A cord of mass m length L, area of cross section A and Young's modulus y is hanging from a ceiling with the help of a rigid support. The elogation developed in the wire due to its own weight is

Options

A. `(mugl^(2))/(YA)`
B. `(mugl^(2))/(2YA)`
C. `(2mugl^(2))/(YA)`
D. `(2mugl^(2))/(3YA)`

Explanation

Consider a small element of length dx at a distance x from the free end of wire as shown in the figure. Tension in the wire at distance x from the lower end is

`T(x) = mugx" "….(i)`
Let dl be increase in lenght of the element . Then `Y = (T(x)//A)/(dl//dx)`
`dl = (T(x)dx)/(YA)= (mugx dx )/(YA)" "["Using (i)"]`
Total extension produced in the wire is
`l = underset(0)overset(L)int(mugx)/(YA) dx =(mug)/(YA) [(x^(2))/(2)]_(0)^(L) = (mugL^(2))/(2YA)`