Question

A solid cube and a solid sphere of the same material have equal surface area. Both are at the same temperature $120{}^\circ C$, then
A. Both the cube and sphere cool down at the same rate
B. The cube cools down faster than the cube
C. The sphere cools down faster than the cube.
D. Whichever is having more mass will cool down faster

Answer 1

Hint:Here a solid cube and sphere which has equal surface area is given. They are made with the same material and are at the same temperature. We have to find out which one cools faster. We use the equation for rate of cooling and finds out which of the factors affect rate of cooling and there by we can predict easily which one cools faster

Formula used:
Rate of cooling of a body is given as:
$R=\dfrac{\Delta \theta }{t}=\dfrac{A\varepsilon \sigma ({{T}^{4}}-T_{0}^{4})}{mc}$
Where A is the area, m is the mass, $\varepsilon $is the emissivity and $\sigma $is the Stefan-Boltzmann constant.

Complete step by step solution:
We have a solid sphere and cube made of the same substance and has the same surface area at a temperature of $120{}^\circ C$. We have to find out whether a solid cube or solid sphere cools faster. We know the equation for the rate of cooling.

It is clear that the rate of cooling depends on mass, area and material. But since it is given that both the cube and sphere are made of the same material and have the same surface area we can say that: In this case, the rate of cooling depends only on volume. That is, rate of cooling,
$R=\dfrac{\Delta \theta }{t}=\dfrac{A\varepsilon \sigma ({{T}^{4}}-T_{0}^{4})}{mc}$

This equation can be simplified as:
$R\propto \dfrac{A}{m}\propto \dfrac{1}{volume}$ (Since surface area is the same.)

We know that the volume of the cube is less than the volume of the sphere. That is, the radius of the cube is greater than the radius of the sphere. In equation we can write as:
${{R}_{cube}}>{{R}_{sphere}}$
Therefore, from the equation of rate of cooling it is clear that the cube cools down at a faster rate than the sphere.

Therefore, the answer is option (B)

Notes: Rate of cooling dependence on material of the substance, surface area, mass and degree of cooling. If you imagine, you may think that a sphere has more volume than a cube due to its shape. But actually the radius of the cube is larger than the sphere for the same surface area. Remember that the radius of the cube is half its body diagonal.