A sphere, a cube and a thin circular plate, all made of the same material and having the same mass are initially heated to a temperature of `3000^(@)`C. Which of these will cool fastest ?

Options

A. sphere
B. cube
C. plate
D. none of these

Answer

To determine which object cools the fastest among a sphere, a cube, and a thin circular plate, all made of the same material and having the same mass, we can analyze the cooling process based on their surface areas.

### Step-by-Step Solution:

1. **Understanding Cooling Rate**: The rate of cooling of an object is influenced by its surface area. According to Newton's Law of Cooling, the rate of heat loss of a body is directly proportional to its surface area when the temperature difference between the body and its surroundings is constant.

2. **Calculating Surface Areas**:
- **Sphere**: The surface area \( A_s \) of a sphere is given by the formula:
\[ A_s = 4\pi r^2 \]
- **Cube**: The surface area \( A_c \) of a cube with side length \( a \) is:
\[ A_c = 6a^2 \]
- **Thin Circular Plate**: The surface area \( A_p \) of a thin circular plate with radius \( r \) is:
\[ A_p = \pi r^2 \]

3. **Comparing Surface Areas**: Since all three objects have the same mass and are made of the same material, we need to compare their surface areas to determine which one will cool fastest:
- The sphere has the smallest surface area.
- The cube has a moderate surface area.
- The thin circular plate has the largest surface area.

4. **Conclusion**: Since the cooling rate is directly proportional to the surface area, the object with the largest surface area will cool the fastest. Thus, the thin circular plate, having the largest surface area, will cool the fastest, followed by the cube, and finally the sphere, which will cool the slowest.

### Final Answer:
The thin circular plate will cool the fastest.