A piece of metal floats on Hg. The coefficient of expansion of metal and H g are Y1 and Y2 respe

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A piece of metal floats on Hg. The coefficient of expansion...

A piece of metal floats on $\mathrm{Hg}$. The coefficient of expansion of metal and $H g$ are $Y_{1}$ and $Y_{2}$ respectively. If the temperature of both $\mathrm{Hg}$ and metal are increased by an amount $\Delta T$, by what factor the fraction of the volume of metal submerged in mercury changes? (a) $\left(\gamma_{2}-\gamma_{1}\right) \Delta T$ (b) $\left(\frac{\gamma_{2}+\gamma_{1}}{2}\right) \Delta T$ (c) $\left(\frac{2 \gamma_{1} \gamma_{2}}{\gamma_{1}+\gamma_{2}}\right) \Delta T$ (d) $\left(\frac{\gamma_{1} \gamma_{2}}{\gamma_{1}+\gamma_{2}}\right) \Delta T$

A piece of metal floats on $\mathrm{Hg}$. The coefficient of expansion of metal and $H g$ are $Y_{1}$ and $Y_{2}$ respectively. If the temperature of both $\mathrm{Hg}$ and metal are increased by an amount $\Delta T$, by what factor the fraction of the volume of metal submerged in mercury changes?
(a) $\left(\gamma_{2}-\gamma_{1}\right) \Delta T$
(b) $\left(\frac{\gamma_{2}+\gamma_{1}}{2}\right) \Delta T$
(c) $\left(\frac{2 \gamma_{1} \gamma_{2}}{\gamma_{1}+\gamma_{2}}\right) \Delta T$
(d) $\left(\frac{\gamma_{1} \gamma_{2}}{\gamma_{1}+\gamma_{2}}\right) \Delta T$

Key Concepts

Archimedes’ Principle

This principle states that a floating object displaces a weight of fluid equal to its own weight. In problems involving buoyancy, the equilibrium condition is set by equating the weight of the object with the buoyant force, which is directly related to the density of the liquid and the volume of fluid displaced.

Thermal Expansion and Density Variation

Thermal expansion causes materials to change their dimensions when subjected to a change in temperature. For a given material, a temperature increase causes its volume to change according to its expansion coefficient. Since density is mass divided by volume, a variation in volume produces a corresponding change in density. When both the solid object and the fluid expand differently, their densities change at different rates, altering the fraction of the solid submerged.

Buoyant Equilibrium under Temperature Change

For a floating body, the condition of equilibrium is maintained by matching the weight of the object with the weight of the displaced fluid. When temperature increases, the densities of both object and fluid change according to their respective expansion coefficients. The ratio of the densities, which determines the submerged fraction, adjusts, and the change can be estimated by comparing the expansion rates of the metal and the fluid.

First-Order Approximation in Thermal Processes

In problems involving small temperature changes, it is common to apply a first-order (linear) approximation, ignoring terms of higher order. This simplification allows us to express the fractional change in submerged volume predominantly as a linear function of the difference between the liquid’s and the metal’s expansion coefficients multiplied by the temperature change.

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