A 10.0 -kg block of metal is suspended from a scale and...

A $10.0$ -kg block of metal is suspended from a scale and immersed in water, as in Figure P9.38. The dimensions of the block are $12.0 \mathrm{~cm} \times 10.0 \mathrm{~cm} \times$ $10.0 \mathrm{~cm}$. The $12.0-\mathrm{cm}$ dimension is vertical, and the top of the block is $5.00 \mathrm{~cm}$ below the surface of the water. (a) What are the forces exerted by the water on the top and bottom of the block? (Take $P_{0}=1.0130 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}$.) (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.

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Key Concepts

Pressure Forces on Submerged Surfaces

The force exerted by a fluid on any surface is given by the product of the pressure at that surface and the area over which it acts. For a submerged object, differences in pressure across its surfaces (due to differences in depth) result in a net force. Specifically, the buoyant force arises from the pressure difference between the bottom and top surfaces of the object, which exactly equals the weight of the displaced fluid.

Hydrostatic Pressure

In a fluid at rest, pressure increases with depth due to the weight of the overlying fluid. This relationship is typically expressed as P = P? + ?gh, where P? is the surface pressure, ? is the fluid density, g is the acceleration due to gravity, and h is the depth below the surface. This concept allows one to determine the force per unit area acting on surfaces submerged at different depths.

Archimedes' Principle

Archimedes' Principle states that any object wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. This principle is fundamental when calculating the buoyant force acting on immersed objects and explains why objects appear lighter when submerged in a fluid.

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