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Ch. 13 - Fluids
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 13 - FluidsProblem 36
Chapter 13, Problem 36

A 0.48-kg piece of wood floats in water but is found to sink in alcohol (SG = 0.79), in which it has an apparent mass of 0.047 kg. What is the SG of the wood?

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Determine the buoyant force acting on the wood in alcohol. The apparent mass of the wood in alcohol is given as 0.047 kg, so the buoyant force is equal to the difference between the actual weight of the wood and the apparent weight in alcohol. Use the formula for weight: \( F_{\text{buoyant}} = m_{\text{wood}} g - m_{\text{apparent}} g \), where \( g \) is the acceleration due to gravity.
Relate the buoyant force to the volume of the wood submerged in alcohol. The buoyant force is equal to the weight of the displaced alcohol: \( F_{\text{buoyant}} = \rho_{\text{alcohol}} V_{\text{wood}} g \), where \( \rho_{\text{alcohol}} \) is the density of alcohol, and \( V_{\text{wood}} \) is the volume of the wood.
Express the density of alcohol in terms of its specific gravity (SG). The specific gravity of alcohol is given as 0.79, so \( \rho_{\text{alcohol}} = 0.79 \times \rho_{\text{water}} \), where \( \rho_{\text{water}} \) is the density of water (approximately \( 1000 \ \text{kg/m}^3 \)). Substitute this into the buoyant force equation.
Determine the volume of the wood using its mass and density. The density of the wood can be expressed as \( \rho_{\text{wood}} = \frac{m_{\text{wood}}}{V_{\text{wood}}} \). Rearrange this equation to solve for \( V_{\text{wood}} \): \( V_{\text{wood}} = \frac{m_{\text{wood}}}{\rho_{\text{wood}}} \).
Calculate the specific gravity (SG) of the wood. The specific gravity is the ratio of the density of the wood to the density of water: \( SG_{\text{wood}} = \frac{\rho_{\text{wood}}}{\rho_{\text{water}}} \). Use the relationships derived in the previous steps to express \( \rho_{\text{wood}} \) in terms of the given quantities and solve for \( SG_{\text{wood}} \).

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Key Concepts

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

Specific Gravity (SG)

Specific Gravity (SG) is a dimensionless quantity that compares the density of a substance to the density of a reference substance, typically water for liquids. It is calculated as the ratio of the density of the substance to the density of water (1 g/cm³ at 4°C). An SG less than 1 indicates that the substance will float in water, while an SG greater than 1 means it will sink.
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Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it, which counteracts the weight of the object. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the object. This principle explains why objects float or sink in different fluids based on their densities relative to the fluid.
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Apparent Weight

Apparent weight refers to the weight of an object when it is submerged in a fluid, which is less than its actual weight due to the buoyant force acting on it. The apparent weight can be calculated by subtracting the buoyant force from the object's weight. This concept is crucial for understanding how objects behave in different fluids, such as water and alcohol in this scenario.
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