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A solid object made of material of density $$0.79\text{ g cm}^{-3}$$ of negligible coefficient of cubical expansion is floating in a liquid of density $$0.8\text{ g cm}^{-3}$$ at $$20^\circ\text{C}$$. When the temperature is raised to $$220^\circ\text{C}$$, the solid object starts sinking in the liquid. Find the coefficient of cubical expansion (per degree Celsius) of the liquid.
Correct Answer: 0.0000633
The object begins to sink when the liquid's density falls to 0.79. Since $$\rho_T = \frac{\rho_0}{1 + \gamma \Delta T}$$, $$0.79 = \frac{0.8}{1 + 200\gamma}$$. So $$1 + 200\gamma = \frac{80}{79}$$ and $$\gamma = \frac{1}{79 \times 200} \approx 6.33 \times 10^{-5}\,^\circ\text{C}^{-1}$$.
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