A steel bolt of cross-sectional area `A_(b) = 5 xx 10^(-5) m^(2)` is passed through a cylindrical tube made of aluminium. Cross-sectional area of the tube material is `A_(t)= 10^(-4) m^(2)` and its length is` l = 50 cm`. The bolt is just taut so that there is no stress in the bolt and temperature of the assembly increases through `/_\theta=10^@C`. Given, coefficient of linear thermal expansion of steel, `alpha_(b) = 10^(-5)//^@C`.

Young's modulus of steel `Y_(b)=2xx10^(11)N//m^(2)`
Young's modulus of `Al, Y_(t)=10^(11) N//m^(2)`, coefficient of linear thermal expansion of `Al alpha_(t)=2xx10^(-5)//^@C`
The compressive strain in tube is

Options

A. `10^(-4)`
B. `5xx10^(-5)`
C. `2xx10^(-3)`
D. `10^(-6)`

Answer

B

Explanation

Since `alpha_(t)gtalpha_(b)` the tube tries to expand more than the bolt when assembly is heated. But the tube is tightened by the bolt therefore its expansion cannot be more than that of the bolt. Hence a compressive stress is developed in the tube and a tensile stress in bolt.
Let the initial length of the assembly be `l` and let is elongation be `/_\l`. Elongation of the tube if it would be free to expanded `=lalpha/_\theta`. But its actual elongation is `/_\l`,therefore elongation prevented in it equal to `(lalpha_(t)/_\theta-/_\l)`.
Therefore compressive strain in tube
`E_(t)=((lalpha_(t)/_\theta-/_\l))/l=5xx10^(5)`