Using the following, data, at what temperature will the wood...

Using the following, data, at what temperature will the wood just sink in benzene? Density of wood at $0^{\circ} \mathrm{C}=8.8 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3}$ Density of benzene at $0^{\circ} \mathrm{C}=9 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3}$ Cubical expansivity of wood $=1.5 \times 10^{-4} \mathrm{~K}^{-1}$ Cubical expansivity of benzene $=1.2 \times 10^{-3} \mathrm{~K}^{-1}$ (a) $27^{\circ} \mathrm{C}$ (b) $21.7^{\circ} \mathrm{C}$ (c) $31^{\circ} \mathrm{C}$ (d) $31.7^{\circ} \mathrm{C}$

Using the following, data, at what temperature will the wood just sink in benzene?
Density of wood at $0^{\circ} \mathrm{C}=8.8 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3}$
Density of benzene at $0^{\circ} \mathrm{C}=9 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3}$
Cubical expansivity of wood $=1.5 \times 10^{-4} \mathrm{~K}^{-1}$ Cubical expansivity of benzene $=1.2 \times 10^{-3} \mathrm{~K}^{-1}$
(a) $27^{\circ} \mathrm{C}$
(b) $21.7^{\circ} \mathrm{C}$
(c) $31^{\circ} \mathrm{C}$
(d) $31.7^{\circ} \mathrm{C}$

Key Concepts

Archimedes' Principle and Buoyancy

Archimedes' principle states that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. An object will just sink when its density becomes equal to the density of the fluid. This principle, combined with the temperature-dependent density changes due to thermal expansion, is used to determine the temperature at which an object transitions from floating to sinking.

Density and Temperature Dependence

Density is defined as mass per unit volume, and because thermal expansion affects volume, density can change with temperature even if the mass remains constant. In a scenario where both an object and a fluid expand, the relative change in their densities as a function of their expansion coefficients determines the conditions under which the object will float or sink.

Coefficient of Cubical Expansivity

The coefficient of cubical expansivity is a measure of how the volume of a substance changes with temperature. It is the fractional change in volume per unit temperature. This concept is crucial in problems involving the buoyancy of objects in fluids, as both the object's and the fluid's densities will change with temperature based on their respective coefficients.

Thermal Expansion

Thermal expansion refers to the change in size (and consequently density) of a material as its temperature changes. It is characterized by the coefficient of expansion, which quantifies how much the material’s volume increases per degree of temperature increase. When temperature changes, both solids and liquids expand or contract, thus altering their densities.

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