An open tube of length L=1.8 m and cross-sectional area A= 4.6 cm^2 is fixed to the top of a c

step 1 So here it's really simple. Because the pressure that is caused by the liquid at the bottom of t

An open tube of length L=1.8 m and cross-sectional area A=...

An open tube of length $L=1.8 \mathrm{~m}$ and cross-sectional area $A=$ $4.6 \mathrm{~cm}^{2}$ is fixed to the top of a cylindrical barrel of diameter $D=1.2 \mathrm{~m}$ and height $H=$ $1.8 \mathrm{~m}$. The barrel and tube are filled with water (to the top of the tube). Calculate the ratio of the hydrostatic force on the bottom of the barrel to the gravitational force on the water contained in the barrel. Why is that ratio not equal to $1.0 ?$ (You need not consider the atmospheric pressure.)

Key Concepts

Gravitational Force (Weight) of the Fluid

The gravitational force on the fluid, commonly referred to as its weight, is determined by multiplying the mass of the fluid (which is the product of its density and volume) by the gravitational acceleration. This force represents the total downward pull acting on the fluid due to gravity and is independent of how the fluid’s pressure is distributed on the container walls or base.

Pressure Distribution versus Weight Distribution

The ratio of the hydrostatic force on the bottom to the gravitational force on the water is not equal to 1.0 because these two quantities are derived from different distributions of force. While the weight of the water considers the entire fluid mass acting uniformly downwards, the hydrostatic force on the bottom arises from a distribution of pressure that varies with depth. Moreover, the container’s geometry, including any additional features like the connected tube, influences the pressure distribution, leading to a net force on the bottom that does not simply match the total weight of the fluid.

Hydrostatic Pressure

Hydrostatic pressure is the pressure that a fluid exerts at any given point within it due solely to the weight of the fluid above that point. It increases linearly with depth according to the relation P = ?gh, where ? is the fluid density, g is the acceleration due to gravity, and h is the depth below the fluid’s surface. This concept is fundamental for determining the pressure on any submerged surface, regardless of the container’s shape.

Integration of Pressure Over a Surface

The net force exerted by a fluid on a surface is calculated by integrating the hydrostatic pressure over the area of that surface. For surfaces where the pressure is not uniform, such as the bottom of a container with a varying depth profile, this integration takes into account the distribution of pressure, which is highest at the deepest points. This process explains why the force due to pressure on a given surface might not equate directly to the weight of the fluid.

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