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

Options

A. 16J
B. 8J
C. 4J
D. `(1)/(4)J`

Answer

A

Explanation

Stretching force, `F= (ypir^(2) DeltaL)/(L)`
where the symbols have their usual meanings.
Both the wires are of same material, so Y will be equal, extension in both the wires is same, so `DeltaL` will be equal.
`therefore" " F prop (r^(2))/(L)`
`therefore" "(F')/(F) = ((2r^(2)))/((L//2)) xx (L)/(r^(2)) = 8`
or F= 8F " ".....(i)
Work done in strecthing a wire,
`W=(1)/(2) xx F xx DeltaL`
For same extension
`W prop F`
`therefore" "(W)/(W) = (F)/(F) = 8 "' "["Using (i)"]`
`W = 8W=8xx 2J = 16 J`