Two spheres, one solid and other hollow are kept in atmosphere at same temperature. They are made of same material and their radii are also same. Which sphere will cool at a faster rate initially?

Explanation

To solve the problem of which sphere will cool at a faster rate initially, we can follow these steps:

### Step 1: Understand the Problem
We have two spheres: one solid and one hollow. Both spheres are made of the same material and have the same radius. They are kept in an environment at the same initial temperature. We need to determine which sphere will cool faster.

### Step 2: Identify Relevant Concepts
The rate of cooling of an object can be described by Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings. However, in this case, we will consider the heat transfer characteristics based on their mass and surface area.

### Step 3: Analyze the Mass of Each Sphere
- **Solid Sphere**: The mass (m) of a solid sphere can be calculated using the formula:
\[ m_{\text{solid}} = \frac{4}{3} \pi r^3 \rho \]
where \( r \) is the radius and \( \rho \) is the density of the material.

- **Hollow Sphere**: The mass of a hollow sphere (thin-walled) can be approximated as:
\[ m_{\text{hollow}} = 4 \pi r^2 \cdot t \cdot \rho \]
where \( t \) is the thickness of the hollow sphere. Since the hollow sphere has a larger volume of air inside, its mass will be less than that of the solid sphere.

### Step 4: Compare the Masses
Since the solid sphere has more material (greater mass) than the hollow sphere, we can conclude that:
\[ m_{\text{solid}} > m_{\text{hollow}} \]

### Step 5: Relate Cooling Rate to Mass
The rate of cooling (change in temperature over time, \( \frac{dT}{dt} \)) is inversely proportional to mass when other factors (like emissivity, area, and specific heat) are constant:
\[ \frac{dT}{dt} \propto \frac{1}{m} \]
Thus, since \( m_{\text{solid}} > m_{\text{hollow}} \), it follows that:
\[ \frac{dT_{\text{solid}}}{dt} < \frac{dT_{\text{hollow}}}{dt} \]

### Step 6: Conclusion
From the analysis, we conclude that the hollow sphere will cool at a faster rate than the solid sphere initially.

### Final Answer
The hollow sphere will cool at a faster rate initially.

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