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 `d_x` of the bar at a distance x from B.
Tension in the bar at this point is `T_x=W_x=((mg)/(l))x`
Elongation in section `d_x` will be `dl=(T_xdx)/(AY)=((mg)/(lAY))xdx`
Total elongation in the bar can be obtained by integrating this expression from `x=o`to` x=l.`
`:.Delta l =int_(x=0)^(x=l)dl=((mg)/(lAY))int_(0)^(l)x dx`
or`Delta l=(mgl)/(2Ay)`