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Q.

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  2ρ by applying a force  F in vertically downward direction as shown. What is the compression in the rod?

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a

3ρgl24Y

b

3ρgl22Y

c

2ρgl23Y

d

ρgl22Y

answer is B.

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Detailed Solution

Considering x length of the rod from bottom, the FBD is as shown.From diagramT=2ρAlg−ρAxg=ρAg2l−x Hence stress  =TA=ρg2l−xHence  strain  =dldx=ρgY2l−xHence net compression is  Δl=∫0lρgY2l−xdx=3ρg l22Y
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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  2ρ by applying a force  F in vertically downward direction as shown. What is the compression in the rod?