Locating the centre of mass by mechanical means

This article discusses moment planimeters, which are mechanical devices with which is it possible to locate the centre of mass of an irregular plane shape by mechanical and graphical methods. They are a type of analogue computing device. In addition to this they may be used to find the static moment (first moment) and moment of inertia (second moment) of a shape about a fixed line. Moment planimeters, sometimes called integrometers or integrators, are direct developments of the planimeter which is a mechanical device used to directly measure the area of a plane shape. While planimeters are reasonably well known, linear planimeters are less common than the polar planimeters of Amsler. Hence in this article we explain how planimeters work through the example of a linear planimeter, and then consider how these may be adapted to find the centre of mass. More detailed comparisons between other types of area measuring planimeters may be found in the comprehensive survey article of [2]. 1 Area and centre of mass Consider the region enclosed by the closed curve in Figure 1, through which we have drawn the x-axis. We consider the area to be split into two regions by this axis, and these regions are described by the the two functions, f1(x) and f2(x). Since a general plane shape cannot be described in thi