A solid sphere, a cube and a plate, all are made of same material and all have same mass. These are heated to a temperature `100^@C` and then allowed to cool at the temperature of room. Which of these will cool down first –

Options

A. Cube
B. Plate
C. Sphere
D. All will cool down simultaneously

Answer

To determine which of the three objects (a solid sphere, a cube, and a plate) will cool down first when made of the same material and having the same mass, we can use the principles of heat transfer and the Stefan-Boltzmann law.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have three different shapes (a solid sphere, a cube, and a plate) made of the same material and having the same mass. They are heated to a temperature of 100°C and then allowed to cool in a room temperature environment.

2. **Applying the Stefan-Boltzmann Law**:
The rate of heat loss (cooling) of an object can be described by the Stefan-Boltzmann law, which states that the power radiated by a body is proportional to the fourth power of its absolute temperature and its surface area:
\[ P = \sigma A e^4 \]
where \( P \) is the power (rate of heat loss), \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( e \) is the emissivity.

3. **Identifying the Surface Areas**:
Since all three objects have the same mass, we need to compare their surface areas:
- **Sphere**: The surface area \( A_s \) of a sphere is given by \( A_s = 4\pi r^2 \).
- **Cube**: The surface area \( A_c \) of a cube is given by \( A_c = 6a^2 \), where \( a \) is the length of a side.
- **Plate**: The surface area \( A_p \) of a plate (assuming it is rectangular) can be maximized by making it very thin, leading to a larger area compared to the cube and sphere for the same mass.

4. **Comparing the Areas**:
For the same mass, the plate will have the largest surface area, followed by the cube, and then the sphere will have the smallest surface area. This is because the plate can be made very thin, thus maximizing its surface area for heat loss.

5. **Conclusion**:
Since the rate of cooling is directly proportional to the surface area, the object with the largest surface area will cool down the fastest. Therefore, the plate will cool down first, followed by the cube, and lastly the sphere.

### Final Answer:
The plate will cool down first.