Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 12: Fluid Mechanics

Problem 26

Chapter 12, Problem 26

A rock has mass 1.80 kg. When the rock is suspended from the lower end of a string and totally immersed in water, the tension in the string is 12.8 N. What is the smallest density of a liquid in which the rock will float?

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Identify the forces acting on the rock when it is immersed in water: the gravitational force (weight), the buoyant force, and the tension in the string.

Use the formula for gravitational force: \( F_g = m \cdot g \), where \( m = 1.80 \text{ kg} \) and \( g = 9.81 \text{ m/s}^2 \). Calculate \( F_g \).

Apply the principle of buoyancy: the buoyant force \( F_b \) is equal to the weight of the displaced water. The tension in the string \( T \) is given as 12.8 N. Use the equation \( F_g = F_b + T \) to find the buoyant force.

The buoyant force can also be expressed as \( F_b = \rho_{water} \cdot V \cdot g \), where \( \rho_{water} \) is the density of water and \( V \) is the volume of the rock. Rearrange to find the volume \( V \) of the rock.

To find the smallest density of a liquid in which the rock will float, set the buoyant force equal to the gravitational force: \( \rho_{liquid} \cdot V \cdot g = F_g \). Solve for \( \rho_{liquid} \) using the volume \( V \) found in the previous step.

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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. Understanding buoyancy is crucial for determining whether an object will float or sink in a fluid, as it directly affects the net force acting on the object.

Density

Density is defined as mass per unit volume and is a key factor in determining whether an object will float in a fluid. An object will float if its density is less than the density of the fluid it is placed in. Calculating the density of the rock and comparing it to the fluid's density helps determine the conditions for flotation.

Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This principle is essential for solving problems related to flotation and helps in calculating the tension in the string when the rock is immersed, as well as determining the conditions for the rock to float in a different liquid.

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