A thin copper rod of length is suspended vertically from the ceiling by one of its ends. Density of copper is , Young's modulus is Y and Poisson's ratio is σ . Then which of the following are correct ?

Correct answer: D. Volume strain of the rod is ( g l)/(2 Y)[1-2 σ] At, P, T=A x g Elongation of element dx is d e=(T(d x))/(A Y) Deformation energy of the element is d U=(1)/(2) T(d e)=(T^2 d x)/(2 A Y) gathered d U=((A x g)^2 d x)/(2 A Y)=(A ^2 g^2 x^2 d x)/(2 Y) u=∫ d u=(A ^2 g^2)/(2 y)=∫_0^ x^2 d x=(A ^2 g^2 ^3)/(6 y) gathered

JEE Advanced

Exam: JEE Advanced
Subject: Physics
Chapter: Mechanical Properties of Solids

JEE Advanced · Physics · Mechanical Properties of Solids

A thin copper rod of length is suspended vertically from the...

A thin copper rod of length

\ell

is suspended vertically from the ceiling by one of its ends. Density of copper is

\rho

, Young's modulus is

Y

and Poisson's ratio is

\sigma

. Then which of the following are correct ? A thin copper rod of length is suspended vertically from the ceiling by one of its ends. Density of copper is , Young's

A thin copper rod of length is suspended vertically from the ceiling by one of its ends. Density of copper is , Young's

Options

AThe strain in the rod due to its own weight is $\frac{\rho g l}{2 Y}$
BThe elastic deformation energy stored in the rod due to its own weight is $\frac{\rho^2 g^2 l^3 A}{6 Y}$ where $A$ is area of cross section of the rod
CVolume strain of the rod is $\frac{\rho g l}{2 Y}[1-\sigma]$
DVolume strain of the rod is $\frac{\rho g l}{2 Y}[1-2 \sigma]$

Correct answer

D. Volume strain of the rod is $\frac{\rho g l}{2 Y}[1-2 \sigma]$

Step-by-step solution

At,

P, T=A \rho x g

Elongation of element dx is

d e=\frac{T(d x)}{A Y}

Deformation energy of the element is

d U=\frac{1}{2} T(d e)=\frac{T^2 d x}{2 A Y}\begin{gathered}d U=\frac{(A \rho x g)^2 d x}{2 A Y}=\frac{A \rho^2 g^2 x^2 d x}{2 Y} \\ u=\int d u=\frac{A \rho^2 g^2}{2 y}=\int_0^\epsilon x^2 d x=\frac{A \rho^2 g^2 \ell^3}{6 y}\end{gathered}

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