Question

Young's modulus of a substance is 18.5 × 10¹¹ dyn/cm² and poisson's ratio is 0.238 . Find Bulk modulus and shear modulus .

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Answer 1

Given :

• Young's modulus of the substance = 18.5 × 10¹¹ dyn / cm².
• Poisson's ratio = 0.238

To find :

• Bulk modulus.
• Shear modulus.

Solution :

★ Formula Used :-

${\boxed{\bold{Y=3K(1-2\sigma)}}}$

• Terms identification :-
• Y = Young's modulus
• K = Bulk modulus
• $\sigma$=Poisson's ratio

Here,

• Y = 18.5 × 10¹¹ dyn/cm²
• $\sigma$=0.238

$\implies\sf{Y=3K(1-2\sigma)}$

$\implies\sf{18.5\times\:10^{11}=3K(1-2\times\:0.238)}$

$\implies\sf{18.5\times\:10^{11}=3K\times\:0.524}$

$\implies\sf{3K=\dfrac{18.5\times\:10^{11}}{0.524}}$

$\implies\sf{K=\dfrac{18.5\times\:10^{11}}{3\times\:0.524}}$

$\implies\sf{K=11.768\times\:10^{11}}$

Therefore, Bulk modulus is 11.768 × 10¹¹ dyn/cm².

________________________

★ Formula Used :-

${\boxed{\bold{Y=2n(1+\sigma)}}}$

• Terms identification :-
• Y = Young's modulus
• n = Shear modulus
• $\sigma$=poisson's ratio

Here,

• Y = 18.5×10¹¹ dyn/cm²
• $\sigma$=0.238

$\implies\sf{Y=2n(1+\sigma)}$

$\implies\sf{18.5\times\:10^{11}=2n(1+0.238)}$

$\implies\sf{18.5\times\:10^{11}=2n\times\:1.238}$

$\implies\sf{2n=\dfrac{18.5\times\:10^{11}}{1.238}}$

$\implies\sf{n=\dfrac{18.5\times\:10^{11}}{2\times\:1.238}}$

$\implies\sf{n=7.47\times\:10^{11}}$

Therefore, Shear modulus is 7.47×10¹¹ dyn / cm².