Question

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 $\theta=45^{\circ} . P_{1}$ and $P_{2}$ are pressures at points $1$ and $2$, respectively located at the base of the tube. Let $\beta=\left(P_{1}-P_{2}\right) /(\rho g d)$, where $\rho$ 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?

• A. $\beta=0$ when $a=g / \sqrt{2}$
• B. $\beta > 0$ when $a=g / \sqrt{2}$
• C. $\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a = g / 2$
• D. $\beta=\frac{1}{\sqrt{2}}$ when $a = g / 2$

Answer 1

Answer: (A), (C)

$\left(P_{1}-P_{2}\right) d s=\rho d s d \sqrt{2}(g \sin 45-a)$
$\left(P_{1}-P_{2}\right)=\rho d(g-a \sqrt{2})$
$\beta=\frac{\left(P_{1}-P_{2}\right)}{\rho g d}=\left(1-\frac{a \sqrt{2}}{g}\right)$
When $a=g / \sqrt{2}, \beta=0$
When $a=g / 2, \beta=\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)$