Question

A metal rod of length Land cross-sectional area A is heated through T °C. What is the force required to prevent the expansion of the rod lengthwise?

(Y = Young's modulus of material of the rod, α = coefficient of linear expansion of the rod.)

Options

• `("YA"alpha"T")/(1-alpha"T")`

• `("YA"alpha"T")/(1+alpha"T")`

• `("YA"alpha)/("T"(1+alpha"T"))`

• `("YA"alpha)/(1-alpha"T")`

Answer 1

`("YA"alpha"T")/(1+alpha"T")`

Explanation:

The increase in length due to heating is given by

L = L₀ (1 + αT) or L - L₀ = `triangle"L"` = L₀ α T

To compress the rod by `triangle"L",` the force required is given by

`"F"=("YA"triangle"L")/"L"`

Substituting the value of L and `triangle"L"` we get

`"F"=("YA"xx"L"_0alpha"T")/("L"_0(1+alpha"T"))=("YA"alpha"T")/(1+alpha"T")`