Question

A sphere, a cube and a thin circular plate, all of the same materials and same mass are initially heated to the same high temperature.
A) Plate will cool fastest and cube the slowest.
B) Sphere will cool the fastest and the cube the slowest.
C) Plate will cool fastest and sphere the slowest.
D) Cube will cool fastest and plate the slowest.

Answer 1

Hint: We need to understand the varying parameters involved in the three different shapes given to us which are made of the same material and have the same mass. The dependence of the heat absorbed has to be related to the surface area of the shapes.

Complete answer:
We are given a sphere, a cube and a thin circular plate all of which are made of the same material and have the same mass. It is said that the three shapes are heated to the same temperature. We know that the heat absorbed by three bodies depends on factors other than the mass and the material property but also the surface area of the shape.
For a given mass of the same material the surface areas of the three shapes are given in the increasing order (from least to the maximum) as –
\[{{A}_{sphere}}<{{A}_{Cube}}<{{A}_{\text{Circular plate}}}\]
Now, let us use the Stefan-Boltzmann constant which will give us the heat related to each of the shapes as –
\[Q=\sigma eA({{T}^{4}}-{{T}_{s}}^{4})\]
Now, we know that that the change in temperature for the three shapes can be given as –
\[\begin{align}
  & Q=ms\Delta T \\
 & \Rightarrow \Delta T=\dfrac{Q}{ms} \\
 & \therefore \Delta T=\dfrac{\sigma eA({{T}^{4}}-{{T}_{s}}^{4})}{ms} \\
\end{align}\]
We can understand from the above relation that the temperature drop is dependent only on the surface area of the three shapes since the shapes are made of the same materials and are of the same mass.
\[\begin{align}
  & {{A}_{sphere}}<{{A}_{Cube}}<{{A}_{Circularplate}} \\
 & \therefore \Delta {{T}_{Sphere}}<\Delta {{T}_{Cube}}<\Delta {{T}_{\text{Circular Plate}}} \\
\end{align}\]
So, the sphere will be cooling the slowest and the circular plate will be cooling the fastest.

The correct answer is option C.

Note:
The rate of temperature drop is directly proportional to the surface area of the material used in the situation. This is applicable to the case of temperature rise for the materials also. It is geometry that is more important than the mass of the material used.