Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 74

Chapter 14, Problem 74

In FIGURE CP14.74, a cone of density ρ₀ and total height l floats in a liquid of density ρ_f. The height of the cone above the liquid is h. What is the ratio h/l of the exposed height to the total height?

Verified step by step guidance

Step 1: Begin by understanding the problem. The cone is floating in a liquid, and we need to find the ratio of the exposed height (h) to the total height (l). This involves applying principles of buoyancy and equilibrium.

Step 2: Use Archimedes' principle, which states that the buoyant force acting on the cone is equal to the weight of the liquid displaced by the submerged part of the cone. The buoyant force is given by:

F

_buoyant = ρ_բgV_submerged, where

ρ

_բ is the density of the liquid,

g

is the acceleration due to gravity, and

V

_submerged is the volume of the submerged part of the cone.

Step 3: The weight of the cone is given by:

F

_gravity = ρ_₀gV_cone, where

ρ

_₀ is the density of the cone and

V

_cone is the total volume of the cone.

Step 4: Set the buoyant force equal to the gravitational force to find the relationship between the submerged volume and the total volume:

ρ

_բV_submerged = ρ_₀V_cone. The volume of a cone is proportional to the cube of its height, so

V

_submerged/V_cone = (H/l)³, where

H

is the submerged height and

l

is the total height.

Step 5: Solve for the ratio

h / l

using the relationship

H = l - h

and the equation derived in Step 4. Substitute the values and simplify to find the final expression for

h / l

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it. This force is equal to the weight of the fluid displaced by the object, as described by Archimedes' principle. In this scenario, the cone experiences a buoyant force that balances its weight, allowing it to float. Understanding buoyancy is crucial for determining how much of the cone is submerged versus exposed.

Density

Density is defined as mass per unit volume and is a key factor in determining whether an object will float or sink in a fluid. The densities of the cone (p₀) and the liquid (pβ) influence the buoyant force acting on the cone. If the density of the cone is less than that of the liquid, it will float, and the ratio of the heights h and l can be derived from their respective densities.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the weight of the fluid above it. In this context, the pressure at the base of the cone is determined by the height of the liquid column above it. This pressure must equal the weight of the cone for it to float, leading to a relationship between the submerged height and the total height of the cone, which is essential for solving the problem.