A metallic rod l cm long, A square cm in cross-section is heated through `t^(@)"C"`. If Young’s modulus of elasticity of the metal is E and the mean coefficient of linear expansion is `alpha` per degree celsius, then the compressional force required to prevent the rod from expanding along its length is

Options

A. `Y alpha t`
B. `(Y alpha t)/(1 + alpha t)`
C. `(Y alpha t)/(1- alpha t)`
D. None of these

Answer

A

Explanation

If the rod is heated, thermal expansion
`Delta l= alpha. l. Delta T= alpha.(1 cm) (t^(@)C)`
If the rod is not allowed to expand by applying a force, it means a mechanical compression equal to the termal expansion has been produced by applying the force.
Now, `Y= ((F)/(A)) + ((Delta l)/(l)), Y = (F.l)/(A Delta l) or F= YA ((Delta l)/(l))`
`F= Y A (alpha t) " As " A = 1 cm^(2)`
`F= Y alpha t rArr` (a) is correct