When finding the centre of gravity of a lamina made up of several simpler shapes, we can take moments about two axis for each individual part and equate the sum of these moments to the total moment of the body.
When a lamina is made by cutting out a part, we can treat the missing part as having a negative oment. Still we add the moments of the individual parts – some negative – and equate this to the whole moment.
For example, the uniform lamina below has a rectangle cut from one corner. We can position the origin at the lower left corner and take moments about the x and y axes for the lamina and missing part, and equate the sum to the moment of the whole lamina.
The centre of gravity of the whole lamina – missing part included - is at
and the mass is 60 units. The centre of gravity of the missing part is at
and the mass is 8 units. The centre of mass of the lamina with the missing part taken away is at
and the mass is 52 units.
Taking moments about the x – axis
Taking moments about the y – axis
The centre of gravity of the lamina is at