Final answer:
The center of mass of the complete ring formed by joining two semicircular rings with linear mass densities λ and 2λ will be at the geometric center. Hence, the distance 'D' from the geometrical center to the center of mass is 0.
Step-by-step explanation:
The question pertains to finding the center of mass of a composite object consisting of two semicircular rings with different linear mass densities, λ and 2λ, joined to form a complete ring. Each semicircular ring has radius 'R'. To find the center of mass of such a system, one should consider the symmetry and distribution of mass across the object.
Due to the symmetrical arrangement of the two semicircular rings, the center of mass will lie along the line that divides the two semicircles, which is also the line of symmetry. Since the semicircular ring with the higher mass density (2λ) will contribute more to the center of mass of the system, the center of mass will be closer to it. However, because the rings are distributed symmetrically about the geometric center, the center of mass of the complete ring will actually coincide with its geometric center, despite the different mass densities.
Therefore, the distance 'D' of the center of mass of the complete ring from its geometrical center is 0.