Question

The SI unit for the coefficients of cubical expansion is
A. $^\circ C$
B. $per^\circ C$
C. $cm/^\circ C$
D. None of these

Answer 1

Hint: It can be calculated by knowing about the factors on which cubical expansion depends. It’s formula can also be used.
Coefficient of cubical expansion,
$r = \dfrac{{\Delta V}}{{V\Delta T}}$

Complete step by step answer:
1. Cubical expansion is the increase in the volume of the block on heating.
2. Coefficient of cubical expansion – Suppose a solid block of initial volume V is heated through a temperature $\Delta T$ and then after heating, its final volume becomes $V'$.
It is found from experiments that
(i) Increase in volume $ \propto $rise in temperature
i.e. $V' - V \propto \Delta T$ ….(1)
(ii) Increase in volume $ \propto $ original volume that is
$V' - V \propto V$ ….(2)
Combining (1) and (2), we get
$V' - V \propto \Delta T$
$ \Rightarrow $ $V' - V = \gamma \,V\Delta T$
Where $\gamma $e $\gamma $ is a proportionality constant which is known as coefficient of cubical expansion and it depends on the nature of the material of solid.
So,
$V' = V + \gamma \,V\Delta T$
$V' = V\;[1 + \gamma \,\Delta T]$
$ \Rightarrow \;\;\;\gamma \; = \;\dfrac{{V' - V}}{{V\Delta T}}$
$ \Rightarrow \;\;\;\gamma \; = \;\dfrac{{\Delta V}}{{V\Delta T}}\;\; = \;\;\dfrac{{Increase\;in\;volume}}{{Original\;volume\;X\;Rise\;in\;temperature}}$
So, SI units of $\Upsilon $ will be
$\gamma = \dfrac{{{m^3}}}{{{m^3} \times ^\circ C}} = ^\circ {C^{ - 1}}$
So, The SI units of coefficient of the cubical expansion is per $^\circ C$.

So, the correct answer is “Option B”.

Note:
Remember both $^\circ C$and K and SI units for temperature. So, the SI of the coefficient of cubical expansion can be either $^\circ {C^{ - 1}}$ or ${K^{ - 1}}$.All the coefficient of thermal expansions i.e. either linear, superficial or cubical have same SI units i.e. $^\circ {C^{ - 1}}$or ${K^{ - 1}}$ and these three are related as
\[\dfrac{1}{\alpha } = \dfrac{2}{\beta } = \dfrac{3}{\gamma }\]
Where $\alpha $ is coefficient of linear expansion, $\beta $is coefficient of superficial expansion and $\gamma $ is coefficient of cubical expansion.