Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass m hangs from P. if the system is in equilibrium, find the magnitude and the direction of frictional force at A and B.

Explanation

`[(M +m)g/2] cot theta`

The various forces acting on the theorem, we get figure. Since the system is in equilibrium, therefore,
`sumF_(Y) =0`
`rArr Mg + mg + Mg = N+N rArr N=((2M+m)g)/2`
Considering the rotational equilibrium of one ladder as shown in the figure. Calculating torques about P
`Mg xx P + f xx PM = N xx OP`
`Mg xx L/2 cos theta +f xx L sin theta = NL cos theta`
`rArr f=[(2N+m)/2 g - (Mg)/2] cot theta rArr f=[(M+m)g/2] cot theta`