Centre of Mass of a Framework

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A framework is a rigid body consisting of a number of rods or wires joined together. As with a lamina formed of different sections, we can find the centre of mass or the famework by equating the moment of the framework about separate axes to sum sum of the moments of the consistuent parts.

Consider the framework below, in which three rods form a triangle.

Since each rod is uniform, the centre of gravity of each rod is in the centre of each rod. Suppose the mass of AB is 5m, the mass of BC is 4m and the mass of AC is 10m. The centre if mass is G(x,y), relative to A as the origin in the diagram below and the mass of the famework is 19m.

Taking moments about the x axis gives

Taking moments about the y axis gives

The centre of gravity is then at