Calculate the compressional force required to prevent the metallic rod of length l cm and cross sectional area `Acm^2` when heated through `t^@C` from expanding lengthwise. Young's modulus of elasticity of the metal is E and mean coefficient of linear expansion is `alpha` per degree celsius.

Options

A. `EA alpha t`
B. `E A alpha t//((1+alpha t)`
C. `E A alpha t//(1- a t)`
D. `E l alpha t`

Answer

B

Explanation

The change in natural length `=Deltal_t=lalphat`
The natural length of rod at temperature `t^@C=l+lalphat`
The decrease in natural length due to developed stress `=Deltal` but the length of rod remains constant,
`:. Deltal_t-Deltal=0`
`:. Deltal=Deltal_t=lalphat`
`:. E=("stress")/("strain")=((F)/(A))/((-Deltal)/(l+Deltal))`
`E=-(EADeltal)/(l+Deltal_t)=(EADeltal_t)/(l+Deltal_t)`
`=-(EAlalphat)/(l+lalphat)=-(EADeltal_t)/((1+alphat))`