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Two wires of the same material and same ...

Two wires of the same material and same length but diameters in the ratio (1 : 2) are stretched by the same force. The potential energy per unit volume of the two wires will be in the ratio .........

A

`1:2`

B

`4:1`

C

`1:1`

D

`16:1`

Text Solution

Verified by Experts

The correct Answer is:
D
The potential energy per unit volume
`u = 1/2 (("stress") ^(2))/( " Young.s modulus") = (S ^(2))/( 2 Y) " "...(1)`
In stress `S = F/A, F` is constant
`therefore S prop (1)/(A)`
`therefore (S_(1))/( S _(2)) = (A_(2))/( A _(1)) = (pi D _(2) ^(2))/( pi D _(1) ^(2)) = ((D _(2))/( D _(1)) ) ^(1) " "...(2)`
Both the wires are of same material so their Young.s modulus is also same
So from equation (1)
`u prop S ^(2)`
`(u _(1))/( u _(2)) = ((S _(1))/( S _(2)) ) ^(2)`
` = ((D _(2))/( D _(!))) ^(4)` (From equation (2))
`= ((2)/(1)) ^(4) = (16)/(1)`
`therefore u _(1) : u _(2) =16:1`
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