Wires `A` and `B` are made from the same material. `A` has twice the diameter and three times the length of `B`. If the elastic limits are not reached, when each is stretched by the same tension, the ratio of energy stored in `A` to that in `B` is

Options

A. `2 :3`
B. `3 : 4`
C. `3 : 2`
D. `6 : 1`

Answer

B

Explanation

Let diameter of B=D, length of B=1,
`:. D_(A) =2D, L_(A)=3 l,F_(A)=F_(B)=F`
As energy stored `=(1)/(2)xx stressxx strai n xxvolume`
`=(1)/(2)xxstress xx(stress)/(Y)xxVolume`
`=((1)/(2)((stress)^(2))/(Y))xxVolume`
`=((1)/(2)((F)/(A_(A))^(2)))/(Y)xxA_(A)l_(A)`
similarly `E_(B)=((1)/(2)xx((F)/(A_(B))^(2)))/(Y)xxA_(B)l_(B)`
`(E_(A))/(E_(B))=((1)/(2)((F)/(A_(A))^(2))/(Y)xxA_(A)l_(A))/((1)/(2)((F)/(A_(B))^(2))/(Y)xxA_(B)l_(B))=(l_(A))/(A_(A))xx(A_(B))/(l_(B))`
`=(3l)/(pi 4 D^(2)//4)xx(pi D^(2))/(4xxl) =(3)/(4)`