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19. Fluid Mechanics

Buoyancy & Buoyant Force

Physics

19. Fluid Mechanics

Buoyancy & Buoyant Force

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5:36 minutes

Problem 40

Textbook Question

(II) A cube of side length 10.0 cm and made of unknown material floats at the surface between water and oil. The oil has a density of 810 kg/m³. If the cube floats so that it is 72% in the water and 28% in the oil, what is the mass of the cube and what is the buoyant force on the cube?

Verified step by step guidance

Step 1: Understand the problem. The cube is floating at the interface of water and oil, with 72% of its volume submerged in water and 28% in oil. This means the buoyant force acting on the cube is equal to its weight, as the cube is in equilibrium. The buoyant force is the sum of the forces due to the displaced water and oil.

Step 2: Write the expression for the buoyant force. The buoyant force is given by the equation:

F

_buoyant = ρ_water × g × V_water + ρ_oil × g × V_oil, where ρ_water is the density of water (1000 kg/m³), ρ_oil is the density of oil (810 kg/m³), g is the acceleration due to gravity (9.8 m/s²), V_water is the volume of the cube submerged in water, and V_oil is the volume of the cube submerged in oil.

Step 3: Calculate the volumes of the cube submerged in water and oil. The total volume of the cube is

V = a³

, where

a = 0.1

m (side length of the cube). The volume submerged in water is

V

_water = 0.72 × V, and the volume submerged in oil is

V

_oil = 0.28 × V.

Step 4: Use the equilibrium condition to find the mass of the cube. The weight of the cube is equal to the buoyant force:

m × g = ρ

_water × g × V_water + ρ_oil × g × V_oil. Simplify this equation to solve for the mass of the cube:

m = ρ

_water × V_water + ρ_oil × V_oil.

Step 5: Calculate the buoyant force. Once the mass of the cube is determined, the buoyant force can be directly calculated using the equation:

F

_buoyant = m × g. This is because the buoyant force is equal to the weight of the cube in equilibrium.

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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 cube's buoyancy allows it to float, with the distribution of its volume in water and oil affecting the overall buoyant force acting on it.

Density

Density is defined as mass per unit volume and is a critical factor in determining whether an object will float or sink in a fluid. The cube's density relative to the densities of water and oil influences how much of it is submerged. In this case, the cube's density can be calculated using the proportions of its volume in each fluid and their respective densities.

Equilibrium

Equilibrium in fluid mechanics refers to the state where the forces acting on an object are balanced, resulting in no net force. For the cube, this means that the buoyant force must equal the weight of the cube. The distribution of the cube's volume in water and oil helps establish this equilibrium, allowing us to calculate both the mass of the cube and the buoyant force acting on it.

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