Consider a long steel bar under a tensile stress due to force F acting at the edges along the length of the bar (as shown in figure). Consider a plane making an angle with the length. What are the tensile and shearing stresses on this plane?
(a) For what angle is the tensile stress a maximum ?
(b) For what angle is the shearing stress a maximum ?

Explanation

Think below figure,

Suppose A is the cross sectional area of the bar. Let for balance of pane aa. angle `pi/2 - theta` between perpendicular ON to the pane and force F. The parallel component of force to the pane,
`F_(P) = F sin ((pi)/(2) - theta) = F cos theta and `
Perpendicular component,
`F_(N) = F cos ((pi)/(2) -theta)= F sin theta`
Suppose A is area of pane (face) aa.
`therefore (A)/(A.) = sin theta`
`therefore A. = (A)/(sin theta )`
Tensile stress `= ("perpendicular force")/("area") = (F _(N))/(A.)= (F sin thea)/(A//sin theta)`
`therefore` Tensile stress `= F/A sin ^(2) theta `
(a) For maximum stress `F/A sin ^(2) theta ` should be maximum.
`sin ^(2) theta =1`
`therefore sin theta =1`
`therefore theta = (pi)/(2)`
For this `theta = 90^(@)`
Shearing stress `= ("parallel force")/("area") = (F _(p ))/(A.)`
` = (F cos theta )/(A //sin theta) = F/A sin theta cos theta= (F)/(2A) xx sin 2 theta`
`[ because 2 sin theta cos theta = sin 2 theta]`
(b) For shearing stress to be maximum `sin 2 theta =1`
`therefore 2 theta =90^(@)`
`therefore theta = 45^(@)`