(a) What is the weight density of a liquid that exerts a...

(a) What is the weight density of a liquid that exerts a total force of $433 \mathrm{N}$ on the sides of a $3.00-\mathrm{m}$ -tall cylindrical tank? The radius of the tank is $0.913 \mathrm{m},$ and it is filled to $1.75 \mathrm{m}$. (b) What liquid might this be?

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Key Concepts

Force Calculation on Submerged Surfaces by Integration

When a fluid exerts pressure over a submerged surface, the total force is determined by integrating the pressure over that surface area. For vertical surfaces, this involves integrating a linearly varying pressure (with depth) across the area. This method yields an expression that involves the weight density and the geometry of the surface, such as dimensions of a tank, enabling a systematic determination of the overall force on the surface.

Hydrostatic Pressure Distribution

Hydrostatic pressure is the pressure exerted by a fluid at rest, which increases linearly with depth. This concept is essential since the force on any submerged surface is not uniform but rises with depth from the surface, meaning that a proper calculation of force requires accounting for the varying pressure over the area of contact.

Weight Density

Weight density (often denoted by ?) is defined as the weight per unit volume of a substance, typically given in newtons per cubic meter (N/m³). It is the product of the fluid’s mass density and the acceleration due to gravity. Understanding weight density is critical for converting between mass density and the hydrostatic forces produced in fluids.

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