We cannot assume that the centre of gravity of a non uniform beam is in the centre of the beam. If we do not also know the centre of the beam, then we have a problem. Fortunately there is quite a neat solution which enables us to find both the position of the centre of gravity and the weight of the beam. We form simultaneous equations for the weight and position of the centre of gravity by lifting the beam at one end then the other, taking moments about hinge point.
Example: Find the weight W and centre of gravity of a beam of length 6m, if when lifted at the end A, a force of 4000N must be exerted, and when lifted at the end B, a force of 5000N must be exerted to lift the beam.
Taking moments about B:
(1)
Taking moments about A:
(2)
+ (2) gives
Substitution of this into (1) gives
The centre of gravity is
from B.