We can find the centre of mass and tension in the strings holding a non uniform rod of known length and mass by resolving and taking moments in the usual way. Consider the non uniform rod shown below.
Take the tension in the string at end A to be
and the tension in the string at end B to be
and take the centre of mass to bex m from A.
Resolving horizontally gives
(1)
Resolving vertically gives
(2)
(1)+(2) gives
Then from (1),
From the diagram below, the perpendicular distance from A to T-2 produced is
Taling moments about A gives