91Ó°ÊÓ

1. The ratio of bulk modulus and density for tube$A=\frac{16B_0}{{\mathrm{ÒÏ}}_0}$
2. The ratio of bulk modulus and density for tube$B=\frac{4B_0}{{\mathrm{ÒÏ}}_0}$
3. The ratio of bulk modulus and density for tube$C=\frac{B_0}{{\mathrm{ÒÏ}}_0}$
4. The distance travelled by the sound in tube$A=6L$
5. The distance travelled by the sound in tube$B=3L$
6. The distance travelled by the sound in tube$C=2L$

The time taken by the pulse of sound will be decided by its speed and the distance that it has to cover. The speed of the sound, in a medium, depends on the bulk modulus and the density of the medium.

Formulae are as follow:

$v=\sqrt{\frac{\text{B}}{\mathrm{ÒÏ}}}$

$v=\frac{d}{t}$

Where, v is speed of sound, d is distance, t is time, ÒÏ is density.

$v_C=\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}$

For tube A:

$v_A=\sqrt{\frac{16{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=4\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}$

$v_A=\frac{6L}{t_A}$

$∴4\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=\frac{6L}{t_A}$

$∴t_A=\frac{6L}{4v_C}where\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=v_c$

$∴t_A=\frac{3L}{2v_C}………………………………………………………….………………………………..\left(1\right)$

For tube B:

$v_B=\sqrt{\frac{4{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=2\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}$

$v_B=\frac{3L}{t_B}$

$∴2\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=\frac{3L}{t_B}$

$∴t_B=\frac{3L}{2v_C}where\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=v_c$

$∴t_B=\frac{3L}{2v_C}………………………………………………………………….……………………….\left(2\right)$

For tube C:

$v_C=\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}$

$v_C=\frac{2L}{t_C}$

$∴\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=\frac{2L}{t_C}$

$∴t_C=\frac{2L}{v_C}where\sqrt{\frac{{\text{B}}_0}{{\mathrm{ÒÏ}}_0}}=v_c……………………………………..……………………….\left(3\right)$

Thus, from equations (1), (2) and (3),

$t_C<t_A=t_B$

Thus, the time taken by the pulse in tube A and B is the same but smaller in tube C.

Hence, the rank of the tubes according to time of arrival of the pulse, (earliest first) is Cand then A and B at the same time.

Therefore, the time taken by the sound pulse to travel from one end of the pipe to the other will be decided by the speed of the sound. The speed of the sound, in the medium, depends on the bulk modulus and the density of the medium. Hence, while ranking the tubes, calculate the time taken by the pulse in each tube.