Question

A very thin metallic shell of radius r is heated to temperature t and then allowed to cool. The rate of cooling of the shell is proportional to
\[A.\,rT\]
\[B.\,{}^{1}/{}_{r}\]
\[C.\,{{r}^{2}}\]
\[D.\,{{r}^{0}}\]

Answer 1

Hint: There are two laws that define the properties of the body depending on the temperature. They are: Newton’s law of cooling – describes the rate of heat loss of a body in terms of temperature and Stefan’s law – describes the power radiated from a black body in terms of temperature.
Formula used:
\[\dfrac{dQ}{dt}\propto \Delta T(t)\]

Complete answer:
From given, we have the data,
The radius of a very thin metallic shell = r
The temperature up to which a very thin metallic shell is heated = t
Newton’s law of cooling states that, the rate of heat flow of a body is proportional to the time dependent temperature difference between the body’s surface and that of the environment.

\[\begin{align}
  & \dfrac{dQ}{dt}\propto \Delta T(t) \\
 & \dfrac{dQ}{dt}\propto T(t)-{{T}_{env}} \\
\end{align}\]
Where, \[T(t)\] is the temperature of the body's surface and \[{{T}_{env}}\] is the environment temperature.

Thus, the rate of the heat of the body is independent of its own mass, weight and height.
Rather, it is dependent on the temperature of the body’s surface and the temperature of the surrounding, that is, the temperature of the environment.

As the rate of cooling of the shell is proportional to the change in the temperature, that is, the temperature difference between the shell's surface and the environment temperature, thus the rate of cooling of the shell is independent of its radius..

So, the correct answer is “Option D”.

Note:
The things to be on your figure tips for further information on solving these types of problems are: There are two ways of heat transfer. They are – (1) Temperature dependent heat transfer \[\left( \dfrac{dQ}{dt}\propto \Delta T(t) \right)\] and (2) Temperature independent heat transfer \[\left( Q=hA\Delta T(t) \right)\]where \[h\]is the heat transfer coefficient.