Choose the correct relationship between Poisson ratio ( $\sigma$ ). bulk modulus ( K ) and modulus of rigidity $(\eta)$ of a given solid object:

(A) $\sigma=\frac{3 K-2 \eta}{6 K+2 \eta}$

(B) $\sigma=\frac{6 \mathrm{~K}+2 \eta}{3 \mathrm{~K}-2 \eta}$

(C) $\sigma=\frac{3 K+2 \eta}{6 K+2 \eta}$

(D) $\sigma=\frac{6 K-2 \eta}{3 K-2 \eta}$

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Elasticity & SHM

Choose the correct relationship between Poisson ratio (

σ

). bulk modulus ( K ) and modulus of rigidity

(η)

of a given solid object:

Answer: (A)

Young's modulus can be written through rigidity and through the bulk modulus as

Y = 2 η (1 + σ)

and

Y = 3 K (1 - 2 σ)

. Equating,

2 η (1 + σ) = 3 K (1 - 2 σ)

, which expands to

2 η + 2 η σ = 3 K - 6 K σ

and rearranges to

σ (6 K + 2 η) = 3 K - 2 η

. The Poisson ratio is therefore

σ = \frac{3 K - 2 η}{6 K + 2 η}

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option