A sphere, a cube and a thin circular plate, all of same material and same mass are initially heated to same high temperature. a) Plate will cool fastest and cube the slowest. b) Sphere will cool fastest and cube the slowest. c) Plate will cool fastest and sphere the slowest. d) Cube will cool fastest and plate the slowest.

Options

A. Plate will cool fastet and cube the slowest.
B. Sphere will cool fastest and cube the slowest.
C. Plate will cool fastest and sphere the slowest.
D. Cube will cool fastest and plate the slowest.

Explanation

To solve the problem, we need to analyze the cooling rates of a sphere, a cube, and a thin circular plate, all made of the same material and having the same mass. The cooling rate of an object is influenced by its surface area, as per Stefan's law, which states that the rate of heat loss is proportional to the surface area and the fourth power of the temperature difference between the object and its surroundings.

### Step-by-Step Solution:

1. **Understand the Cooling Process**:
- According to Stefan's law, the rate of heat loss (cooling) is directly proportional to the surface area of the object. The larger the surface area, the faster the cooling.

2. **Calculate the Surface Areas**:
- For a sphere:
\[ A_{sphere} = 4\pi r^2 \]
- For a cube (with side length \(a\)):
\[ A_{cube} = 6a^2 \]
- For a thin circular plate (with radius \(R\) and negligible thickness):
\[ A_{plate} = \pi R^2 \]

3. **Compare Surface Areas**:
- Since all three objects have the same mass and are made of the same material, we can infer that their volumes are equal.
- The volume of the sphere, cube, and plate can be related to their dimensions, but we need to focus on their surface areas for cooling.
- Generally, for a given mass, the thin circular plate has a larger surface area compared to the sphere and cube.

4. **Determine the Cooling Order**:
- Since the thin circular plate has the largest surface area, it will cool the fastest.
- The sphere, having the smallest surface area, will cool the slowest.
- The cube will have a surface area that is in between that of the plate and the sphere.

5. **Conclusion**:
- Based on the analysis, we conclude that:
- The thin circular plate will cool fastest.
- The sphere will cool slowest.
- Therefore, the correct option is **C**: Plate will cool fastest and sphere the slowest.

### Final Answer:
The correct option is **C**: Plate will cool fastest and sphere the slowest.