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.

Solution

Physics sir · 3 steps

Write Y in terms of η and K through Poisson's ratio, then eliminate σ. Only one option survives.

1. Two standard links. With Poisson's ratio σ, Young's modulus connects to the other two moduli as Y = 2η(1+σ) and Y = 3K(1-2σ).
2. Eliminate σ. From the first, 2σ = Y/η - 2, so 1 - 2σ = 3 - Y/η = (3η - Y)/η. Put this into the second: Y = 3K(3η - Y)/η.

   9/Y = 3/η + 1/K ⇒ Y = 9Kη/(3K+η), K = Yη/(9η - 3Y), η = 3YK/(9K - Y)

   Option	Given form	Correct form	Verdict
   (a)	Y = 9Kη/(3K - η)	Y = 9Kη/(3K + η)	sign wrong
   (b)	Y = 9Kη/(2η + 3K)	Y = 9Kη/(3K + η)	extra factor 2
   (c)	η = 3YK/(9K + Y)	η = 3YK/(9K - Y)	sign wrong
   (d)	K = Yη/(9η - 3Y)	K = Yη/(9η - 3Y)	matches

3. So. Solving Yη = 3K(3η - Y) for K gives K = Yη/(9η - 3Y), which is option (d).

The idea to keep. Remember 9/Y = 3/η + 1/K. Every other form, for Y, K or η, is just this one equation rearranged.

Answer from the NTA final answer key. Solution by PrepWiser.