A rod PQ of mass, area of cross section A, length l and young's modulus of elasticity Y is lying on a smooth table as shown in figure. A force F is applied at P. Find
(a) tension at a distance x from end P,
(b) longitudinal stress at this point,
(c) total change in length and
(d) total strain the rod.

Explanation

(a)Acceleration of the rod ,` a=F/m`
`F-T_(x)=(m_(pm))a=(m/lx)(F/m)`
`:.T_x =F(1-x/l)`

(b) Stress `sigma=F/A=T_(x)=F/A(1-x/l)`
(c) Change in length `Delta l =int_0^(l)(T_xdx)/(AY)`
` =int_0^(l) (F(1-x/l)dx)/(AY)`
`Deltal = (Fl)/(2AY)`
(d) `=(Deltal)/(l)=(F)/(2AY)`