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 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?
$\beta=0$ when $a=\frac {g}{\sqrt{2}}$
$\beta > 0$ when $a=\frac{g}{\sqrt{2}}$
$\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a =\frac{g}{2}$
$\beta=\frac{1}{\sqrt{2}}$ when $a = \frac{g}{2}$
The Correct Option is C
Solution and Explanation
The correct answer is option (C): $\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a =\frac{g}{2}$
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Concepts Used:
Mechanical Properties of Fluid
The science of the mechanical properties of fluids is called Hydrostatics. A fluid is a substance that relents to the slightest pressure. Fluids are categorized into two classes famed by the names of liquids, and elastic fluids or gases, which later comprehend the air of the atmosphere and all the different kinds of air with which chemistry makes us acquainted.
Streamline Flow:
A streamline is a curve the tangent to which at any point provides the direction of the fluid velocity at that point. It is comparable to a line of force in an electric or magnetic field. In steady flow, the pattern of the streamline is motionless or static with time, and therefore, a streamline provides the actual path of a fluid particle.
Tube of Flow:
A tubular region of fluid enclosed by a boundary comprises streamlines is called a tube of flow. Fluid can never cross the boundaries of a tube of flow and therefore, a tube of flow acts as a pipe of the same shape.
Surface Tension and Viscosity:
The surface tension of a liquid is all the time a function of the solid or fluid with which the liquid is in contact. If a value for surface tension is provided in a table for oil, water, mercury, or whatever, and the contacting fluid is unspecified, it is safe to consider that the contacting fluid is air.