{1}$ , closed at one end is kept in a chamber of gas of density $ρ_{1}$ . A second pipe open at both ends is placed in a second chamber of gas of density $ρ_{2}$ . The compressibility of both the gases is equal. Calculate the length of the second pipe if frequency of first overtone in both the cases is equal. [2004]

Smart Achievers · Accepted Answer

(2)

Frequency of first overtone in closed organ pipe,

$ν = \frac{3 v}{4 ℓ_{1}} \sqrt{\frac{P}{ρ_{1}}}$

Frequency of first overtone in open organ pipe,

$ν' = \frac{1}{ℓ_{2}} \sqrt{\frac{P}{ρ_{2}}}$

Here, $ν = ν'$

$\frac{3 v}{4 ℓ_{1}} \sqrt{\frac{P}{ρ_{1}}} = \frac{1}{ℓ_{2}} \sqrt{\frac{P}{ρ_{2}}}$

$∴$ $ℓ_{2} = \frac{4}{3} ℓ_{1} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$

Solution

1 $\frac{4}{3} ℓ_{1} \sqrt{\frac{ρ_{2}}{ρ_{1}}}$

2 $\frac{4}{3} ℓ_{1} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$

3 $ℓ_{1} \sqrt{\frac{ρ_{2}}{ρ_{1}}}$

4 $ℓ_{1} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$

Want Us to Email you About Special Offers & Updates?

Solution

1 43ℓ1ρ2ρ1

2 43ℓ1ρ1ρ2

3 ℓ1ρ2ρ1

4 ℓ1ρ1ρ2

Ans. (2) Frequency of first overtone in closed organ pipe, ν=3v4ℓ1Pρ1 Frequency of first overtone in open organ pipe, ν'=1ℓ2Pρ2 Here, ν=ν' 3v4ℓ1Pρ1=1ℓ2Pρ2 ∴ ℓ2=43ℓ1ρ1ρ2

Want Us to Email you About Special Offers & Updates?

1 $\frac{4}{3} ℓ_{1} \sqrt{\frac{ρ_{2}}{ρ_{1}}}$

2 $\frac{4}{3} ℓ_{1} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$

3 $ℓ_{1} \sqrt{\frac{ρ_{2}}{ρ_{1}}}$

4 $ℓ_{1} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$