A bar of mass m and length l is hanging from point A as shown in figure.Find the increase in its length due to its own weight. The young's modulus of elasicity of the wire is Y and area of cross-section of the wire is A.

Explanation

Consider a small section dx of the bar at a distance x from B. The weight of the bar for a length x is,

`w=((mg)/(l))x`
Elongation in section dx will be, `dl=((w)/(AY))dx=((mg)/(lAY))x dx`
Total elongation in the bar can be obtained by integrating this expression for x=0 to x=l.
`therefore" "Deltal=int_(x=0)^(x=1)dl=((mg)/(lAY))int_(0)^(1)x dx`
or Increase in length, `Deltal=(mgl)/(2AY)`