A solid cube is subjected to identical normal forces on all the faces. The ratio of volumetric strain to linear strain in any of the three axes isa) 1b) 2c) 3d) √3Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Explanation:

Concept:

Volumetric strain is the strain produced in a body due to a change in its volume.

Linear strain is the strain produced in a body due to a change in its length.

Young's modulus is the ratio of linear stress to linear strain.

Bulk modulus is the ratio of volumetric stress to volumetric strain.

Given:

A solid cube is subjected to identical normal forces on all the faces.

To find:

The ratio of volumetric strain to linear strain in any of the three axes.

Solution:

Let the force applied on each face be 'F'.

Let the original length of the cube be 'L'.

Let the original volume of the cube be 'V'.

Due to the force, each face will experience a normal stress of 'F/A', where 'A' is the area of each face.

The volumetric stress will be equal to the average of the normal stresses, i.e., 'F/A'.

The volumetric strain can be calculated using the formula:

Volumetric strain = (change in volume)/(original volume) = (change in length)^3/(original length)^3

The linear strain can be calculated using the formula:

Linear strain = (change in length)/(original length)

From Hooke's Law, we know that:

Linear stress = Young's modulus * linear strain

Volumetric stress = Bulk modulus * volumetric strain

As the normal forces are identical on all faces, the cube will experience uniform deformation.

Hence, the linear strain in any of the three axes will be the same.

Let the linear strain in any of the three axes be 'ε'.

The volumetric strain will be equal to 3 times the linear strain, i.e.,

Volumetric strain = 3*ε

From the equations above, we can write:

F/A = Young's modulus * ε

F/A = Bulk modulus * 3*ε

Dividing the second equation by the first equation, we get:

Bulk modulus/Young's modulus = 3

Hence, the ratio of volumetric strain to linear strain in any of the three axes is 3.

Therefore, option (c) is the correct answer.

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