Question

The interatomic distance for a metal is \[3 \times {10^{ - 10}}m\]. If the interatomic force constant is \[3.6 \times {10^{ - 9}}N/m\]then the young's modulus in\[N/{m^2}\] will be:
A) \[1.2 \times {10^{11}}\]
B) \[4.2 \times {10^{11}}\]
C) \[10.8 \times {10^{ - 19}}\]
D) \[2.4 \times {10^{18}}\]

Answer 1

Hint: Hooke's law is defined as the force (F) needed to extend or compress a elastic material by some distance (x). Where k is defined as a constant factor. It is an expression as an important characteristic of elastic material. Interatomic force constant is calculated by finding the ratio of interatomic force to that of change in interatomic distance. It is also given as \[K = Y \times r\].Here Y is known as the Young's modulus and r is described as interatomic distance.
In question only force constant and interatomic distance is given. So we use a formula which has a relation between young’s modulus, interatomic distance and force constant.

Formula used:
We use \[Force{\text{ }}constant{\text{ }}\left( K \right){\text{ }} = {\text{ }}Young's{\text{ }} modulus {\text{ }} \times \;interatomic\;distance\] to calculate young’s modulus.

Complete step by step solution:
Given: \[Force{\text{ }}constant{\text{ }}\left( K \right)\]= \[3.6 \times {10^{ - 9}}N/m\] and interatomic distance =\[3 \times {10^{ - 10}}m\]
We know that, \[Force{\text{ }}constant{\text{ }}\left( K \right){\text{ }} = {\text{ }}Young's{\text{ }}modulus{\text{ }} \times \;interatomic\;distance\]
\[\therefore Young's{\text{ }}modulus = \dfrac{{Force{\text{ }}constant{\text{ }}\left( K \right)}}{{interatomic\;distance}}\]
\[Young's{\text{ }}modulus = \dfrac{{3.6 \times {{10}^{ - 9}}N/A}}{{3 \times {{10}^{ - 10}}m}}\]
\[ \Rightarrow Young's{\text{ }}modulus = 1.2 \times {10^{11}}\]\[N/{m^2}\]
Hence Young’s modulus of a metal is \[1.2 \times {10^{11}}\]\[N/{m^2}\].

Hence , the correct option is (A).

Additional information: Young’s Modulus is also known as the Elastic Modulus or Tensile Modulus. It is a measure of mechanical properties of linear elastic solids. Young’s modulus describes the relationship between stress and strain in an object. Stress is given by force per unit area and strain is given by proportional deformation. But young modulus also has a relation with force constant. So it can be calculated in terms of force constant.Interatomic distance is defined by distance between two atoms. It is generally defined by distance between two bonded atoms in a material (in solid forms).

Note: Students should know the relation between Young’s modulus, force constant and interatomic distance. Because, according to the question there is no need to use \[Young's{\text{ }}modulus = \dfrac{{stress}}{{strain}}\] formula. In question stress and strain is not given.