Calculate the compressional force required to prevent the metallic rod of length l cm and cross-sectional area A cm2 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 α per degree celsius.

Question

Calculate the compressional force required to prevent the metallic rod of length l cm and cross-sectional area A cm2 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 α per degree celsius. (A)  textitEAα textitt (B) ( textitEA α textitt/(1 + α textitt)) (C) ( textitEAα textitt/(1 - α textitt)) (D)  textitEl α textitt

Tardigrade Admin · Accepted Answer

The change in natural length = Δ l textt = l â¡ α textt The natural length of rod at temperature textt° textC = l + l â¡ α textt The decrease in natural length due to developed stress = Δ l But the length of rod remains constant. â´ Δ l textt - Δ l â¡ = 0 â´ Δ l = Δ l â¡ textt = l â¡ α textt â´ textE = ( textstress/ textstrain) = ( textF/A/(- Δ l )l â¡ + Δ l â¡ textt) â´ textE = - ( textEA Δ l /l â¡ + Δ l â¡ textt) = - ( textEA Δ l â¡ textt/l â¡ + Δ l â¡ textt) = - ( textEA l α textt/l â¡ + l â¡ α textt) = - ( textEA α textt/(1 + α textt))

$EA α t$

$\frac{EA α t}{( 1 + α t )}$

$\frac{EA α t}{( 1 - α t )}$

$E l α t$

Q. Calculate the compressional force required to prevent the metallic rod of length l cm and cross-sectional area Acm2 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 α per degree celsius.

EAαt

(1+αt)EAαt​

(1−αt)EAαt​

Elαt

$EA α t$

$\frac{EA α t}{( 1 + α t )}$

$\frac{EA α t}{( 1 - α t )}$

$E l α t$