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Chapter 7: Problem 939

If the young's modulus of the material is 3 times its modulus of rigidity. Then what will be its volume elasticity? (A) zero (B) infinity (C) \(2 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (D) \(3 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

Short Answer

Expert verified
The volume elasticity (K) of the material is equal to \(\frac{2}{3}\) times the modulus of rigidity (G).

Step by step solution

01

Write down the given relationships

We are given: - Young's modulus (E) = 3 times modulus of rigidity (G) - The relationship between E, G, and K: \(E = 2G (1 + \frac{G}{3K})\)
02

Substitute the given relationship into the formula

Substitute E = 3G into the formula: \(3G = 2G (1 + \frac{G}{3K})\)
03

Solve for K

To solve for K, let's first simplify the equation: \(3G = 2G + \frac{2G^2}{3K}\) Now, let's isolate the term with K: \(\frac{2G^2}{3K} = G\) Now, let's solve for K by first multiplying both sides by \(3K\): \(2G^2 = 3GK\) Finally, divide both sides by \(3G\): \(K = \frac{2G^2}{3G}\) Simplify: \(K = \frac{2}{3}G\) Hence, the volume elasticity (K) is equal to \(\frac{2}{3}\) times the modulus of rigidity (G).

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