A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration a along a fixed inclined plane with angle $θ = 45^{\circ}$ . P₁and P₂ are pressure at points 1 and 2, respectively, located at the base of the tube Let $β = (P_{1} - P_{2}) / (ρgd)$ , where $ρ$ is density of water, d is the inner diameter of the tube and g is the acceleration due to gravity. Which of the following statement(s) is(are) correct ?

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$β = 0 when a = g / \sqrt{2}$

$β > 0 when a = g / \sqrt{2}$

$β = \frac{\sqrt{2} - 1}{\sqrt{2}} when a = g / 2$

$β = \frac{1}{\sqrt{2}} when a = g / 2$

answer is A.

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Detailed Solution

$\begin{array}{l} P_{1} + ρ (\sqrt{2} d) a - ρ \frac{g}{\sqrt{2}} (\sqrt{2} d) = P_{2} \\ P_{1} - P_{2} = ρd (- \sqrt{2} a + g) \\ β = \frac{P_{1} - P_{2}}{ρgd} = (- \frac{\sqrt{2} a}{g} + 1) \end{array}$

$if β = 0 \Rightarrow a = \frac{g}{\sqrt{2}}$

$if a = \frac{g}{2} then β = \frac{\sqrt{2} - 1}{\sqrt{2}}$

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