Mechanical Properties of Solids

JEE Main level, single correct, medium

59 of 118

Question 59 of 118

The value of tension in a long thin metal wire has been changed from T1{T}_{1} to T2{T}_{2}. The lengths of the metal wire at two different values of tension T1{T}_{1} and T2{T}_{2} are ℓ1{\ell }_{1} and ℓ2{\ell }_{2}, respectively. The actual length of the metal wire is:

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Answer: (A) T1ℓ2−T2ℓ1T1−T2\dfrac{{T}_{1}{\ell }_{2}-{T}_{2}{\ell }_{1}}{{T}_{1}-{T}_{2}}

Y=FLAΔLY=\dfrac{FL}{A\Delta L}
⇒Y=T1ℓ0A(ℓ1−ℓ0)=T2ℓ0A(ℓ2−ℓ0)\Rightarrow Y=\dfrac{{T}_{1}{\ell }_{0}}{A\left({\ell }_{1}-{\ell }_{0}\right)}=\dfrac{{T}_{2}{\ell }_{0}}{A\left({\ell }_{2}-{\ell }_{0}\right)}
1=T1(ℓ2−ℓ0)T2(ℓ1−ℓ0)1=\dfrac{{T}_{1}\left({\ell }_{2}-{\ell }_{0}\right)}{{T}_{2}\left({\ell }_{1}-{\ell }_{0}\right)}
T2ℓ1−T2ℓ0=T1ℓ2−T1ℓ0{T}_{2}{\ell }_{1}-{T}_{2}{\ell }_{0}={T}_{1}{\ell }_{2}-{T}_{1}{\ell }_{0}
(T1−T2)ℓ0=T1ℓ2−T2ℓ1\left({T}_{1}-{T}_{2}\right){\ell }_{0}={T}_{1}{\ell }_{2}-{T}_{2}{\ell }_{1}
ℓ0=(T1ℓ2−T2ℓ1T1−T2){\ell }_{0}=\left(\dfrac{{T}_{1}{\ell }_{2}-{T}_{2}{\ell }_{1}}{{T}_{1}-{T}_{2}}\right)

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