Question

The coefficient of cubical expansion of copper is $5.1 \times {10^{ - 5}}$ per C. Coefficient of linear expansion is $K \times {10^{ - 5}}$ per C. Find the value of K.

Answer 1

As we know that the coefficient of cubical expansion is three times the coefficient of linear expansion.
That is $\gamma = 3\alpha$
Where $\gamma =$ coefficient of cubical expansion
$\alpha =$ Coefficient of linear expansion
So, we will substitute the value of $\alpha$ and $\gamma$ in the given equation to find the value of K which is required in the question.

Complete step by step solution:
As we know that coefficient of cubical expansion is three times the coefficient of linear expansion, we have been given the values of coefficient of linear expansion and coefficient of cubical expansion, that is $\gamma = 3\alpha$
So, putting the given values in the equation $\gamma = 3\alpha$ we get
Here coefficient of linear expansion $= K \times {10^{ - 5}}$ per C
Coefficient of cubical expansion $= 5.1 \times {10^{ - 5}}$ per C,
So, substituting the given values in the equation we get
$5.1 \times {10^{ - 5}} = 3(K \times {10^{ - 5}})$
$\dfrac{{5.1 \times {{10}^{ - 5}}}}{3} = K \times {10^{ - 5}}$
$\Rightarrow K = \dfrac{{5.1}}{3} = 1.7$
Hence the value of K= 1.7

Additional information: The relation between coefficient of linear expansion is in one dimension, coefficient of areal expansion is in two dimension and coefficient of volume expansion is in three dimensions so to compensate that factor the relation between them will be like, given as follows
$\alpha = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$

Note: One should keep in mind that the coefficient of cubical expansion is nearly three times the coefficient of linear expansion. So accordingly, the given values should be applied so as to get the correct solution.
Also, one should keep in mind the proper units regarding the coefficient of linear expansion and coefficient of cubical expansion to avoid calculation mistakes.