Question

There is a cuboid of dimensions $l\times b\times h$ and thermal capacity ’k’ units. On doubling the dimensions of the cuboid. Find thermal capacity.

Answer 1

Hint: The thermal capacity of a body is the ability of the body to absorb heat. First we will have to know how the thermal capacity of a body is defined. Accordingly we will determine on what factors the thermal capacity of the material depends upon and on verifying whether it changes with the change in dimensions of the body will enable us to obtain the correct answer.
Formula used:
$k=mc$

Complete answer:
The thermal capacity of the body is defined as the amount of heat required to raise the temperature of the substance by one degree. If there is a body of mass ‘m’ and the specific heat of the body is equal to ‘c’, than the thermal capacity ‘k’ of the body is given by,
$k=mc$
Now if we observe the above equation, the heat capacity of the body is proportional to the amount of the substance i.e. the mass and the specific heat. In the question it is mentioned to us that all the dimensions of the body are doubled i.e. twice their actual value. This means that the volume of the body is increased implies that there is addition of mass to the body. If ‘d’ is the density of the substance and ‘V’ is the volume occupied by the body, the mass of the substance is given by,
$\begin{align}
  & m=d\times V \\
 & \because V=l\times b\times h \\
 & \therefore m=d(l\times b\times h) \\
\end{align}$
Therefore the thermal capacity of the body is given by,
$k=d(l\times b\times h)c....(1)$
In the questions it is given that the dimensions of the cuboid are doubled. Hence the mass of the body is now equal to,
$\begin{align}
  & m=d\times (2l\times 2b\times 2h) \\
 & \therefore m=8d(l\times b\times h) \\
\end{align}$
Hence the thermal capacity of the body is now equal to,
$\begin{align}
  & k=mc \\
 & \because m=8d(l\times b\times h) \\
 & \therefore k=8d(l\times b\times h)c....(2) \\
\end{align}$
Comparing equation 1 and 2 we can imply that the thermal capacity becomes 8 times the initial when the dimensions of the cuboid are doubled.

Note:
It is to be noted that specific heat of a substance is the amount of heat required to raise the temperature of unit mass of the substance by one degree. If the mass of the body is kept constant, the thermal capacity of the body does not increase despite changing the dimension of the body. If the mass of the body is kept constant then the above equations do not hold valid.