Question

For an anisotropic solid a values in x, y, z
direction is 2:3:7. If coefficient of
volume expansion is 2.4 x 10-6/°Cfind
areal expansion coefficient of the solid in
XZ plane.​

Answer 1

Given:

For an anisotropic solid a values in x, y, z

direction is 2:3:7. Coefficient of

volume expansion is 2.4 x 10-6/°C.

To find:

Areal expansion coefficient of the solid in the XZ plane.

Calculation:

Let us assume "a" is the constant of proportionality.

So , the linear expansion coefficient along the respective axes will be :

• X axis = 2a
• Y axis = 3a
• Z axis = 7a

We know that algebraic summation of the linear expansion coefficient along the three axes will give us the volume expansion coefficient.

$2a + 3a + 7a = 2.4 \times {10}^{ - 6}$

$= > 12a = 2.4 \times {10}^{ - 6}$

$= > a = 2 \times {10}^{ - 7}$

So, Areal Expansion Coefficientcan be calculated by algebraic summation of the linear expansion coefficient along the X axis and the Z axis.

$\therefore \: \beta _{XZ} = 2a + 7a$

$= > \: \beta _{XZ} = 9a$

$= > \: \beta _{XZ} = 9 \times (2 \times {10}^{ - 7} )$

$= > \: \beta _{XZ} = 1.8 \times {10}^{ - 6} \: \degree {C}^{ - 1}$

So final answer is:

$\boxed{ \bold{\: \beta _{XZ} = 1.8 \times {10}^{ - 6} \: \degree {C}^{ - 1} }}$