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Question: If the interatomic spacing in a steel wire is \(2.8 \times {10^{ - 10}}m\) and \(Y{\text{steel}}\,{\...

If the interatomic spacing in a steel wire is 2.8×10−10m2.8 \times {10^{ - 10}}m and Ysteel  =  2×1011Nm−2Y{\text{steel}}\,{\text{ = }}\,{\text{2}} \times {\text{1}}{{\text{0}}^{11}}N{m^{ - 2}}, then force constant in Nm−1N{m^{ - 1}} is-
(A)5.6\left( A \right)5.6
(B)56\left( B \right)56
(C)0.56\left( C \right)0.56
(D)560\left( D \right)560

Explanation

Solution

By using the interatomic force constant formula we can solve this problem.Where interatomic force constant is the multiplication of the Young’s modulus of the material with the interatomic distance between the atoms of that material.

Complete step by step solution:
Interatomic force Constant:
When an external force is applied on a solid, this distance between its atom changes and interatomic force works to restore the original dimension.The ratio of interatomic force to that of change in interatomic distance is defined as the interatomic force constant. It is also given as the product of young’s modulus and the interatomic distance.
Hence we can write,
F=Y×rF = Y \times r
Where,
Force constant or interatomic force constant = F
Young’s Modulus = Y
Interatomic distance = r
As per the problem there is an interatomic spacing in a steel wire is 2.8×10−10m2.8 \times {10^{ - 10}}m and Ysteel  =  2×1011Nm−2Y{\text{steel}}\,{\text{ = }}\,{\text{2}} \times {\text{1}}{{\text{0}}^{11}}N{m^{ - 2}} .
We need to calculate the force constant or we can say it as an interatomic force constant.
We know the formula,
F=Y×rF = Y \times r
Here we have to put the Young’s modulus of steel wire
So in place of YY we can write Ysteel Y{\text{steel}}\,
Now the force constant will become,
F=Ysteel×rF = Y{\text{steel}} \times r
By putting the given values from the problem in the above equation we will get,
F= 2×1011Nm−2×2.8×10−10mF = \,{\text{2}} \times {\text{1}}{{\text{0}}^{11}}N{m^{ - 2}} \times 2.8 \times {10^{ - 10}}m
⇒F= 5.6×1011−10Nm−2+1\Rightarrow F = \,5.6 \times {\text{1}}{{\text{0}}^{11 - 10}}N{m^{ - 2 + 1}}
On further solving we will get,
F= 5.6×10Nm−1F = \,5.6 \times {\text{10}}N{m^{ - 1}}
F=56Nm−1F = 56N{m^{ - 1}}
Therefore the correct option is (B)\left( B \right).

Note:
Remember when an external force is applied on a solid which creates a disturbance inside the solid by changing its interatomic distance then this force constant is applied to restore its original dimension. Note the dimensional formula for force constant is MLT−2ML{T^{ - 2}}. This force constant is also called an interatomic force constant.