First slide

Elastic behaviour of solids

Question

A thin uniform elastic rod of natural length l, area of cross-section A, density $ρ$ and young’s modulus Y is just completely immersed in vertical position in a liquid of density $2 ρ$ by applying a force F in vertically downward direction as shown.
What is the compression in the rod?

Difficult

A

$\frac{3 ρ g l^{2}}{4 Y}$
B

$\frac{3 ρ g l^{2}}{2 Y}$
C

$\frac{2 ρ g l^{2}}{3 Y}$
D

$\frac{ρ g l^{2}}{2 Y}$

Solution

Considering x length of the rod from bottom, the FBD is as shown.
From diagram
$T = 2 ρ A \lg - ρ A x g = ρ A g (2 l - x)$
Hence stress $= \frac{T}{A} = ρ g (2 l - x)$
Hence strain $= \frac{d l}{d x} = \frac{ρ g}{Y} (2 l - x)$
Hence net compression is $Δ l = \int_{0}^{l} \frac{ρ g}{Y} (2 l - x) d x$ = $\frac{3 ρ g l^{2}}{2 Y}$

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