If the work done in stretching a wire by 1 mm is 2J, then work necessary for stretching another wire of same material but with double radius of corss-section and half the length by 1 mm is

Options

A. 2 J
B. 4 J
C. 8 J
D. 16 J

Answer

D

Explanation

Given that, work done, W = 2J
As, we have two wires of same material and they are elongated to the same length.
So, this means, their Young's modulus (Y) is same and `Deltal_(1) = Deltal_(2) = Deltal` ( say)
The work done in stretching a wire ,
`W=1/2FDeltal`
Now, for both the wires we have
`(W_(1))/(W_(2))=(1/2F_(1)Deltal)/(1/2F_(2)Deltal)`
`!W_(2)=2xx(F_(2))/(F_(1))` ...(i)
The forces by which the wires are stretched can be expressed as the ratio,
`(F_(1))/(F_(2))=(YA_(1)(Deltal)/(L_(1)))/(YA_(2)(Deltal)/(L_(2)))`
`=(r_(1)^(2)L_(2))/(r_(2)^(2)L_(1))` .....(ii)
Given, `r_(2)=2r_(1)` and `L_(2)=(L_(1))/2`
Now, substituting the given values in Eq. (ii), we get
`(F_(1))/(F_(2))=(r_(1)^(2)L_(1))/(4r_(1)r^(2)2L_(2))=1/8` ....(iii)
From Eqs. (i) and (iii), we get
`W_(2)=2xx8`
=16J