16. (AIEEE 2012)

A cylindrical tube, open at both ends, has a fundamental frequency, $f,$ in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now :

A. f

B. f/2

C. 3/4

D. 2f

Correct Answer is Option (A)

The fundamental frequency of open tube

$v_{0} = \frac{v}{2 l_{0}} . . . (i)$

That of closed pipe

$v_{c} = \frac{υ}{4 l_{c}} . . . (i i)$
According to the problem $l_{c} = \frac{l_{0}}{2}$

Thus $v_{c} = \frac{υ}{l_{0} / 2} \Rightarrow v_{c} \frac{υ}{2 l} . . . (i i i)$

From equations $(i)$ and $(i i i)$

$v_{0} = v_{c}$

Thus, $v_{c} = f$ $($ as $v_{0} = f$ is given $)$

17. (AIEEE 2008)

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of $18$ $c m$ during winter. Repeating the same experiment during summer, she measures the column length to be $x$ $c m$ for the second resonance. Then

A. $18 > x$

B. $x > 54$

C. $54 > x > 36$

D. $36 > x > 18$

Correct Answer is Option (B)

For first resonant length $v = \frac{v}{4 ℓ_{1}} = \frac{v}{4 \times 18}$ (in winter)

For second resonant length

$v^{'} = \frac{3 v^{'}}{4 ℓ_{2}} = \frac{3 v^{'}}{4 x}$ (in summer)

$∴$ $\frac{v}{4 \times 18} = \frac{3 v^{'}}{4 \times x}$

$∴$ $x = 3 \times 18 \times \frac{v^{'}}{v}$

$∴$ $x = 54 \times \frac{v^{'}}{v} c m$

$v^{'} > v$ because velocity of light is greater in summer as compared to winter

$(v \propto \sqrt{T})$

$∴$ $x > 54 c m$

18. (AIEEE 2002)

Tube $A$ has bolt ends open while tube $B$ has one end closed, otherwise they are identical. The ratio of fundamental frequency of tube $A$ and $B$ is :

A. 1 : 2

B. 1 : 4

C. 2 : 1

D. 4 : 1

Correct Answer is Option (C)

KEY CONCEPT : The fundamental frequency for closed organ pipe is given by $υ_{c} = \frac{v}{4 ℓ}$ and

For open organ pipe is given by $υ_{0} = \frac{v}{2 ℓ}$

$∴$ $\frac{υ_{0}}{υ_{c}} = \frac{v}{2 ℓ} \times \frac{4 ℓ}{v} = \frac{2}{1}$

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