Two wires A and B are of the same material their lengths are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If they are pulled by the same force, their increase in length will be in the ratio ........

Options

A. `2:1`
B. `1:4`
C. `1:8`
D. `8:1`

Answer

C

Explanation

We know that Young.s modulus
`Y = (F)/(pi r ^(2)) xx (L)/(l)`
Here Y and Fare constants for both wires.
`therefore (1)/( r _(1) ^(2)) . (L_(1))/( l _(1)) = ( 1)/( r _(2)^(6)). (L_(2))/( l _(2))`
`therefore ( l _(1))/( l_(2)) = (r _(2) ^(2) xx L _(1))/( r _(1) ^(2) xx L _(2)) = (((D _(2))/( 2 )) ^(2) xx L _(1))/(( (D _(1))/( 2 )) ^(2) xx L _(2))`
`therefore ( l _(1))/( l _(2)) = ( D _(2) ^(2) xx L _(1))/( D _(1) ^(2) xx L _(2)) = ( D _(2) ^(2))/( ( 2D _(2)) ^(2)) xx (L _(2))/( 2 L _(2)) = 1/8`
Hence `l _(1) : l_(2) = 1:8`