A sphere a cube and thin circular plate, all made of the same material and having the same mass are initially heated to a temperature of `1000^(@)C`. Which one of these will cool first?

Options

A. plate
B. Sphere
C. Cube
D. None of these

Explanation

To determine which object (sphere, cube, or thin circular plate) will cool first when all are made of the same material, have the same mass, and are initially heated to a temperature of \(1000^\circ C\), we can analyze their cooling rates based on their surface areas. The cooling rate of an object is influenced by its surface area, as a larger surface area allows for more heat to be lost to the environment.

### Step-by-Step Solution:

1. **Understanding the Cooling Process**:
- The rate of cooling of an object can be described by Newton's Law of Cooling, which states that the rate of heat loss of a body is directly proportional to the difference in temperature between the body and its surroundings, provided this difference is small.

2. **Heat Transfer Equation**:
- The heat transfer due to radiation can be described by the equation:
\[ q = \sigma \epsilon A (T^4 - T_0^4) \]
where:
- \(q\) = heat lost,
- \(\sigma\) = Stefan-Boltzmann constant,
- \(\epsilon\) = emissivity of the material,
- \(A\) = surface area,
- \(T\) = temperature of the object,
- \(T_0\) = ambient temperature.

3. **Cooling Rate Dependence**:
- Rearranging the equation to find the change in temperature (\(\Delta \theta\)):
\[ \Delta \theta \propto \frac{q}{mc} \]
where \(m\) is mass and \(c\) is specific heat capacity. Since \(m\) and \(c\) are constant for all three objects, the cooling rate (\(\Delta \theta\)) is directly proportional to the surface area \(A\).

4. **Surface Area Calculation**:
- For a given mass, the surface areas of the three shapes are different:
- **Sphere**: The surface area \(A_s = 4\pi r^2\).
- **Cube**: The surface area \(A_c = 6a^2\) (where \(a\) is the length of a side).
- **Thin Circular Plate**: The surface area \(A_p = \pi r^2\) (where \(r\) is the radius).

5. **Comparing Surface Areas**:
- For a fixed mass, the sphere has the smallest surface area to volume ratio, followed by the cube, and the thin circular plate has the largest surface area relative to its volume.
- Therefore, the order of surface areas from smallest to largest is:
- Sphere < Cube < Thin Circular Plate.

6. **Conclusion**:
- Since the cooling rate is directly proportional to the surface area, the object with the largest surface area will cool the fastest.
- Hence, the thin circular plate will cool first, followed by the cube, and finally the sphere will cool the slowest.

### Final Answer:
The thin circular plate will cool first.