Final answer:
The question asks for the center of mass of a planar lamina using different coordinate systems. Appropriate systems such as Cartesian, polar, cylindrical, and spherical coordinates are chosen based on the object's symmetry and are used to find the COM by integrating the mass distribution.
Step-by-step explanation:
The subject question is asking about finding the center of mass of a planar lamina. The first step to solve this is to introduce an appropriate coordinate system that would simplify the computation of the center of mass based on the object's geometry and mass distribution. In general, the center of mass (COM) is calculated by integrating over the mass distribution, or by finding the weighted average position of discrete masses. For Cartesian coordinates, the COM is found by integrating the product of the density function and the position vector over the area of the lamina. Polar and cylindrical coordinates are often useful when dealing with objects that have circular symmetry, whereas spherical coordinates are more appropriate for three-dimensional, radially symmetric objects. For instance, to find the COM of a semicircular wire or plate, which has circular symmetry, polar or cylindrical coordinates might be chosen with the origin at the center of the semicircle. The mass elements are then integrated over the defined coordinate system to find the COM. On the other hand, the COM of a rectangular material with non-uniform density, described by a density function that varies along both dimensions, will likely be easier to determine using Cartesian coordinates.