A smooth uniform string of natural length `L_0`, cross-sectional area `A` and Young's modulus `Y` is pulled along its length by a force `F` on a horizontal smooth surface. The elastic potential energy stored in the string is

Options

A. `(2F^2L_0)/(AY)`
B. `(F^2L_0)/(2AY)`
C. `(F^2L_0)/(2AY)`
D. `(F^2L_0)/(6AY)`

Answer

D

Explanation

Tension, `T=(F)/(L_0).xx`
stress,`sigma=(T)/(A)=(F)/(AL_0)x`
`dU=(1)/(2)(sigma^2)/(Y)Adx=(1)/(2)(F^2)/(A^2L_0^2)x^2(A)/(Y)dx`
or `dU=(F^2)/(2A^2L_0^2Y)x^2dx`
`impliesU=(F^2)/(2AYL_0^2)int_0^(L0)x^2dx`
`U=(F^2)/(2AYL_0^2)(L_0^3)/(3)=(F^2L_0)/(6AY)`