An engineer is designing a transparent safety window for a deep-sea research vessel

PHYSICS · Multi-part · CBSE · Physics · Mechanical Properties of Fluids · Mechanical Properties of Fluids

Problem

An engineer is designing a transparent safety window for a deep-sea research vessel. The window consists of a flat, circular pane of radius

0.20\,\text{m}

that must withstand external seawater pressure at a depth of

25\,\text{m}

. To reduce the net force on the pane, the inner side of the window will be pressed against by a sealed column of two immiscible fluids: a layer of fluorocarbon liquid (density

1.80\times10^{3}\,\text{kg}\,\text{m}^{-3}

) of adjustable thickness

h

floating atop a layer of distilled water (density

1.00\times10^{3}\,\text{kg}\,\text{m}^{-3}

) of fixed thickness

0.10\,\text{m}

. The column is closed at the top by a rigid cap that is internally vented to the vessel’s cabin air, maintained at standard atmospheric pressure. The bottom of the water layer is flush with the inner surface of the window. (a) Derive an expression, in terms of

h

, for the absolute pressure exerted by the two-fluid column on the inner surface of the window. (b) The external seawater exerts an absolute pressure of approximately

3.45\times10^{5}\,\text{Pa}

at the operating depth. Determine the minimum thickness

h_{\text{min}}

of the fluorocarbon layer required so that the net force on the window does not exceed

1.2\times10^{4}\,\text{N}

outward (from the sea toward the cabin). (c) Using your result from part (b), calculate the corresponding gauge pressure at the fluorocarbon–air interface inside the column. (d) Discuss qualitatively how increasing the cabin air pressure (while keeping

h

fixed at

h_{\text{min}}

) would affect the net force on the window. Support your answer with reasoning based on pressure differences.

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