Askiitians Tutor Team
Last Activity: 5 Months ago
When considering the cooling rates of a solid sphere and a hollow sphere made of the same material and size, we need to delve into the principles of heat transfer and how they apply to these two different geometries. The cooling process can be analyzed using Newton's Law of Cooling, which states that the rate of heat loss of a body is proportional to the difference in temperature between the body and its surroundings, provided this temperature difference is small.
Understanding Heat Transfer
Heat transfer occurs through conduction, convection, and radiation. In this scenario, we primarily focus on convection, as both spheres are cooling in the same environment. The rate of heat loss depends on the surface area and the volume of the object, as well as the temperature difference (T) between the object and the surrounding air.
Surface Area and Volume Considerations
For a solid sphere, the entire volume is solid, while a hollow sphere has an inner cavity. Both spheres have the same outer dimensions, which means they share the same surface area for heat exchange. However, their volumes differ significantly:
- The solid sphere has a volume that is entirely filled with material.
- The hollow sphere has a significant portion of its volume as air, which does not contribute to heat storage.
Cooling Rate Analysis
When both spheres are heated to the same temperature and then placed in the same environment, the cooling rate can be influenced by their thermal mass. The solid sphere, having more material, has a greater thermal mass compared to the hollow sphere. This means:
- The solid sphere can store more heat energy, which results in a slower rate of temperature decrease.
- The hollow sphere, with less thermal mass, will lose heat more quickly due to its lower capacity to store heat.
Temperature Difference Impact
As the temperature difference (T) increases, the rate of cooling for both spheres will initially follow Newton's Law of Cooling. However, the solid sphere will consistently cool at a slower rate than the hollow sphere because of its greater thermal mass. This trend holds true for both small and large values of T, as the fundamental properties of heat transfer remain consistent.
Conclusion on Cooling Rates
In summary, while both spheres will cool in accordance with the temperature difference, the solid sphere will always cool at a slower rate compared to the hollow sphere due to its greater thermal mass. Therefore, the correct understanding is that the solid sphere will cool at a slower rate for all values of T when compared to the hollow sphere.
