A metal rod of length 'L' and cross-sectional area 'A' is heated through `'T'^(@)C` What is the force required to prevent the expansion of the rod lengthwise ?

Options

A. `(YAalphaT)/((1-alphaT)`
B. `(YAalphaT)/((1+alphaT))`
C. `((1-alphaT))/(YA alphaT)`
D. `((1+alphaT))/(Y A alphaT)`

Answer

B

Explanation

Increase iin the length of the rod is given as,
`Delta L = alpha L_(0)T`
`L = L_(0)[1 + alphaT] "…."(i)`
Also, Young's modulus,
`Y = (FL_(0))/(ADeltaL)"….."(ii)`
Using relation (i) and substituting in the relation (ii), we get
`Y = (F)/(A) . (L_(0) (1+alphaT))/(DeltaL)`
or `DeltaL = (FL_(0)(1+alphaT))/(A.Y) "...."(iii)`
From Eqs. (i) and (ii), we get
`(FL_(0)(1+alphat))/(AY) = alphaL_(0)T`
`:. F = (Y A alphaT)/((1+alphaT))`