Question

v) The ratio relation between coefficient of linear, areal and volume expansion of a solid is
a) 1:3:2
b) 1:2:3
c) 2:3:1
d) 3:1:2
​

Answer 1

To find:

Ratio relation between coefficient of linear, areal and volume expansion of a solid is?

Calculation:

Let's consider a solid with length, breadth and height each equal to l_(0):

Now, with change in temperature:

$l = l_{0} (1 + \alpha \: \Delta t)$

Now, area change will be :

$a = a_{0}(1 + \beta \Delta t)$

$\implies {l}^{2} = {l_{0}}^{2} (1 + \beta \Delta t)$

$\implies { \bigg \{l_{0}(1 + \alpha \Delta t) \bigg \} }^{2} = {l_{0}}^{2} (1 + \beta \Delta t)$

$\implies {(1 + \alpha \Delta t) }^{2} = (1 + \beta \Delta t)$

Using binomial approximations:

$\implies (1 + 2\alpha \Delta t)= (1 + \beta \Delta t)$

$\implies 2\alpha \Delta t= \beta \Delta t$

$\implies \beta = 2 \alpha$

Similarly, we can say:

$\implies \gamma = 3 \alpha$

So, the ratio is :

$\boxed{ \alpha : \beta : \gamma = 1 : 2 : 3}$

Hope It Helps.