A bar of mass M and length L is hanging from point S as shown in figure. The Young's modulus of elasticity of the wire is Y and the area of cross-section of the wire is A.

(i) Find the stress at x distance from bottom end.
(ii) Consider a samll section dx of the bar at a distance x from lowest point of bar. Find elongation (dL) in section dx.
(iii) Find total elongation in bar.
(iv) Find energy density at x distance from bottom and.
(v) Find total elastic potential energy stored in bar.

Explanation

(i) The weight x length of the bar is

`W=((Mg)/(L))x`
So stress at x distance from bottom.
`(W)/(A)=(Mgx)/(AL)`
(ii) Stress = Y (strain)
`(Mgx)/(AL)=Y(dL)/(dx)`
`rArr dL=(Mgxdx)/(ALY)`
(iii) Total elongation in wire
`DeltaL=int_(0)^(L)dL`
`rArr DeltaL=(Mg)/(ALY)int_(0)^(L)x dx`
`=(MgL)/(2AY)`
(iv) Energy density at x distance
`=(1)/(2)"(stress) (strain)"`
`=(1)/(2)("(stress)"^(2))/(Y)`
`=(1)/(2Y)[(Mgx)/(AL)]^(2)=(M^(2)g^(2)x^(2))/(2YA^(2)L^(2))`
(v) Energy stored in Adx volume = `"Energy density"xx"Adx"=(M^(2)g^(2)x^(2))/(2YA^(2)L^(2))"(Adx)"`
`"Total energy "=int_(0)^(L)(M^(2)g^(2)x^(2))/(2YAL^(2))dx`
`=(M^(2)g^(2)L)/(6AY)`