Mechanical Properties of Solids

JEE Main level, single correct, medium

57 of 118

Question 57 of 118

Two wires of same length and radius are joined end to end and loaded. The Young's moduli of the materials of the two wires are Y1{Y}_{1} and Y2{Y}_{2}. The combination behaves as a single wire then its Young's modulus is:

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Answer: (B) Y=2Y1Y2Y1+Y2Y=\dfrac{2{Y}_{1}{Y}_{2}}{{Y}_{1}+{Y}_{2}}

In series combination, Δl=ℓ1+ℓ2\Delta l={\ell }_{1}+{\ell }_{2}
Y=F/AΔℓ/ℓ⇒Δℓ=FℓAYY=\dfrac{F/A}{\Delta \ell /\ell }\Rightarrow \Delta \ell =\dfrac{F\ell }{AY}
⇒Δℓ∝ℓY\Rightarrow \Delta \ell \propto \dfrac{\ell }{Y}
Equivalent length of rod after joining is =2ℓ=2\ell
As, lengths are same and force is also same in series
Δℓ=Δℓ1+Δℓ2\Delta \ell =\Delta {\ell }_{1}+\Delta {\ell }_{2}
ℓeqYeq=ℓY1+ℓY2⇒2ℓY=ℓY1+ℓY2\dfrac{{\ell }_{eq}}{{Y}_{eq}}=\dfrac{\ell }{{Y}_{1}}+\dfrac{\ell }{{Y}_{2}}\Rightarrow \dfrac{2\ell }{Y}=\dfrac{\ell }{{Y}_{1}}+\dfrac{\ell }{{Y}_{2}}
∴Y=2Y1Y2Y1+Y2∴Y=\dfrac{2{Y}_{1}{Y}_{2}}{{Y}_{1}+{Y}_{2}}

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