If Y, K and eta are the values of Young's mod | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The relationship between Young's modulus $(Y)$,Bulk modulus $(K)$,and modulus of rigidity $(\eta)$ is derived using Poisson's ratio $(\sigma)$.We

Vedclass · Accepted Answer

$Y = \frac{9K\eta}{3K + \eta}$

Vedclass · Answer

(C) The relationship between Young's modulus $(Y)$,Bulk modulus $(K)$,and modulus of rigidity $(\eta)$ is derived using Poisson's ratio $(\sigma)$.We know the relations:$Y = 3K(1 - 2\sigma) \implies 1 - 2\sigma = \frac{Y}{3K} \implies 2\sigma = 1 - \frac{Y}{3K} \implies \sigma = \frac{1}{2} - \frac{Y}{6K} \dots (i)$$Y = 2\eta(1 + \sigma) \implies 1 + \sigma = \frac{Y}{2\eta} \implies \sigma = \frac{Y}{2\eta} - 1 \dots (ii)$Equating $(i)$ and $(ii)$:$\frac{1}{2} - \frac{Y}{6K} = \frac{Y}{2\eta} - 1$$1 + \frac{1}{2} = \frac{Y}{2\eta} + \frac{Y}{6K}$$\frac{3}{2} = \frac{3KY + Y\eta}{6K\eta}$$9K\eta = 3KY + Y\eta$$9K\eta = Y(3K + \eta)$$Y = \frac{9K\eta}{3K + \eta}$Thus,the correct relation is $Y = \frac{9K\eta}{3K + \eta}$.

If $Y, K$ and $\eta$ are the values of Young's modulus,bulk modulus,and modulus of rigidity of any material respectively,choose the correct relation for these parameters.

Similar Questions

For a given material, the Young's modulus is $2.4$ times its modulus of rigidity. Its Poisson's ratio is ......

$A$ metal wire having Poisson's ratio $1/4$ and Young's modulus $8 \times 10^{10} \, N/m^2$ is stretched by a force,which produces a lateral strain of $0.02 \%$ in it. The elastic potential energy stored per unit volume in the wire is [in $J/m^3$]

$A$ material has a Poisson's ratio of $0.50$. If a uniform rod made of this material suffers a longitudinal strain of $2 \times 10^{-3}$,what is the percentage change in its volume (in $\%$)?

The Poisson's ratio cannot have the value:

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module