A tank with a flat bottom of area A and vertical sides is...

A tank with a flat bottom of area $A$ and vertical sides is filled to a depth $h$ with water. The pressure is $P_{0}$ at the top surface. (a) What is the absolute pressure at the bottom of the tank? (b) Suppose an object of mass $M$ and density less than the density of water is placed into the tank and floats. No water overflows. What is the resulting increase in pressure at the bottom of the tank?

Key Concepts

Force Distribution and Pressure Increase

This principle explains how an additional force (such as an object’s weight) distributed over a given area results in an increase in pressure. In a container of constant cross-sectional area, adding an object that displaces fluid will raise the fluid level; the extra weight of the displaced fluid per unit area directly translates to an increase in pressure at the bottom of the container (calculated as the object's weight divided by the area over which it is distributed).

Buoyancy and Archimedes’ Principle

Buoyancy is the upward force exerted by a fluid on an object immersed in it, and Archimedes’ Principle states that this buoyant force is equal to the weight of the fluid displaced by the object. When an object floats, it displaces a volume of fluid whose weight is equal to its own, which in turn causes an increase in the fluid level and affects the pressure distribution within the fluid.

Hydrostatic Pressure

This concept refers to the pressure exerted by a fluid due to the weight of the fluid above a given point. In a static fluid, the pressure increases linearly with depth because of the gravitational force acting on the fluid mass. It is expressed as the sum of the initial pressure at the fluid's surface along with the product of the fluid’s density, gravitational acceleration, and the depth below the surface (i.e. P = P? + ?gh).

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