Mechanical Properties of Solids

JEE Main level, single correct, medium

61 of 118

Question 61 of 118

The length of metallic wire is l1{l}_{1} when tension in it is T1{T}_{1}. It is l2{l}_{2} when the tension is T2{T}_{2}. The original length of the wire will be :

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Answer: (C) T2l1−T1l2T2−T1\dfrac{{T}_{2}{l}_{1}-{T}_{1}{l}_{2}}{{T}_{2}-{T}_{1}}

Assuming Hooke's law to be valid
T∝(Δℓ)T\propto \left(\Delta \ell \right)
T=k(Δℓ)T=k\left(\Delta \ell \right)
Let,ℓ0={\ell }_{0}= natural length (original length) ⇒T=k(ℓ−ℓ0)\Rightarrow T=k\left(\ell -{\ell }_{0}\right)
so, {T}_{1}=k\left({\ell }_{1}-{\ell }_{0}\right)&{T}_{2}=k\left({\ell }_{2}-{\ell }_{0}\right)
⇒T1T2=ℓ1−ℓ0ℓ2−ℓ0\Rightarrow \dfrac{{T}_{1}}{{T}_{2}}=\dfrac{{\ell }_{1}-{\ell }_{0}}{{\ell }_{2}-{\ell }_{0}}
⇒ℓ0=T2ℓ1−T1ℓ2T2−T1\Rightarrow {\ell }_{0}=\dfrac{{T}_{2}{\ell }_{1}-{T}_{1}{\ell }_{2}}{{T}_{2}-{T}_{1}}

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