An iron cube floating in mercury is a problem involving thermal expansion and buoyancy. As the temperature increases, both the cube and the mercury will undergo thermal expansion. We need to analyze how these expansions affect the floating behavior of the cube.
Concept Explanation:
- The expansion of materials with temperature can be described using the coefficient of linear or volume expansion.
- The volume of the cube and the volume of mercury will change when the temperature is increased.
- Buoyancy is described by Archimedes' principle, which states that the upward buoyant force experienced by a body immersed in a fluid is equal to the weight of the fluid displaced by the body.
- For the cube to float, the volume of mercury displaced must have a weight equal to the weight of the cube.
Step-by-Step Analysis:
- The volume expansion of a solid (e.g., the iron cube) is given by the formula: \(V = V_0(1 + \beta \Delta T)\), where \(\beta\) is the coefficient of volume expansion, \(V_0\) is the initial volume, and \(\Delta T\) is the change in temperature.
- The expansion of mercury, also a liquid, is similarly characterized by its volume expansion coefficient.
- Given that the coefficient of volume expansion for mercury is higher than that for iron, using: \({\beta}_{\text{mercury}} \gt {\beta}_{\text{iron}}\). This means mercury expands more than iron for the same temperature increase.
- As temperature increases, the whole volume of mercury expands more than the volume of the iron cube.
- The greater expansion of mercury means that the quantity of mercury displaced must increase to keep the effective buoyancy unchanged, but since mercury expands more, the cube will effectively float lower.
Conclusion:
- The iron cube will float lower when the temperature is increased by 100°C, due to the higher volume expansion of mercury compared to iron.
Thus, the correct answer is: lower.