A solid sphere and a hollow sphere are heated the same temperature . Point out the true statement.

Options

A. hollow sphere cools more quickly
B. both hollow and solid sphere cools equally
C. solid sphere cools more quickly .
D. none of the statement is true.

Explanation

To solve the question regarding the cooling rates of a solid sphere and a hollow sphere when both are heated to the same temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two spheres - a solid sphere and a hollow sphere, both heated to the same temperature. We need to determine which one cools faster.

2. **Identify Key Concepts**:
- The rate of heat transfer (cooling) is influenced by the mass and surface area of the objects.
- The formula for heat transfer can be expressed as \( \frac{dQ}{dt} = ms \frac{dT}{dt} \), where:
- \( dQ \) is the heat transferred,
- \( dt \) is the time,
- \( m \) is the mass,
- \( s \) is the specific heat capacity,
- \( dT \) is the change in temperature.

3. **Compare Masses**:
- For a solid sphere, the mass \( m_s \) is greater than that of the hollow sphere \( m_h \) if both have the same radius.
- This is because the solid sphere has mass distributed throughout its volume, while the hollow sphere has mass only at its surface.

4. **Rate of Heat Transfer**:
- The rate of heat transfer for both spheres can be expressed as:
\[ \frac{dQ}{dt} \propto m \]
- Since \( m_s > m_h \), we conclude that:
\[ \frac{dQ_s}{dt} > \frac{dQ_h}{dt} \]
- This means the solid sphere transfers heat more quickly than the hollow sphere.

5. **Cooling Rate**:
- The cooling rate can be expressed as:
\[ \frac{dT}{dt} \propto \frac{dQ}{dt} \]
- Since the solid sphere has a greater mass, it will have a smaller change in temperature over time compared to the hollow sphere. Thus:
\[ \frac{dT_s}{dt} < \frac{dT_h}{dt} \]
- This indicates that the hollow sphere cools faster than the solid sphere.

6. **Conclusion**:
- Therefore, the true statement is that the hollow sphere cools faster than the solid sphere.

### Final Answer:
The hollow sphere cools faster than the solid sphere.