A soft plastic bottle, filled with water of density 1 gm / cc , carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of 5 gm , and it

Question

A soft plastic bottle, filled with water of density $1 gm / cc$ , carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of $5 gm$ , and it is made of a thick glass of density $2.5 gm / cc$ . Initially the bottle is sealed at atmospheric pressure $p_{0} = 1 0^{5} Pa$ so that the volume of the trapped air is $v_{0} = 3.3 cc$ . When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure $p_{0} + Δ p$ without changing its orientation. At this pressure, the volume of the trapped air is $v_{0} - Δ v$ .
Let $Δ v = X$ cc and $Δ p = Y \times 1 0^{3} Pa$ .

The value of $Y$ is _____.

TestHub · Accepted Answer

When all the forces on tube are balanced then the tube will start sinking.

$F_{u p w a r d} = F_{d o w n w a r d}$

$(ρ_{w}) (V_{tube} + V_{air}) g = (m_{tube}) g$

$(2 {cm}^{3} + V_{air}) g = (5 gm) g$

$V_{air} = 3 cm$ , while the initial volume of the air was $3.3 {cm}^{3}$ so the decrease in the volume $Δ V = 3.3 - 3 = 0.3 {cm}^{3}$

$V_{tube} = \frac{m_{tube}}{ρ_{tube}} = \frac{5 gm}{2.5 gm {cm}^{- 3}}$

$V_{tube} = 2 {cm}^{3}$

Since the temperature of the tapped air remains constant, so according to the ideal gas equation,

$P V =$ constant

Take $\ln$ on both side.

$\Rightarrow l n P + l n V = l n$ (constant)

Differentiate both sides,

$\frac{d P}{P} + \frac{d V}{V} = 0 (∵ \frac{d}{d t} constant = 0) \Rightarrow \frac{Δ P}{P} = - \frac{Δ V}{V}$

$\Rightarrow \frac{Δ P}{10^{5}} = - \frac{- 0.3}{3.3} \Rightarrow Δ P = \frac{1}{11} \times 10^{5} Pa = \frac{100}{11} \times 10^{3} \Rightarrow Δ P = 9.09 \times 10^{3} Pa$

Compare it with, $Δ P = Y \times 10^{3} Pa$

So, $Y = 9.09$

Physics - Fluid Question with Solution | TestHub

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