Final answer:
The center of mass of a uniform thin semicircular plate of radius R with the origin at the center is at (0,0, 4R/3π), as the center of mass lies on the axis of symmetry of the plate and is a common result derived using calculus.
Step-by-step explanation:
The center of mass of a uniform thin semicircular plate of radius R with the origin at the center of the semicircle can be found using the principles of symmetry and integration. Since the plate is uniform and symmetrical about the y-axis, the center of mass along the x-axis is at 0. The center of mass along the y-axis can be determined by setting up an integral that averages the y-coordinates of all the differential mass elements dm that make up the semicircle. However, this integral is a standard result in mechanics and the center of mass for a semicircle lies along the axis of symmetry at a distance of 4R/(3π) from the flat edge of the semicircle. Therefore, if we are asked to find the center of mass of a uniform thin semicircular plate of radius R in the z direction (perpendicular to the plate), the correct answer would be the same as along the y-axis due to the plate lying in the x,y plane and being symmetrical. Hence, the answer is (0,0, 4R/3π).