If Y, K and η are the values of Young’s modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

a. \(k = \frac{Y\eta }{9\eta -3Y} N/m^{2}\)

b. \(\eta = \frac{3YK}{9K + Y} N/m^{2}\)

c. \(Y = \frac{9K\eta}{3K – \eta} N/m^{2}\)

d. \(Y = \frac{9K\eta}{2\eta + 3K} N/m^{2}\)

Answer: (a)

⇒ Y = 3K (1 – 2σ)

⇒ σ = ½ (1 – Y / 3K) …. (1)

⇒Y = 2η (1 + σ)

⇒ σ = Y / 2η – 1 …. (2)

By comparing equation (1) and (2), we get

⇒ Y / 2η – 1 = ½ (1 – Y / 3K)

⇒ Y / η – 2 = 1 – Y / 3K

⇒ Y / η = 1 + 2 – Y / 3K ⇒ Y / η = 3 – Y / 3K

⇒ Y / 3K = 3 – Y / η ⇒ Y / 3K = 3η – Y / η

⇒ K = ηY / 9η – 3Y

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