A rigid rod is pivoted at point O and caries a body of weight w1 at the left end A.Find the weight w2 of another body which must be attached at...

The arrangement described in the question forms a lever
system in which the the fulcrum or the pivot is in the center with the load (w1) and the
effort (w2) are on opposite side of the fulcrum.

A system
like this is in equilibrium when:

Load x (Distance of load
from fulcrum) = Effort x (Distance of effort from
fulcrum)

Therefore, for the given
question:

w1 x (Distance of w1 from pivot) = w2 x (Distance
of w2 from pivot)

Therefor:

w2
= w1 x [(Distance of w1 from pivot)/(Distance of w2 from
pivot)]

Further, both the load and effort will act in the
same direction. Therefor. the total force on the pivot will be the sum of force exerted
by the sum of two
weights.

Thus:

Force acting on
the pivot = (w1 + w2) x (acceleration due to gravity)