A wire having a length `L` and cross- sectional area `A` is suspended at one of its ends from a ceiling . Density and young's modulus of material of the wire are `rho` and `Y`, respectively. Its strain energy due to its own weight is `(rho^(2)g^(2)AL^(3))/(alphaY)`. Find the value of `alpha `

Answer

6

Explanation

Consider an element as shown in the figure.
Stress in the element `=("Force")/("Area")=(XArhog)/(A)=Xrhog`
Now, elastic potential energy stored in the wire is
`=(1)/(2).(("Stress")^(2))/(Y)`(Volume)
`dU=(1)/(2).((Xrhog)^(2))/(Y)Adx=(1)/(2).(rho^(2)g^(2)A)/(Y)X^(2)dX`
Total elastic potential energy `=(1)/(2).(rho^(2)g^(2)A)/(Y)int_(0)^(L)X^(2)dX=(rho^(2)g^(2)A)/(6Y)`