10. A beam is constructed of 3 steel plates which are welded to form a shape as in Figure Q10 (a). Each plate has a thickness of \( 10 \mathrm{~mm} \).
Figure Q10 (a)
a) Calculate the centre of mass of the beam.
\( [7 \) marks]
b) The beam is at rest in limiting equilibrium on a rough plane. The angle that plane makes with the horizontal begins to increase from 0 to \( \theta \) as in Figure Q10 (b). The coefficient of friction between the beam and the surface is 0.4 .
Figure Q10 (b)
i) Derive an expression relating the forces acting on the beam to the angle of elevation and find the angle, \( \theta \) at which the beam will start to slide.
\( [4 \) marks]
ii) If the beam is prevented from sliding, find the angle, \( \theta \) at which the beam will start to topple.
[4 marks]