Askiitians Tutor Team
Last Activity: 14 Days ago
To analyze the situation you've described, we need to consider how the temperature of the sphere changes over time in relation to its surroundings. The rate at which the sphere's temperature drops is influenced by several factors, including the temperature difference between the sphere and the enclosure, the sphere's density, specific heat capacity, and its radius. Let's break down the options you've provided to determine which one correctly describes the relationship.
Understanding the Factors Involved
When a sphere is placed in an environment with a lower temperature, heat transfer occurs from the sphere to the surrounding air. The rate of this heat transfer can be modeled using Newton's Law of Cooling, which states that the rate of heat loss of a body is directly proportional to the difference in temperature between the body and its surroundings. However, in this case, we also need to consider the physical properties of the sphere.
Key Variables
- Density (d): This affects how much mass is present in a given volume, influencing how much heat is stored in the sphere.
- Specific Heat Capacity (c): This indicates how much heat energy is required to change the temperature of the material by a certain amount.
- Radius (r): The size of the sphere plays a role in the surface area available for heat transfer.
Analyzing the Options
Now, let's evaluate the options you've provided in terms of their relevance to the rate of temperature change:
1. c/r^3d
This option suggests that the rate of temperature change is inversely proportional to the cube of the radius and directly proportional to the specific heat capacity and density. This doesn't quite align with the principles of heat transfer, as it doesn't adequately account for the temperature difference.
2. 1/r^3dc
This expression indicates that the rate of temperature change is inversely proportional to the cube of the radius and directly proportional to the specific heat capacity. While the radius does affect the surface area for heat transfer, this option lacks a clear connection to the temperature difference.
3. 3r^3dc
This option suggests a direct proportionality to the radius cubed, density, and specific heat capacity. However, it does not reflect the cooling process accurately, as it does not incorporate the temperature difference.
4. 1/rdc
This expression indicates that the rate of temperature change is inversely proportional to the radius and directly proportional to both the density and specific heat capacity. This option is the most plausible because it suggests that as the radius increases, the rate of temperature change decreases, which aligns with the idea that larger spheres have more mass and thus retain heat longer.
Conclusion
Based on the analysis, the most reasonable choice that reflects the relationship between the temperature drop of the sphere and the factors involved is option 4: 1/rdc. This option effectively captures the interplay between the sphere's physical properties and the cooling process in relation to the surrounding environment. Understanding these relationships is crucial for predicting how objects behave thermally in different conditions.
